---
title: "Spacekime Analytics (Time Complexity and Inferential Uncertainty)"
subtitle: "Complex-Valued Wavefunctions, Complex-time and Inference Functions"
author: "SOCR Team (Yueyang Shen, Milen V. Velev, Ivo D. Dinov)"
date: "`r format(Sys.time(),'%m/%d/%Y')`"
output:
  html_document:
    theme: spacelab
    highlight: tango
    includes:
      before_body: TCIU_header.html
    toc: yes
    number_sections: yes
    toc_depth: 3
    toc_float:
      collapsed: no
      smooth_scroll: yes
    code_folding: hide
  word_document:
    toc: yes
    toc_depth: '3'
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

This [TCIU section](https://http://tciu.predictive.space/) outlines the synergies 
between *wavefunctions*, probability *distributions*, probability *densities*, 
complex-time *(kime) representations* of repeated-measurement
spatiotemporal data, and *inference functions*. There is a direct duality between (1) the quantum mechanics interpretation that *measurements collapse wavefunctions*, bringing *quantum* reality into *physical* existence; and (2) the spacekime interpretation that *measurements represent random draws* (samples) from the kime-phase distribution. Just like geodesic curves (generally nonlinear twisted lines) in spacetime represent the shortest paths traversed by light, in a multiverse, 4D hyperplanes foliating the 5D spacekime represent alternative ("parallel") curvilinear spacetime projections (rather than continuously branching and bifurcating universes at each time point). 

# Manifold Foliation

A *foliation* $\mathcal{F}$ of *dimension* (or *rank*) $r$ of a differentiable manifold
$M^n$ is a decomposition of $M^n$ into a *disjoint union* of immersed 
sub-manifolds (*leaves* or *plaques*) of dimension $r\lt n$ that locally and jointly 
cover $M^n$, like fibers of a *submersion.* Symbolically, the *foliation* $\mathcal{F}$
is defined by an atlas $\{\psi_{\alpha} : U_{\alpha}\to M^n\},$ such that the *charts* are

$$U_{\alpha}\subset \underbrace{\mathbb{R}^r}_{leaves}\times \underbrace{\mathbb{R}^{n-r}}_{foliation\\ co-dimension\\ leaf-parametrization}$$
and the *differentiable transition functions* are

$$\psi_{\alpha}: \mathbb{R}^n\longrightarrow {M}^n \ \ \ 
\psi_{\alpha}(x,y)\equiv \left (f(x,y),g(y)\right ),\ \ \forall (x,y)\in 
\mathbb{R}^r \times  \mathbb{R}^{n-r}$$

The *foliation leaves* are locally given by the fibers of the projection 
$U_i\to\mathbb{R}^{n-r}$. When the transition functions $\psi_{\alpha}(x,y)\in C^{\infty},$
the $\mathcal{F}$ is a $C^{\infty}$-foliation.

The simplest foliation example is the [foliation of the complex plane](https://www.ams.org/notices/202207/rnoti-p1137.pdf) where lower-dimensional real lines or non-intersecting curves (*leaves*) are parameterized by a co-dimension $1$ *foliation parameter* (*foils*) $\left(\mathbb{C} \cong \underbrace{\mathbb{R}}_{leaves}\times \underbrace{\mathbb{R}}_{foils}\right)$. Similarly, in aggregate, the infinite set of all lower dimensional *4D Minkowski spacetime leaves* forms a co-dimension $1$ foliation of *5D spacekime.*

The figure below shows 2D examples of three foliations including:

 - A flat *complex plane* $\mathbb{C}$, a lower-dimensional mock up of the flat 4D Minkowski spacetime, 
 - And a pair of curved manifolds, *sphere* ($\mathbb{S}^2$) and a *torus* ($\mathbb{T}^2\cong \mathbb{R}^2/\mathbb{Z}^2$), as mock ups of a more realistic 4D spacetime manifold, curved by matter (mass or energy).

```{r planeSphereTorusFoliation, warning=FALSE, message=FALSE, error=FALSE, fig.width=8, fig.height=8}
library(plotly)

# sweep or define (u,v) spherical coordinate parameter ranges
phi <- seq(from = 0, to = 2*pi, by = ((2*pi - 0)/(200 - 1)))
psi <- seq(from = 0, to = pi, by = ((pi - 0)/(200 - 1)))

#p <- plot_ly(x = ~x, y = ~y, z = ~z, type = 'surface', opacity=1,
#             contour=list(show=TRUE, color="#000", width=15, lwd=10))  %>%
#  layout(title = paste("Layout ", shape), 
#         scene = list(xaxis=x_label,yaxis=y_label, zaxis=z_label))
#p

shapes <- c("Complex Plane", "Sphere", "Torus")
shapeNames <- c(TeX("\\text{Complex Plane }\\mathbb{C}\\cong\\mathbb{R}^2"), 
                TeX("\\text{Sphere }\\mathbb{S}^2"), 
                TeX("\\text{Torus }\\mathbb{T}^2\\cong\\mathbb{R}^2 / \\mathbb{Z}^2"))
nCurves <- 5  # number of toral solenoid curves
                # https://mathcurve.com/courbes3d/solenoidtoric/solenoidtoric.shtml
curveGap <- 4 # index gap between solenoids curve rotational shifts
samplingRate <- 200  # curve sampling rate

# shape=="complexPlane"
    supportWidth <- 2
    x2D <- seq(-supportWidth*pi, supportWidth*pi, length.out=samplingRate)
    y2D <- array(0, c(nCurves, samplingRate))
    x0 = x2D %o% x2D   # Complex-Plane Grid
    y0 = x2D %o% x2D
    z0 <- array(1, c(4*samplingRate, samplingRate))
    x1D <- c(1:samplingRate)
    y1D <- seq(1:(4*samplingRate))
    # p <- plot_ly(x=x1D, y=y1D, z=z0, type="surface", name="Complex Plane", opacity=0.3)
    # for (curve in 1:nCurves) {  # for each solenoid curve on torus
    #   y2D[curve, ] <- (curve-nCurves/2)/curveGap + (2+sin(x2D)) + x2D/(supportWidth*pi)
    #   p <- p %>% add_trace(x=c(1:samplingRate), y=(y2D[curve,]*samplingRate),
    #                        z=1, name=paste0("Curve: ", curve), opacity=1,
    #                        type="scatter3d", mode="lines")
    # }
    # p
    
# shape=="torus"
    # h2= 10   # cone height
    # r2 = seq(from = 0, to = h2, by = ((h2 - 0)/(200 - 1)))  # r = radius
    # x2 = 3* ((h2 - r2)/h2 ) %o% rep(1, 200)             # x = 3*r
    # y2 = 3* ((h2 - r2)/h2 ) %o% sin(phi)   # y = r*sin(phi)
    # z2 = 3* ((h2 - r2)/h2 ) %o% cos(phi)   # z = r*cos(phi)

    ### Kime tube on torus
#### Parametric curve on torus
# k1 <- phi -pi
# k2 <- ((phi-1)^3 + 5*(phi-1)^2 + 2*(phi-5) - 8)/10
  a <- 6   # Torus radius (from torus gravitational center)
  r <- 2   # Tube radius (from torus circular core)
  # When n is a rational p / q , and R > r , the curve is closed and simple, and the node 
  # associated with the corresponding toroidal solenoid is the toroidal node T ( p, q ), 
  # p windings around the torus for q turns around of the axis, which is always a prime node . The nodes T ( p, q ) and T ( q, p ), are equivalent (to go from ( p, q ) to ( q, p ), passing a needle in the core of the torus).
  p <-  5 # p winding around the torus
  q <-  1# q turns around of the toral axis
  n <- p/q

  xOuter <- outer(phi, psi, function(phi, psi) { cos(phi)*(a+r*cos(n*psi)) })
  yOuter <- outer(phi, psi, function(phi, psi) { sin(phi)*(a+r*cos(n*psi)) })
  zOuter <- outer(phi, psi, function(phi, psi) { r*sin(n*psi) })

  phi <- seq(from = 0, to = 2*pi, by = ((2*pi - 0)/(50 - 1)))
  psi <- seq(from = 0, to = 2*pi, by = ((2*pi - 0)/(50 - 1)))
  psi2 <- seq(from= 0, to = 2*pi, by = ((2*pi - 0)/(50 - 1)))

  xOuter <- outer(phi, psi, function(phi, psi) { cos(phi)*(a+r*cos(psi)) })
  yOuter <- outer(phi, psi, function(phi, psi) { sin(phi)*(a+r*cos(psi)) })
  zOuter <- outer(phi, psi, function(phi, psi) { r*sin(psi) })

  xCurveOuter <- array(0, c(nCurves, length(phi)))
  yCurveOuter <- array(0, c(nCurves, length(phi)))
  zCurveOuter <- array(0, c(nCurves, length(phi)))
  
  # for (curve in 1:nCurves) {  # for each solenoid curve on torus
  #   for (ind in 1:length(phi)) {  # for each point on the curve
  #     xCurveOuter[curve, ind] <-
  #       cos(phi[1 + ((ind -1 + curveGap*(curve-1)) %% length(phi))]) *
  #         (a+r*cos(n*psi2[1 + ((ind -1 + curveGap*(curve-1)) %% length(phi))]))
  #     yCurveOuter[curve, ind] <-
  #       sin(phi[1 + ((ind -1 + curveGap*(curve-1)) %% length(phi))]) *
  #         (a+r*cos(n*psi2[1 + ((ind -1 + curveGap*(curve-1)) %% length(phi))]))
  #     zCurveOuter[curve, ind] <-  r*sin(n*psi2[1 + ((ind -1 + curveGap*(curve-1)) %% length(phi))])
  #   }
  # }

  for (curve in 1:nCurves) {  # for each solenoid curve on torus
    gamma <- (curve-1)/(2*pi)
    rotMatZ <- # 3-by-3 rotation matrix around Z axis
        matrix(c(cos(gamma), sin(gamma),0,  -sin(gamma),cos(gamma),0,   0,0,1), ncol=3)
    for (ind in 1:length(phi)) {  # for each point on the curve
      xCurveOuter[curve, ind] <- cos(phi[ind]) * (a+r*cos(n*psi2[ind])) 
      yCurveOuter[curve, ind] <- sin(phi[ind]) * (a+r*cos(n*psi2[ind])) 
      zCurveOuter[curve, ind] <- r*sin(n*psi2[ind])
      # apply rotation matrix
      xCurveOuter[curve, ind] <- 
        (rotMatZ %*% c(xCurveOuter[curve, ind], yCurveOuter[curve, ind], zCurveOuter[curve, ind]))[1]
      yCurveOuter[curve, ind] <-
        (rotMatZ %*% c(xCurveOuter[curve, ind], yCurveOuter[curve, ind], zCurveOuter[curve, ind]))[2]
      zCurveOuter[curve, ind] <-  
        (rotMatZ %*% c(xCurveOuter[curve, ind], yCurveOuter[curve, ind], zCurveOuter[curve, ind]))[3]
    }
  }
  
# # plot_ly() %>%
#     add_trace(x=~xOuter, y=~yOuter, z=~zOuter, type='surface', 
#               opacity=0.5, visible=TRUE, showlegend= FALSE, showscale = FALSE) %>%
#      add_trace(x=xCurveOuter[1,], y=yCurveOuter[1,], z=zCurveOuter[1,], 
#                type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
#      add_trace(x=xCurveOuter[2,], y=yCurveOuter[2,], z=zCurveOuter[2,], 
#                type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
#      add_trace(x=xCurveOuter[3,], y=yCurveOuter[3,], z=zCurveOuter[3,], 
#                type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
#      add_trace(x=xCurveOuter[4,], y=yCurveOuter[4,], z=zCurveOuter[4,], 
#                type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
#      add_trace(x=xCurveOuter[5,], y=yCurveOuter[5,], z=zCurveOuter[5,], 
#                type="scatter3d", mode="markers+lines", line=list(width=15))
   

# shape=="sphere" 
    phi <- seq(from = 0, to = 2*pi, by = ((2*pi - 0)/(samplingRate - 1)))
    psi <- seq(from = 0, to = pi, by = ((pi - 0)/(samplingRate - 1)))

    xCurveOuterSphere <- array(0, c(nCurves, length(phi)))
    yCurveOuterSphere <- array(0, c(nCurves, length(phi)))
    zCurveOuterSphere <- array(0, c(nCurves, length(phi)))
  
    r3 = 1                           # r = 1
    x3 = r3 * cos(phi) %o% sin(psi)   # x = r*cos(phi)*sin(psi)
    y3 = r3 * sin(phi) %o% sin(psi)   # y = r*sin(phi)*sin(psi)
    z3 = r3 * rep(1, samplingRate) %o% cos(psi) # still need z to be 200*200 parameterized tensor/array

    for (curve in 1:nCurves) {  # for each solenoid curve on torus
      gamma <- (curve-1)/(2*pi)
      rotMatZSphere <- # 3-by-3 rotation matrix around Z axis
          matrix(c(cos(gamma), sin(gamma),0,  -sin(gamma),cos(gamma),0,   0,0,1), ncol=3)
      for (ind in 1:length(phi)) {  # for each point on the curve
        xCurveOuterSphere[curve, ind] <- r3*cos(phi[ind]) * sin(psi[ind])
        yCurveOuterSphere[curve, ind] <- r3*sin(phi[ind]) * sin(psi[ind])
        zCurveOuterSphere[curve, ind] <- r3*cos(psi[ind])
        # apply rotation matrix
        xCurveOuterSphere[curve, ind] <- 
          (rotMatZSphere %*% c(xCurveOuterSphere[curve, ind], yCurveOuterSphere[curve, ind],
                               zCurveOuterSphere[curve, ind]))[1]
        yCurveOuterSphere[curve, ind] <-
          (rotMatZSphere %*% c(xCurveOuterSphere[curve, ind], yCurveOuterSphere[curve, ind],
                               zCurveOuterSphere[curve, ind]))[2]
        zCurveOuterSphere[curve, ind] <-  
          (rotMatZSphere %*% c(xCurveOuterSphere[curve, ind], yCurveOuterSphere[curve, ind],
                               zCurveOuterSphere[curve, ind]))[3]
      }
    }
  # plot_ly() %>%
  #    add_trace(x=x3, y=y3, z=z3, type='surface', 
  #             opacity=0.5, visible=TRUE, showlegend= FALSE, showscale = FALSE) %>%
  #    add_trace(x=xCurveOuterSphere[1,], y=yCurveOuterSphere[1,], z=zCurveOuterSphere[1,], 
  #              type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
  #    add_trace(x=xCurveOuterSphere[2,], y=yCurveOuterSphere[2,], z=zCurveOuterSphere[2,], 
  #              type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
  #    add_trace(x=xCurveOuterSphere[3,], y=yCurveOuterSphere[3,], z=zCurveOuterSphere[3,], 
  #              type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
  #    add_trace(x=xCurveOuterSphere[4,], y=yCurveOuterSphere[4,], z=zCurveOuterSphere[4,], 
  #              type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
  #    add_trace(x=xCurveOuterSphere[5,], y=yCurveOuterSphere[5,], z=zCurveOuterSphere[5,], 
  #              type="scatter3d", mode="markers+lines", line=list(width=15))

# https://plot.ly/r/custom-buttons/

# updatemenus component
updatemenus <- list(
  list(active = -1, type = 'buttons', # active = -1
    buttons = list(
      list(label = shapes[1], method = "update",  # Complex Plane
        args = list(list(visible = c(TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
                                     FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, 
                                     FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE)), 
                    list(title = shapeNames[1]))),
      list(label = shapes[2], method = "update",  # Sphere
        args = list(list(visible = c(FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE,
                                     TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
                                     FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE)), 
                    list(title = shapeNames[2]))),
      list(label = shapes[3], method = "update",  # Torus
        args = list(list(visible = c(FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, 
                                     FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, 
                                     TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE)), 
                    list(title = shapeNames[3])))
    )
  )
)

# Complex Plane
p <- plot_ly(x=x1D, y=y1D, z=z0, type="surface", 
             name="Complex Plane", opacity=0.3)
    for (curve in 1:nCurves) {  # for each solenoid curve on torus
      y2D[curve, ] <- (curve-nCurves/2)/curveGap + (2+sin(x2D)) + x2D/(supportWidth*pi)
      p <- p %>% add_trace(x=c(1:samplingRate), y=(y2D[curve,]*samplingRate),
                           z=1, name=paste0("Curve: ", curve), opacity=1,
                           type="scatter3d", mode="lines")
    }
p <- p %>% add_trace(type='scatter3d', mode='text', x=150, y=200, z=1, 
              text="C=R^2", showlegend=F, inherit=F) 
# Sphere 
p <- p %>%
     add_trace(x=x3, y=y3, z=z3, type='surface', name="Sphere",
              opacity=0.5, visible=FALSE, showlegend= FALSE, showscale = FALSE) %>%
     add_trace(x=xCurveOuterSphere[1,], y=yCurveOuterSphere[1,], z=zCurveOuterSphere[1,], 
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15), visible=FALSE) %>%
     add_trace(x=xCurveOuterSphere[2,], y=yCurveOuterSphere[2,], z=zCurveOuterSphere[2,], 
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15), visible=FALSE) %>%
     add_trace(x=xCurveOuterSphere[3,], y=yCurveOuterSphere[3,], z=zCurveOuterSphere[3,], 
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15), visible=FALSE) %>%
     add_trace(x=xCurveOuterSphere[4,], y=yCurveOuterSphere[4,], z=zCurveOuterSphere[4,], 
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15), visible=FALSE) %>%
     add_trace(x=xCurveOuterSphere[5,], y=yCurveOuterSphere[5,], z=zCurveOuterSphere[5,], 
               type="scatter3d", mode="markers+lines", line=list(width=15), visible=FALSE) %>%
     add_trace(type='scatter3d', mode='text', x=0.35, y=-0.9, z=0.9, 
                    text='Sphere S^2', showlegend=F, inherit=F, visible=FALSE) %>%
  # Torus
     add_trace(x=~xOuter, y=~yOuter, z=~zOuter, type='surface', name="Torus",
              opacity=0.5, visible=FALSE, showlegend= FALSE, showscale = FALSE) %>%
     add_trace(x=xCurveOuter[1,], y=yCurveOuter[1,], z=zCurveOuter[1,], visible=FALSE,
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
     add_trace(x=xCurveOuter[2,], y=yCurveOuter[2,], z=zCurveOuter[2,], visible=FALSE,
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
     add_trace(x=xCurveOuter[3,], y=yCurveOuter[3,], z=zCurveOuter[3,], visible=FALSE,
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
     add_trace(x=xCurveOuter[4,], y=yCurveOuter[4,], z=zCurveOuter[4,], visible=FALSE,
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
     add_trace(x=xCurveOuter[5,], y=yCurveOuter[5,], z=zCurveOuter[5,], visible=FALSE,
        opacity=1, type="scatter3d", mode="markers+lines", line=list(width=15)) %>%
     add_trace(type='scatter3d', mode='text', x=6.5, y=-6.5, z=2.5, 
                    text='Torus T^2', showlegend=F, inherit=F, visible=FALSE) %>%
  layout(title = "Foliations of 2D Euclidean and Non-Euclidean Spaces", showlegend = FALSE,
         scene = list(xaxis=list(title="X"),yaxis=list(title="Y"),zaxis=list(title="Z")),
         updatemenus=updatemenus) %>% config(mathjax = 'cdn')
p
```

# Probability Density (Mass) Functions

The relation between *probability density* (or for discrete processes, 
*probability mass*) *functions* and *probability distributions* resemble the
dichotomy between solid *density* (mass per unit volume) and
*mass* (integral over the density), which are related by

$$mass= density \times volume\ .$$

Suppose the density $\rho(x,y,z, t):\mathbb{R}^3\times \mathbb{R}^+ \to \mathbb{R}^+$ 
is a function of position and time, i.e., the density is spatiotemporally dependent. At a
given time $t_o\in \mathbb{R}^+$, a tiny cube (solid volume in $\mathbb{R}^3$) 
with sides $dx$, $dy$ and $dz$ will correspond to a volume $dV=dx\times dy\times dz$ of *mass*  

$$dm(t_o)=\rho(x,y,z, t_o)\times dV \equiv \rho(x,y,z, t_o)\times dx\times dy\times dz, \ 
\forall\ t_o \in \mathbb{R}^+\ .$$

To estimate the total mass for any volume $V$, we integrate the mass against
the density over the spatial extent of the volume $V$

$$m(t_o)=\int_V { \rho(x,y,z,t_o) dx dy dz} $$

The relation between *probability density* and *probability* resembles the 
relation between *density* and *mass.*
Given a probability *density*, $p(x,y,z, t):\mathbb{R}^3\times \mathbb{R}^+ \to \mathbb{R}^+$
as the probability per unit 3D solid volume, the *probability*, $P$ of finding 
the particle in the *tiny solid cube* of size $dx\times dy\times dz$ at time $t_o$ is:

$$dP(t_o) =p(x,y,z, t_o) dx dy dz, $$

and in general, the probability of finding the particle in any finite volume $V$
is also estimated by integrating against the density over the spatial region

$$P(t_o)=\int_V{ p(x,y,z, t_o) dx dy dz} .$$

**Note**: At any given time $t_o$, the *probability density* is typically estimated 
using the (complex-valued) *wavefunction* of the system, which can be defined 
over different coordinates, e.g., position space, momentum space, etc. 

$$\psi(x,y,z, t): \mathbb{R}^3\times \mathbb{R}^+ \to \mathbb{C} \ .$$

The wavefunction assigns a complex number to each spacetime location and 
the [Born rule](https://en.wikipedia.org/wiki/Born_rule) allows us to map its 
*complex probability amplitude values* into actual *probability values* 
in $[0,1]\subseteq \mathbb{R}$. This mapping squares the modulus 
of the wavefunction to produce the *probability density* of observing
a particle in a given spatiotemporal region. Symbolically, using the 
*complex conjugation* notation 
$\left (z=x+iy\in \mathbb{C}, z^*\equiv \bar{z}=x-iy\right )$, 
the *probability density* is defined by

$$p(x,y,z,t)=\psi^*(x,y,z,t)\ \psi(x,y,z,t)\ .$$

And the corresponding *probability* of finding the particle in a solid volume $V$ is

$$P(t)=\int_V {p(x,y,z,t) dx dy dz} =\int_V {\psi^*(x,y,z,t)\psi(x,y,z,t) dx dy dz}\ .$$
Probability density functions arise when we model likelihoods, chances, or probabilities
of events connected with experimental observations of continuous or discrete variables, 
e.g., particle positions, momenta, spins, energies, etc. In situations with discrete probability 
mass functions, we have a finite number of outcomes, each with non-trivial chances 
of being observed. In continuous outcomes with infinitely many non-trivial possibilities
it is impractical to directly assign a probability value to each individual outcome. 

Recall the [three axioms of probability](https://en.wikipedia.org/wiki/Probability_axioms)

 - All probabilities are in $[0,1]\subset \mathbb{R}$, 
 - All probabilities integrate to $1$, and 
 - Probabilities are additive. 
 
In the continuous outcome case, there are infinitely many 
possible states of the particle (e.g., positions) and generally, observing the particle
in most specific points in spacetime would be trivial. 

Therefore, in general, its most interesting to quantify the chance of finding the 
particle within some non-trivial spacetime *hypervolume*, and we expect
to compute or associate a nonzero probability with such an event. 


# Wavefunctions

## Matter waves

In [Chapter 3 (Time Complexity)](https://www.socr.umich.edu/TCIU/HTMLs/Chapter3_Introduction.html), we show examples of 1D and 2D matter waves. Below is an example of a complex wave function of constant amplitude.

```{r matterWaves, warning=FALSE, message=FALSE}
t <- seq(-6*pi, 6*pi, length.out=400)
r <- 3
x <- r * cos(t)
y <- r * sin(t)
z <- 2*t
c <- t%%(2*pi)
  
data1 <- data.frame(x, y, z)

p <- plot_ly(data1, x = ~x, y = ~y, z = ~z, type = 'scatter3d', mode = 'lines',
        showlegend = F, name="complex wave",
        line=list(width=12, color=~c, colorscale=list(c(0,'red'), c(1,'blue')))) %>%
  # trace the z-axis
  add_trace(data1, x = 0, y =0, z = ~z, type="scatter3d", mode="lines", 
              line = list(width = 10, color = ~c, colorscale = list(c(0,'red'), c(1,'blue'))),
            name="Z", hoverinfo="none") %>%
  # add projection on plane x=-3
  add_trace(data1, x = -3, y = ~y, z = ~z, type="scatter3d", mode="lines", 
              line = list(width = 2, dash="solid", color = ~c, 
                          colorscale = list(c(0,'gray'), c(1,'black'))),
            name="Z", hoverinfo="none") %>%
  # add projection on plane y=-3
  add_trace(data1, x = ~x, y = -3, z = ~z, type="scatter3d", mode="lines", 
              line = list(width = 2, dash="solid", color = ~c, 
                          colorscale = list(c(0,'gray'), c(1,'black'))),
            name="Z", hoverinfo="none") %>%
  # add a z-line at x=-3,y=0
  add_trace(data1, x = -3, y = 0, z = ~z, type="scatter3d", mode="lines", 
              line = list(width = 5, dash="solid", color = "gray"),
            name="Z", hoverinfo="none") %>%
  # add a z-line at x=0,y=-3
  add_trace(data1, x = 0, y = -3, z = ~z, type="scatter3d", mode="lines", 
              line = list(width = 5, dash="solid", color = "gray"),
            name="Z", hoverinfo="none") %>%
  # add a x-line at y=0, z=-40
  add_trace(data1, x = ~x, y = 0, z = -40, type="scatter3d", mode="lines", 
              line = list(width = 15, dash="solid", color = "gray"),
            name="Z", hoverinfo="none")
  # add a few annotations/arrows
p
```


## General QM wavefunctions

In quantum mechanics, we often estimate the probability density for a particle's 
position (or momentum, energy, spin, etc.) by using a dynamical model equation 
describing how $p(x,y,z, t)$ changes over time. Despite being highly quantitative, 
probabilities are always associated with uncertainty, i.e., they are stochastic
representations of event likelihoods. In general, probabilistic models represent
*lossy information compression*, as they do not contain all information 
needed to completely reconstruct, describe, or predict the exact past and future 
of a particle system. There will be situations where particles start with 
identical densities $p(x,y,z, t)$ under fixed initial conditions, however
the systems evolve over time into completely different states.

**Note**: Much like the duality between time-limited and band-limited functions
describe different characterizations of the same signals via the Fourier transform,
particle systems are completely described by their corresponding *wavefunctions*, 
which may be difficult to work with. Wavefunctions are *approximately described* by 
their corresponding *probability functions* - squared magnitudes of the wavefunctions.

Using the probability functions we have sufficient information to predict likelihoods
by using the system's complex-valued *wavefunction* $\psi(x)$ to express
the corresponding *probability density* $p(x,y,z, t)=\psi^*(x,y,z,t)\psi(x,y,z,t)$.

In essence, even though both are descriptors of the same particle system,
the *wavefunctions are richer information theoretic objects*, whereas the 
*probability densities are more computationally efficient (realistic) object representations*.
The complex phase of $\psi(x,y,z,t)$
varies dynamically across spacetime and captures the complete
information about the particle's placement, momentum, spin, energy, etc.,
which are useful to characterize and predict its future behavior. 

The measuring unit of a wavefunction can be thought of as 
*square root of a probability density*, and the particle system wavefunction 
is a solution the [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)

$$i\hbar {\frac {\partial }{\partial t}}\psi (x,t)=\left[
\underbrace{-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}}_{kinetic}+\underbrace{V(x,t)}_{potential}\right]\psi (x,t)\ ,$$

where in this one-spatial dimension example, $\psi (x,t)$ is a wavefunction
assigning a complex number to each spatial location $x$ and each time point $t$,
$m$ is the mass of the particle, and $V( x , t )$ is the *potential energy*
of the system environment, $i$ is the imaginary unit, and $\hbar$ is the reduced 
[Planck constant](https://en.wikipedia.org/wiki/Planck_constant) measured
in units of action ($energy \times time$).

The physical interpretation of a single spatial dimension wavefunction 
$\psi(x,t)$ may be explicated in diffraction patterns of electromagnetic waves
observed in two-slit interference of monochromatic light. 
The *wavefunction* $\psi(x,t)$ of a light wave represents the electric field strength
with an *energy density* $\psi^*(x,t)\psi(x,t)$. Since the 
[Planck constant](https://en.wikipedia.org/wiki/Planck_constant) $h$ is fixed,
the energy of an individual photon $\epsilon_{photon}=hf$ only depends 
on the *light frequency* $f$. Hence, $|\psi(x,t)|^2=\psi^*(x,t)\psi(x,t)$ 
is proportional to the number of photons. When light waves travel through two
space-separated slits, there is *interference* reflecting the superposition of the 
waves that shows on the diffraction screen used to observe (collect) the 
measurements. Patterns of *bright fringes* correspond to points of *constructive* 
(*agglomerative*) light wave interference, whereas *dark fringes* correspond 
to points in the diffraction plane reflecting *destructive* (discriminative)
interference of the light waves.

When the image detection screen where the wave image is projected is exposed to 
very weak light (few photons at a time), the observed interference pattern 
appears random, and less characteristic of light wave interference. 
Each individual photon hitting the detection screen appears as a single point. 
The overall *point density* is expected to be *high* at locations of 
*constructive* wave interference, whereas correspondingly *low (point density)* 
at locations of *destructive* wave interference. 

Per unit area, the probability that a single photon will strike a particular 
spot $(x)$, on the screen is proportional to the *square of the total electric field*, 
$|\psi(x,t)|^2=\psi^*(x,t)\psi(x,t)$.

[This PHET interactive demonstration shows the light waves passing through the slits
and forming overlapping circular waves that interact and project on the detection screen](https://phet.colorado.edu/sims/cheerpj/quantum-wave-interference/latest/quantum-wave-interference.html?simulation=quantum-wave-interference). 
High-density points (*maxima*) correspond to overlapping wave crests, whereas low-density
points (*minima*) correspond to locations where the *crests* of the wave from one slit
overlap the *troughs*, which cancels them. 

The interpretations of *square of the matter waves* $|\psi(x,t)|^2=\psi^*(x,t)\psi(x,t)$ 
is similar to the interpretation of the *square of the electric field* $|E(x,t)|^2$, where $E(x,t)$ is the *electric field strength.* Specifically, the probability density
$|\psi(x,t)|^2$ is the probability that a particle will be found at a particular 
spatial position ($x$) and time ($t$) per unit length.

More generally, the probability that at time $t$ a particle is located in a 
narrow interval $(x, x + dx)$ is

$$P(x,x+dx)=|\psi(x,t)|^2dx\ .$$

For *smooth wavefunctions*, which vary slowly over the spatial interval, we can 
approximate the probability that a particle is found in the interval $\Delta x$ by

$$P(x,x+\Delta x)\approx |\psi(x,t)|^2 \Delta x\ .$$

The square of the complex-valued wavefunction guarantees the resulting probability
is in $\mathbb{R}^+$, even though the wavefunction is complex-valued, $\psi(x,t):\mathbb{R}\times \mathbb{R}^+\to \mathbb{C}$. 

For *rapidly changing wavefunctions* over the interval, we use 
[integration to estimate
the same probability](https://socr.umich.edu/BPAD/BPAD_notes/Biophysics430_Chap01_MathFoundations.html) that that a particle is found in the interval $\Delta x$

$$P(x,x+\Delta x)=\int_{x}^{x+\Delta x} |\psi(x',t)|^2\ dx'\ .$$

This probability is just the area under the function $|\psi(x,t)|^2$ between $x$ and
$x+\Delta x$. Clearly, to satisfy the unitarity axiom of probability, we need
a *normalization condition*.

$$P(-\infty, +\infty)=\int_{-\infty}^{\infty} |\psi(x',t)|^2\ dx'=1\ .$$

Of course, for particles in 3D spatial dimensions the integration extends to cover a
hypervolume (solid volume) of interest.

## Example 1: Linear Density

Suppose a *particle*, i.e., a *moving ball*, is constrained along a linear tube of length $L$.
Assume at some time $t$, the probability of finding the ball increases linearly from 
$0$ (at the start of the tube $0$) to $1$ (at the end of the tube $L$). 
Find the probability of finding the ball in the left half of the tube at that time $t$.

*Solution*: Using the probabilistic interpretation of the quantum mechanical 
wavefunction we can express the wavefunction as a function of spacetime $(x,t)$. 

```{r triangleProb, warning=FALSE, message=FALSE}
library(plotly)    
L <- 5
linFunct <- function (x) {
  return (x/L)
}
int_val_0.5L   <- round(integrate(linFunct, lower=0, upper=L/2)$value, 3)
t <- seq(from=0, to=L, length.out = 1000)

# plots
f_val <- t/L

plot_ly(x = t[1:500], y = f_val[1:500], type = 'scatter', 
        name=paste0("Left Half Density y=f(x)=x; Area=", int_val_0.5L / L),
        mode = 'lines', fill = 'tozeroy', opacity=0.3) %>%
  add_trace(x = t[501:1000], y = f_val[501:1000], type = 'scatter', 
        name=paste0("Right Half Density y=f(x)=x; Area=", (L-int_val_0.5L)/L),
        mode = 'lines', fill = 'tozeroy', opacity=0.3) %>%
  layout(title="Probability Density of a Ball Traveling in a Linear Tube of Length L (fixed time point)", 
         xaxis=list(title="Space", tickvals=c(0, L/5, 2*L/5, L/2, 3*L/5, 4*L/5, L), 
                    automargin = T, #seq(0, L, length=L+2), 
                    ticktext = c("0", "L/5", "2L/5", "L/2", "3L/5", "4L/5", "L")), 
         yaxis=list(title="Probability"), legend = list(orientation = 'h', y=-0.5))
```

The wave function of the ball can be expressed as

$$\psi \, (x,t) = \overbrace{C}^{const.} \times \overbrace{x}^{linear} \times 
\overbrace{\mathbb{I}_{\{0\leq x \leq L\}}(x)}^{support}, \ 
\forall (x,t)\in\mathbb{R}\times\mathbb{R}^+ \ .$$

Using the probability normalization condition (total probability must be $1$),
we can estimate the constant $C=\frac{\sqrt{3}}{L^{3/2}}$, 

$$1=P(\infty, +\infty) = \int_{-\infty}^{\infty} |C\times x \times 
\mathbb{I}_{0\leq x \leq L}(x)|^2 dx = \\
\int_{0}^{L} |C\ x |^2 dx = C^2 \frac{x^3}{3}\bigg |_{0}^{L/2}=
\frac{C^2 L^3}3 = 1\ \ \Longrightarrow C=\frac{\sqrt{3}}{L^{3/2}}.$$


Next, we can integrate the *squared wavefunction magnitude* over the *left-half* of 
the tube to obtain the *probability* that the particle is in that left-half of the tube. The graph shows that the probability increases linearly with the position $x$.
Hence, the chance that the particle is in the left-half is much smaller than the 
probability that the particle is in the right-half of the tube

$$P\left (x = 0, \frac{L}{2}\right ) = \int_{0}^{L/2} \left| C x \right| ^2 dx
= C^2\frac{x^3}{3} \bigg |_{x=0}^{L/2} = \frac{3}{L^{3}}\ \frac{L^3}{24}
=\frac{1}{8}= 0.125\ . $$

Clearly the odds that the particle is in the *left half* of the tube $p=0.125$
are much smaller than finding the particle in the *right half* of the biased (loaded)
tube, $p=1 - 0.125 = 0.875$.


## Example 2: Non-linear density

Suppose now the a moving ball is constrained along a linear tube of length $L$, where 
instead of the linear density we saw in the prior example, we have a trigonometric $\sin()$ 
density function. In this case, we will compute the probability of finding the 
particle in the *first one-quarter of the tube*, $[0,L/4]$.

*Solution*: A similar strategy to the earlier linear density *Example 1*, however, now the wavefunction has *two unknown parameters*, instead of a sole *scaling* parameter $C$ corresponding to *amplitude* of the wavefunction. The second parameter is associated with the *wavelength* ($\kappa$) of the `sine` wave. The amplitude parameter can be estimated using the given boundary conditions. Whereas the estimation of the wavelength parameter can be accomplished by normalizing the wavefunction to ensure the resulting probability is unitary. 

To compute the probability that the particle is in the *first one-quarter of the tube*, 
we again integrate the square of the wavefunction over the first quarter of the tube.

The wavefunction of the ball can be written

$$\psi(x,t)=C\ \sin(\kappa x)\ \mathbb{I}_{\{0\leq x \leq L\} }(x), 
\forall (x,t)\in\mathbb{R}\times\mathbb{R}^+ \ , $$

where the parameters $C$ and $\kappa=\frac{2\pi}{\lambda}$ represent the 
wavefunction *amplitude* and *wavenumber*. As the particle is restricted inside
the tube, the amplitude of the wavefunction is trivial outside the tube of length $L$

$$\psi(x=0,t=0)\equiv \psi(x=L,t=0) =0 \Longrightarrow \\
\underbrace{C \sin(\kappa L)}_{\psi(x,t)} = 0 \Longrightarrow 
\kappa L= m\pi, \ \forall m\in \mathbb{Z} \ .$$

For $m=1$, one solution for the *wavenumber* is $\kappa=\frac{\pi}{L}$.

For estimating the wavefunction *amplitude* we use the normalization 
condition 

$$\begin{align} 1=P(x\in [0, L]) &= \int_0^{L} \left|
\underbrace{C \sin(\overbrace{\kappa}^{\frac{\pi}{L}} x)}_{\psi(x,t)}\right|^2dx \\[5pt] &\underbrace{=}_{x'=\kappa x} \frac{C^2}{\kappa}
\int_0^{\pi} \sin^2(x')dx' = \frac{C^2}{\kappa}\ \frac{\pi}{2}\approx 1.5708
\ \frac{C^2}{\kappa}\\[5pt] &
\underbrace{\Longrightarrow}_{\kappa=\frac{\pi}{L}} C=\sqrt{\frac{2}{L}} . \end{align} $$

Therefore, the wavefunction describing the spatiotemporal
behavior of the particle is

$$\psi(x,t)= \sqrt{\frac{2}{L}} \sin\left (\frac{\pi}{L} x\right )\ .$$

Finally, we can compute the probability of finding the particle in the *first one-quarter of the tube* by squaring the wavefunction amplitude and integrating over the interval $[0,L/4]$

$$\begin{align} P(x\in [0, L/4]) &= \frac{2}{L} \int_0^{L/4} 
 \sin^2\left (\frac{\pi}{L} x\right ) dx =
\frac{\pi -2}{4\pi}\approx 0.090845\ . \end{align} $$

```{r sineTube, warning=FALSE, message=FALSE}
# library(plotly)    
L <- 5
kappa <- pi/L
C <- 
sinFunct2 <- function (x) { 
  #|    | psi(x,t=0)|^2 = (2/L) * sin ( (pi/L) * x ) * * sin ( (pi/L) * x )
  return ( (2/L) * (sin(x * (pi/L)))^2 )
}
int_val_0.25L   <- round(integrate(sinFunct2, lower=0, upper=L/4)$value, 3)
t <- seq(from=0, to=L, length.out = 1000)

# plots
f_val <- sqrt(2/L) * sin(t * (pi/L))

plot_ly(x = t[1:250], y = f_val[1:250], type = 'scatter', 
        name=paste0("First Quarter Density y=f(x)=C sin(k x); Area=", int_val_0.25L),
        mode = 'lines', fill = 'tozeroy', opacity=0.3) %>%
  add_trace(x = t[251:1000], y = f_val[251:1000], type = 'scatter', 
        name=paste0("Right 3 Quarter Density y=f(x)=C sin(k x); Area=", 1-int_val_0.25L),
        mode = 'lines', fill = 'tozeroy', opacity=0.3) %>%
  layout(title="Nonlinear Probability Density of a Particle\n Traveling in a Tube of Length L (fixed time point)", 
         xaxis=list(title="Space", 
              tickvals=c(0, L/8, 2*L/8, 3*L/8, 4*L/8, 5*L/8, 6*L/8, 7*L/8,L), 
              automargin = T,      #seq(0, L, length=L+2), 
              ticktext = c("0","L/8", "L/4", "3L/8", "L/2", "5L/8", "3L/4",
                           "7L/8", "L")), 
         yaxis=list(title="Probability"), legend=list(orientation='h', y=-0.5))
```

Hence, the probability of finding the particle in the *first quarter of the tube* 
is $\approx 9.1\%$. According the periodicity of the
trigonometric function $\sin()$, the particle wavelength is $\lambda=2 L$. 
For [matter waves, the relationship](https://labs.phys.utk.edu/mbreinig/phys222core/modules/m10/matter_waves.html) between *momentum* $p$ and *wavelength* 
$\lambda$ is $p = \frac{h}{\lambda}$ and the relationship between *energy* $E$ and 
*frequency* $f$ is $E = hf$. For our *macroscopic tube* of length $L=1m$ ($1$ meter), 
we can estimate the *momentum* of the particle 

$$p=\frac{h}{\lambda}=\frac{h}{2 L}\approx 10^{-36}m/s\ ,$$
which is too tiny to be *directly* sensed, detected, or observed using currently available instruments or technologies.

#  Copenhagen Quantum Mechanics Interpretation of Wavefunctions

Suppose we are modeling the dynamics of a propagating sound wave whose *wavefunction*, $\phi(x,t)=C \sin(\kappa x - \omega t)$, is a mathematical function modeling the spatial extents and temporal dynamics of the particle
and quantify the likelihood of the particle to be within certain spacetime
domain. A [harmonic wavefunction of the particle *position*](https://socr.umich.edu/BPAD/BPAD_notes/Biophysics430_Chap02_PhysicsMechanics.html) $x$ and *time* $t$ can be used to a model for the particle displacement $\psi(x, t)= C \sin(\kappa x - \omega t)$, where the model parameters $\kappa  = \frac{2\pi}{\lambda}$ is the *wavenumber* and $\omega$ is the *frequency* of the sound wave propagating in the horizontal direction. 

This harmonic wave function represents a solution of the 1D wave equation 

$${\frac {\partial^{2} \psi}{\partial t^{2}}} = c^{2} {\frac {\partial^{2} \psi}{\partial x^{2}}},$$

where $c=\frac{\omega}{\kappa}\equiv \sqrt{\frac{E}{\rho}}$ is the *phase velocity* (speed) of the sound wave, $E$ is Young's modulus (compression modulus), $\rho$ is the density of the ambient material. 

```{r soundWave, warning=FALSE, message=FALSE}
# library(plotly)
size=300
t <- seq(from=0, to=2*pi, length.out=size)
pressure <- cos(5*t)
displacement <- sin(5*t)
y <- seq(from=-1, to=1, length.out=size)
z <- pressure %o% rep(1, size)
plot_ly(x=~t, y=~y, z=~t(z), type="heatmap", colors="Greys", showscale= F) %>%
  add_trace(x=~t, y=~pressure, type="scatter", mode="lines", name="Pressure",
            line = list(color = 'blue', width = 5)) %>%
  add_trace(x=~t, y=~displacement, type="scatter", mode="lines",
            name="Displacement", line = list(color = 'red', width = 5)) %>%
  layout(title = "Displacement and Pressure of a Propagating Sound Wave", 
         xaxis=list(title="time"), 
         yaxis=list(title="Value", scaleanchor="x", scaleratio=2.2),
        legend = list(title=list(text='<b>Functions</b>'), orientation='h'))
```

This sound wave *wavefunction model* can also be used for *predictions.*
In other words, *data-driven predictions* can be complemented by *model-based predictions* via the wavefunction $\psi (x,t)$ without assuming the availability of observations or measurements of the process. 

This theoretical-model based inference scheme avoids potential measurement-intervening
effects on the actual physical particle spatiotemporal dynamics. The particle model
provides a rigorous quantification of the spatial localization 
of the particle using the probabilistic interpretation of the wavefunction.

In quantum mechanical reality, the particle is in a superposition of all of its base states and simultaneously occupies each position, everywhere all the time, $(x,t)\in\mathbb{R}^3\times \mathbb{R}^+$. It's just
that its probabilistic localization is *stochastic.* During the process of *instantaneously*
measuring or detecting observations, the *wavefunction collapse interpretation*
suggests that the particle may be in one specific
fixed position state $(x,x+dx)$ with a corresponding *probability*

$$P([x,x+\Delta x], t)=|\psi(x,t)|^2 \Delta x\ .$$

## Example 3: Quantum Computing and Qubits

A binary two-state systems, such as $\pm$, $left/right$, $\uparrow\ /\ \downarrow$, etc., 
are used to illustrate basic quantum mechanics principles. Applications of two-state 
system models include *electron spin* and *mixed states of particles*, *atoms*, *molecules*, and *quantum computing.* Contemporary digital computers store, encode, 
and compute information in *binary digits* ($0$'s and $1$'s).

*Quantum computers* store, encode, and process information and data in 
the form of *quantum bits* (**qubits**). Qubits are more than a binary
two-state system. They encode information using superposition of mixed
$0$'s and $1$'s states. This duality between binary and quantum systems is reminiscent to the duality between *arithmetic* over the reals $x,y, (x+y)\in\mathbb{R}$ and *complex-arithmetic*
over the significantly *enriched complex plane*
$(\alpha x+i\beta y)\in\mathbb{C}\cong \mathbb{R}\times \mathbb{R}$. 


In general, the form of the wavefunction depends on the specifics of the physical system. 
Most *wavefunctions are complex-valued*, whereas all *physical reality experimental measurements* only produce real number outcomes. The probability that a particle is found in the 
narrow interval $(x,x+\Delta x)$ at time $t$ is

$$P(x,x+\Delta x)=|\psi(x,t)|^2dx=\underbrace{\psi^*(x,t)}_{complex\\ conjugate}
\psi(x,t)\Delta x\ .$$

For instance, if the value of the wavefunction at a fixed spatiotemporal location
$(x_o,t_o)$ is $\psi(x_o,t_o)=1.5-4i$, its conjugate is $\psi^*(x_o,t_o)=1.5+4i$ and
$\psi^*(x_o,t_o)\times \psi(x_o,t_o)=18.25$.

The wavefunction represents a quantum mechanical model for the motion of a free 
particle moving with a constant velocity along the x-axis without influence of 
external forces

$$\psi(x,t)= \underbrace{C}_{amplitude}\cos(\kappa x - \omega t) + 
i C\sin(\kappa x - \omega t) =
Ce^{i\ \overbrace{(\kappa x - \omega t)}^{\phi,\ phase\ angle}},$$
where $C$ is the *amplitude*, $\kappa$ is the *wavenumber*, $\omega$
is the *angular frequency*, and $\phi=(\kappa x - \omega t)$ is the *phase angle.*  

Over an infinitesimal interval of length $\Delta x$, the probability of finding the particle in that interval is

$$P(x,x+\Delta x)\approx \psi^*(x,t)\psi(x,t)\Delta x=
(C e^{-i\phi})(C e^{i\phi})\Delta x = (C^* C)\Delta x = 
|C|^2\Delta x\in\mathbb{R}\ .$$

Note that the constant $C\in\mathbb{C}$, $|C|^2\in\mathbb{R}$,
and $P(x,x+\Delta x)\geq 0$ is the *probability density*
that leads to *quantum mechanics predictions* that can be checked 
using real physical world experimental observations.

**Example**. Consider a particle with energy $E$ moving along the 
x-axis restricted to the interval $[0,L]$. Let's assume 
$\psi(x,t) = C e^{-i\frac{Et}{\hbar}} \sin(\pi x) \mathbb{I}_{\{0\leq x\leq L\}}(x)$ is the wavefunction model for this system.

As we showed earlier, to ensure the probabilistic interpretation of
the model, the normalization constant has to be $C=\sqrt{\frac{2}{L}}$.

# Expectation Values

The classical mechanics solutions to the [equations of motion](https://en.wikipedia.org/wiki/Equations_of_motion) are functions
of a *measurable quantity*, such as the *particle position* $x(t)$ at time $t$. 

Here are the equations of motion for constant acceleration, $a$:

$$v=u+at,\\ s=\frac{u+v}{2}t,\\ v^2=u^2+2as, \\ s=ut+\frac{1}{2}at^2, 
\\s=vt-\frac{1}{2}at^2 \ .$$
where the *variables* are

 - $t$ - time (seconds, $s$),
 - $s$ - distance (or displacement in meters, $m$), 
 - $u$ - initial velocity ($m/s$), 
 - $v$ - velocity at time t  ($m/s$),
 - $a$ - the constant acceleration ($m/s^2$).

Note that all the measurements are in base *SI* units and the quantities $s$, 
$u$, $v$, and $a$ are all *vector quantities*
and their signs reflect the *direction of motion.*

In *classical Newtonian mechanics*, the particle has one value of position $x(t)$
for any time $t$. Whereas in *quantum mechanics*, the solution to the equations of motion
are *wavefunctions*, $\psi(x,t)$ and the particle has a *distribution associated
with all positions* at a given time $t$, not a single value, but a distribution
quantifying the *probability density* of finding the particle in certain areas
using the square wavefunction amplitude $|\psi(x,t)|^2$.

To relate these QM distributional representations to discrete observable classical
mechanics measurements we can compute the corresponding *position expected value*,
which represents the *average value of position over a large number of particles (repeated sampling connection to spacekime representation) with the same wavefunction*:

$$\overbrace{\langle \underbrace{x}_{position}\rangle}^{Expected\ value\ of \\ 
position\ at\ time\ t} = \int_{-\infty}^{\infty} x\ \overbrace{P(x,t)}^{density}dx=
\int_{-\infty}^{\infty} x\ \psi^*(x,t) \psi(x,t)\ dx\\ =
\int_{-\infty}^{\infty} \underbrace{\psi^*(x,t)\ x\ \psi(x,t)}_{\langle \psi | 
\underbrace{\hat{x}}_{Position\\ operator} | \psi \rangle}\ dx=
\langle \psi | \hat{x} | \psi \rangle\ .$$

Just like we have multiple *coordinate representations* of describing functions
$f:\mathbb{R}^2\to \mathbb{R}$, e.g., Cartesian $(x,y)$, polar $(\rho, \varphi)$,
and conjugate pairs $(z,{\bar{z}})$, the wavefunction can also be expressed in 
different bases, i.e., in terms of quantities other than position. Examples
of such alternative bases for representing the wavefunction include *velocity* 
$(v)$, *momentum* $(p)$, *kinetic energy* $(K)$, etc. It's often useful to express
the *expectation value of momentum*

$$\overbrace{\langle \underbrace{p}_{momentum}\rangle}^{Expected\ value\ of \\ 
momentum\ at\ time\ t} = \int_{-\infty}^{\infty} p\ \overbrace{P(p,t)}^{density}dp=
\int_{-\infty}^{\infty} p\ \psi^*(p,t) \psi(p,t)\ dp\\ =
\int_{-\infty}^{\infty} \underbrace{\psi^*(p,t)\ p\ \psi(p,t)}_{\langle \psi | 
\hat{p} | \psi \rangle}\ dp=
\langle \psi | \underbrace{\hat{p}}_{Momentum\\ operator} | \psi \rangle\ ,$$

where $dp$ is an infinitesimal interval in momentum. When the wavefunction is
expressed in position basis, $\psi(x,t)$, rather than in momentum space, $\psi(p,t)$,
we can compute the *expectation value of momentum in position-space*

$$\overbrace{\langle \underbrace{p}_{momentum}\rangle}^{Expected\ value\ of \\ 
momentum\ at\ time\ t} = 
\int_{-\infty}^{\infty} {\psi^*(x,t)\ \underbrace{\hat{p}}_{Momentum\\ operator} 
\ \psi(x,t)}\ dx\\ =
\int_{-\infty}^{\infty} {\psi^*(x,t)\ \underbrace{\left (-i\hbar\frac{d}{dx}\right )
}_{Momentum\ operator\\ in\ direction\ x}\ \psi(x,t)}\ dx\equiv -i\hbar\frac{d}{dx}\ .$$

Explicitly, to compute the expectation value of the momentum in position space,
the integrand $\langle \psi | \hat{p} | \psi \rangle$ involves two steps.
First, the momentum operator operates on the wavefunction to the right, 
$\hat{p} | \psi \rangle$. And second, the result is multiplied by the complex 
conjugate of the wavefunction on the left, 
$$\langle \psi | \hat{p} | \psi \rangle\equiv \underbrace{\langle \psi |}_{second\\ step} \overbrace{\left (\hat{p} | 
\psi \rangle\right )}^{first\\ step}\ .$$

There many alternative strategies to derive the *momentum operator in position space*
(e.g., in the 1D $x$-direction). One approach is by using the [Fourier transform](https://socr.umich.edu/BPAD/BPAD_notes/Biophysics430_Chap12_Imaging.html#2_Fourier_Transform) 
relation between position and momentum representations. We can express

$$\psi (x)=\langle x|\psi \rangle =\int \langle x|p\rangle \langle p|\psi \rangle dp=
\frac{1}{{\sqrt {2\pi \hbar }}} \int {e^{ixp/\hbar }{\hat {\psi }}(p) }\ dp\ ,$$

where $\hat {\psi }(p)$ is the Fourier transform of the wavefunction $\psi(x)$, 
transforming from coordinate *position-space* ($x$) to *momentum space* ($p$). 
Than,

$${\hat {p}}=\int |p\rangle p\langle p| \ dp=-i\hbar \int |x\rangle 
\left( {\frac {d}{dx}}\right ) \langle x|\ dx\ .$$

In other words, the momentum acting in coordinate space corresponds to spatial frequency,

$$\langle x|{\hat {p}}|\psi \rangle =-i\hbar {\frac {d}{dx}}\psi (x)\ .$$

Similarly, the forward and the inverse [Fourier transforms](https://socr.umich.edu/HTML5/Fourier_Wavelet_app/?Fourier) allows us to express
the *position operator in the momentum basis*,

$$\langle p|{\hat {x}}|\psi \rangle =i\hbar {\frac {d}{dp}}\psi (p)\ .$$

For the other spatial dimensions ($y,z$), the momentum operators are defined analogously.


**Example**: Consider the one-spatial dimension ($x$) *kinetic energy operator* 

$$\hat{K} =\frac{1}{2}m(\hat{v}_x)^2=\frac{(\hat{p}_x)^2}{2m}=
\frac{\left (-i\hbar \frac{d}{dx}\right )^2}{2m} =-\frac{\hbar^2}{2m}
\frac{d}{dx}\left (\frac{d}{dx} \right )\ .$$

In essence, computing the *expectation value of the kinetic energy* of a particle in 1D
requires a pair of derivatives of the wavefunction prior to its integration
(to find the overall average). When the wavefunction is *even*, *odd*, or *symmetric*,
the calculations of operator expectation values are often simplified.

## Example: Laplace wavefunction

Consider a particle whose behavior is modeled by a
*Laplace wavefunction* $\psi(x)=\frac{1}{\sqrt{x}}\ e^{-\frac{|x|}{x_o}}$.
Compute the *expectation value of position*.

*Solution*: Plug in the Laplace wavefunction in the definition of expectation value
for the position operator. Recalling that an integral of an *odd function* over
a symmetric interval is trivial:

$$\begin{align*} \langle x \rangle &= \int_{-\infty}^{\infty} x|\psi(x)|^2 dx = 
\int_{-\infty}^{\infty} x\left |\dfrac{e^{-\frac{|x|}{x_0}}}{\sqrt{x_0}}\right |^2\ dx \\[4pt] &= \dfrac{1}{x_0} \int_{-\infty}^{\infty} \underbrace{x e^{-2\frac{|x|}{x_0}}}_{odd\ function}\ dx = 0  \end{align*}  \ .$$

The result that $\langle x\rangle =0$ is to be expected as the Laplace probability density
function is symmetric about $x=0$.

### Connection between Expectation Value, Laplace Transform, and MGF

In probability and statistics, there is a direct connection between the *expectation value*, the *Laplace transform*, and the *moment generating function* (MGF).

Suppose $X$ is a *random variable*, then its *Laplace transform* (LT), more specifically, the LT of its *probability density function* $f_X(\cdot)$, is given by the expectation of an exponential: 

$$\mathcal{L}(X)\equiv \mathcal{L}(f_X)(z)=
\mathbb{E}(e^{-zX})\ .$$

The [moment generating function](https://en.wikipedia.org/wiki/Moment-generating_function) of a random variable $X$ is

$$MGF_{X}(t)=\mathbb{E} \!\left[e^{tX}\right]= \int_0^{\infty}f_X(x) e^{tx}dx\ , \forall t\in\mathbb{R}^+\ .$$
The Laplace transform of a distribution with probability density function $f_X(\cdot)$ over time $t\in\mathbb{R}^+$ is

$${\mathcal{L}}(f_X)(t)=\mathbb{E}\left[e^{-tX}\right]=
\int_0^{\infty}f_X(x) e^{-tx}dx\ .$$

The double-sided Laplace transform of the density $f_X$ is

$$\mathcal{L}(f_{X})(t)=\int_{-\infty }^{\infty }e^{-tx}f_{X}(x)\ dx\ ,$$

and the moment-generating function is

$$MGF_{X}(t)=\mathbb{E} \left[e^{tX}\right]=
\int_{-\infty }^{\infty }e^{tx}f_{X}(x)\ dx\ .$$

Hence, the relation between a random variable's moment generating function $MGF_{X}(t)$ and the double-sided Laplace transform of its probability density function 
$f_{X}(x)$ is

$$M_{X}(t)={\mathcal {L}}(f_{X})(-t),$$

In other words, when the moment generating function exists, the characteristic function of $X$ is a [Wick rotation](https://en.wikipedia.org/wiki/Wick_rotation) of $MGF_{X}(t)$. The characteristic function of a continuous random variable $X$ is the Fourier transform of its  density $f_{X}(x)$. The Fourier transform of an exponential function $f(x)$ is a Wick rotation of its double-sided Laplace transform. 


## Exponential Function Uniqueness

**Theorem**: The exponential function $f(x)\equiv C\ e^{x}$ is the only function
that is its own derivative.

*Proof*: First, confirm that if $f(x)\equiv C\ e^{x}$, then 
$f'(x)\equiv C\ e^{x}=f(x),\ \forall x$. Next, suppose another function 
possesses the same property $g'(x)\equiv g(x),\ \forall x$. 
Consider their *quotient* $h(x)=\frac{g(x)}{f(x)}=\frac{g(x)}{e^{x}}=g(x)e^{-x}$,
which is well-defined, since $e^x\gt 0,\ \forall x$. Realize that the quotient 
function is constant as $h'(x)=g'(x)e^{-x} - g(x)e^{-x}=0$, since we assumed 
the property $g'(x)\equiv g(x)$. Therefore, $h(x)$ is a constant and 
$g(x)=Cf(x)$, and the exponential function is the only possible function that is
its own derivative.

This *uniqueness of the exponential function* is important as assuming a 
constant Hamiltonian (time-invariant energy), the Schrodinger equation 
$\hbar \frac{d}{d t}|\psi(t)\rangle = H|\psi(t)\rangle$ has a solution

$$|\psi (t)\rangle =\underbrace{e^{-i\frac{{\hat {H}}t}{\hbar} }}_{exponential}\times 
\underbrace{|\psi (0)\rangle
}_{constant}.$$

The operator $\hat{U}( t ) = e^{-i\frac{{\hat {H}}t}{\hbar} }$ is the 
*unitary time-evolution operator*, it preserves the inner product between
vectors in the Hilbert space. If the initial state is 
$| \psi( 0 )\rangle$, then the state at a later time $t$ is given by
$|\psi(t)\rangle = \hat{U}( t ) |\psi(0)\rangle$. 

## Example: Time-dependent wavefunction

Let's consider a particle restricted to an interval $[0,L]$ whose
behavior is governed by *time-dependent wavefunction* 

$$\psi(x,t) = C\ e^{-i\omega t}\ \sin \left (\frac{\pi x}{L}\right )
\ \mathbb{I}_{\{0\leq x\leq L\}}(x)\ ,$$

where $\omega$ is the *angular frequency*. The $\sin()$ periodic trigonometric 
function ensures the wavefunction confines the particle to the interval 
$[0,L]$ and $\psi(0,t) =\psi(L,t) =0\ \forall\ t\in\mathbb{R}^+$.
Calculate the expectation values of the particle's *position*, *momentum*, 
and *kinetic energy.*

*Solution*: First, to ensure probabilistic interpretation of the wavefunction square
magnitude, we normalize the wavefunction to estimate the amplitude constant $C$.
Then, we will expand the three operators to estimate the corresponding
expectation values.

 - Estimation of the *normalization constant* ($C$):

$$\begin{align*} 1 =\langle \psi |\psi \rangle &= \int_0^L \psi^* (x) \psi(x)\ dx = 
\int_0^L \left(C e^{+i\omega t} \sin \, \dfrac{\pi x}{L}\right) \left(C e^{-i\omega t} 
\sin \, \dfrac{\pi x}{L}\right) dx \\[4pt] &= C^2 \int_0^L \sin^2 \, 
\dfrac{\pi x}{L}\ dx = C^2 \dfrac{L}{2}\ \  \Longrightarrow\ \  C = \sqrt{\dfrac{2}{L}}.
\end{align*} $$

 - Expectation value of *position*

$$\begin{align*}\langle x \rangle &= \int_0^L \psi^* (x) x \psi(x) \ dx = 
\int_0^L \left(C e^{+i\omega t} \sin \, \dfrac{\pi x}{L}\right) x \left(C e^{-i\omega t} 
\sin \, \dfrac{\pi x}{L}\right) \ dx \\[4pt] &= 
C^2 \int_0^L x \, \sin^2 \, \dfrac{\pi x}{L}\ dx = C^2 \dfrac{L^2}{4} 
\ \ \Longrightarrow \ \  \langle x \rangle = \dfrac{L}{2}. \nonumber \end{align*}  \nonumber$$

 - Expectation value of *momentum* in the $x$-direction
 
$$\underbrace{\hat{p}}_{operator}|\psi(x)\rangle \equiv -i\hbar\frac{d}{dx} \psi(x) = -i\hbar \dfrac{d}{dx}
\left (C e^{-i\omega t}\sin \frac{\pi x}{L} \right ) = 
- i\frac{C\ \hbar}{2L} e^{-i\omega t} \cos\frac{\pi x}{L}.$$

$$\begin{align*} \Longrightarrow \ \  \langle p \rangle &= \int_0^L 
\underbrace{\left(C e^{+i\omega t}\sin \frac{\pi x}{L}\right)}_{\underbrace{\langle\psi(x)|}_{bra}}
\underbrace{\left(-i \frac{Ch}{2L} e^{-i\omega t} \cos\frac{\pi x}{L}\right)}_{\hat{p}\underbrace{|\psi(x)\rangle}_{ket}}
\ dx \\[4pt] &= 
-i \dfrac{C^2 \hbar}{4L} \int_0^L \underbrace{\sin \frac{2\pi x}{L}}_{odd\ function\\ 
about\ x=\frac{L}{2}}\ dx =0\ . \end{align*}$$

 - Expectation value of the *kinetic energy* in the $x$-direction 
 
$$\begin{align} \underbrace{-\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}}_{\hat{K}} 
\psi (x) &= 
- \dfrac{\hbar^2}{2m} \dfrac{d^2}{dx^2} \left (C e^{-i\omega t} \, \sin \, 
\dfrac{\pi x}{L}\right ) \\[4pt] &= - \dfrac{\hbar^2}{2m} C\ e^{-i\omega t} 
\frac{d^2}{dx^2} \, \sin \, \dfrac{\pi x}{L} = 
\frac{C\ \hbar ^2}{8mL^2} e^{-i\omega t} \, \sin \, \dfrac{\pi x}{L}. \end{align}$$


$$\begin{align*} \Longrightarrow \langle K \rangle &= \int_0^L \left( C\ e^{+i\omega t} 
\sin \frac{\pi x}{L}\right) \left(\dfrac{C\ \hbar^2}{8mL^2} e^{-i\omega t} 
\sin \frac{\pi x}{L}\right) \ dx \\[4pt] &= \dfrac{C^2\ \hbar^2}{8mL^2} 
\int_0^L  \sin^2 \, \dfrac{\pi x}{L}\ dx = \dfrac{C^2\ \hbar^2}{8mL^2} 
\dfrac{L}{2} = \dfrac{\hbar^2}{8mL^2}. \end{align*}$$

*Summarizing*:

 - The *average position* of a **large number of repeated measurements** of particles in this state is $\frac{L}{2}$.
 - Even though all the particles are constantly moving (not at rest) since their individual kinetic energy is not zero, the *average momentum* of all these particles in aggregate is trivial because the particles are equally likely to be moving to the right as they are to be moving to the left,
 which cancels their *aggregate momenta.*  
 - The *average kinetic energy* is *inversely proportional* to the particle *mass*, $m$, and the *squared length* of the tube, $\langle K \rangle \propto \dfrac{1}{m\ L^2}$.
 - The *probability density* function quantifying the particle's localization, $|\psi|^2 = \frac{2}{L} \, \sin^2 \left (\pi \frac{x}{L}\right )$ is *maximized* at location $\frac{L}{2}$ and *minimized* at the ends, $x=0$ and $x=L$.

For this specific wavefunction, these expected values are *time-independent*. However, 
in general, time dynamics can be encoded in the wavefunction, which may result in 
*time-dependent expectation estimates.* 


```{r TBD, echo=F, eval=F, warning=FALSE, error=FALSE, message=FALSE, fig.width=9, fig.height=9}
library(plotly)
x <- seq(from=-4, to=4, length.out=2000)
indexN <- seq(1:10)
y <- list()

p <- plot_ly(type = "scatter", mode="lines")
for (i in indexN) {
  y[[i]] <- dnorm(x, mean=0, sd=1/indexN[i])
  p <- p %>% add_trace(x=x, y=y[[i]], 
                       hovertemplate = paste0("Distribution=N(mu=0,sd=1/", indexN[i], ")\n", "(%{x}, %{y})"),
                       name=paste0("N(mu=0,sd=1/", indexN[i], ")"))
}
p <- p %>% 
  layout(title="Gaussian Test Functions Approximating Dirac Delta",
     xaxis=list(title="Critical Values"), yaxis=list(title="Densities"))
p
```

# Time Evolution

Some quantum mechanics tasks require prediction of properties of atoms and
nuclei in equilibrium states, i.e., *time-independent quantum-mechanical systems*.
However, certain QM models describe temporally-dynamic processes, such as 
*radioactive decay*, *scattering*, and *nuclear reactions*, where the 
quantum mechanical systems evolve over time, i.e., 
*time-dependent quantum-mechanical systems*.

## Time-dependent Schrodinger Equation

The fourth postulate of QM states:

 - The evolution of a closed system is unitary (reversible). The time evolution
of the quantum system is given by the *time-dependent Schrodinger equation* 

$$i\hbar \frac{\partial |\psi\rangle}{\partial t} = H(t)| \psi\rangle\ ,$$
where $H=H(t)$ is the time-varying energy operator (Hamiltonian of the system) and 
the reduced Planck constant ($\hbar= \frac{h}{2\pi}$) facilitates the
conversion from *energy* to *frequency* units.

The Schrodinger equation describes the time-evolution of a quantum-mechanical system.
However, the axiom that the *evolution of any closed quantum system is unitary*
is more general and suggests that the state of a system at a later time $t$ is 
given by 
$$|\psi(t)\rangle = \underbrace{U(t)}_{unitary\ } \underbrace{| 
\psi(t=0)\rangle}_{initial\\ condition}\ ,$$

where $U(t)$ is a *unitary operator*, i.e., its *adjoint* 
$U^\dagger\equiv \bar{U}^t$, *transpose* of the complex *conjugate* of the operator, 
is identical to its inverse, i.e., $U^\dagger\equiv U^{-1}$ and 
$U\ U^\dagger\equiv \underbrace{\mathbb{I}}_{identity\\ operator}$.

The integral equation $| \psi(t)\rangle = U(t) | \psi(0)\rangle$ relates the 
state at the initial time $t=0$ to the state at time $t$. 

**Example**: A first order linear approximation allows us to represent
a function (e.g., a position) $x(t) = x(0) + vt$, where the speed $v=x'(0)$ is constant. 
Another way to express this local linear model as a solution to the problem of quantifying the 
temporal dynamics of position is via a *differential equation*, 
$x'(t)=\frac{d x(t)}{d t} = v$. 

More generally the [Newton's Second Law of Motion](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion) links *acceleration* to the *force*.

*The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.*

$$\underbrace{a}_{acceleration} = \frac{\overbrace{F}^{force}}{\underbrace{m}_{mass}}\ \ \ 
\Longleftrightarrow m\frac{d^2x(t)}{dt^2}=F\ .$$

In QM, the Schrodinger equation describes the evolution of closed systems. 

$$i\hbar \frac{\partial \psi(x,t)}{\partial t} = H \psi(x,t) ,$$

where $H$ is the system's Hamiltonian. The solution to the Schrodinger PDE
is the wavefunction $\psi(x,t)$ which agrees with the initial condition $\psi(x,0)$.

## Solutions to the Schrodinger Equation

### Time-independent Hamiltonian

The TISE is a (narrow) special case of the real Schrodinger equation
describing a quantum system whose Hamiltonian $H$ has no explicit time dependence.
To find the solutions in this case it makes sense to try and find *stationary states*,
whose time evolution is given by 
$|\psi(t)\rangle = e^{-i\frac{Et}{\hbar}}|\psi_0\rangle$. 
Plugging this model solution in the more general (time-dependent) Schrodinger equation,
yields the *time-independent Schrodinger equation* 
$\hat{H}|\psi(0)\rangle = E|\psi(0)\rangle$, which mathematically is an *eigenvalue problem*.
Solving the TISE for the energy $E$ and $|\psi(0)\rangle$ yields the 
*energies* (eigenvalues, $E$) for the stationary states and the corresponding
eigenfunctions.

The connection between *TISE* (corresponding to a system with time-independent Hamiltonian) and the *TDSE* (where the system Hamiltonian is time-dependent) is
rooted in the fact that finding the stationary states (solutions of the TISE) 
naturally leads to formally solving the full TDSE. Indeed, since $H$ is a [Hermitian operator](https://en.wikipedia.org/wiki/Self-adjoint_operator),
the [spectral theorem](https://en.wikipedia.org/wiki/Spectral_theorem) suggests the existence of an orthonormal basis $|n\rangle$ of
*stationary eigenstates* with corresponding energies $E_n$ (eigenvalues). 
Hence, any state $|\psi(0)\rangle$ can be expanded as a superposition of the eigenstate 
basis 

$$|\psi(0)\rangle = \sum_n a_n |n\rangle \ \ \  \text{with} \quad a_n = \langle n |\psi(0)\rangle\ .$$
Then, a corresponding solution of the *time-dependent Schrodinger equation* is:

$$|\psi(t)\rangle = \sum_n a_ne^{-i\frac{E_n t}{\hbar}}|n\rangle\ .$$

Suppose the Hamiltonian $H$ is *time independent* 

$$\underbrace{H}_{total\\ energy} = \underbrace{\frac{\hat{p}}{2m}}_{kinetic} +
\underbrace{V(x,t)}_{potential} \overbrace{=}^{time\\ independent}
\frac{\hat{p}}{2m} + \underline{V(x)}\ .$$

The solutions of the TISE can be obtained using *separation of variables*.

$$\psi(x,t) = \varphi(x)f(t)\ .$$

Taking the partial derivatives and using the product rule for differentiation
we get

$$\frac{\partial \psi(x,t)}{dt} = \frac{df(t)}{dt}\ \varphi(x)\\
\frac{\partial \psi(x,t)}{dx} = \frac{d\varphi(x)}{dx}\ f(t)\\
\frac{\partial^2 \psi(x,t)}{dx^2} = \frac{d^2 \varphi(x)}{d^2x}\ f(t)\ .$$

Thus, the Schrodinger equation can also be expressed as

$$i\hbar \frac{\partial \psi(x,t)}{\partial t} = H \psi(x,t) ,$$
$$i\hbar \frac{\partial f(t)}{\partial t}\varphi(x) = -\frac{\hbar^2}{2m} 
\frac{d^2\varphi(x)}{dx^2} f(t) +V(x)\varphi(x)f(t)\ .$$

Dividing both sides by $\psi(x,t)=\varphi(x)f(t)$ yields

$$\underbrace{i\hbar \underbrace{\frac{\partial f(t)}{\partial t}\frac{1}{f(t)}}_{solely\ a
\ function\ of\ t} = -\frac{\hbar^2}{2m} \underbrace{\frac{d^2\varphi(x)}{dx^2} \frac{1}{\varphi(x)} +V(x)}_{solely\ a \ function\ of\ x}}_{both\ hand\ sides\ are\ a\ constant,\ \ E}\ .$$

Therefore,

$$i\hbar \frac{\partial f(t)}{\partial t}\frac{1}{f(t)}= 
E\ \Longrightarrow \frac{\partial f(t)}{\partial t}=
\underbrace{-\frac{i}{\hbar}E}_{constant}\  f(t) \ \Longrightarrow 
f(t)=f(0) e^{-i\frac{Et}{\hbar}} \ ,\\ 
-\frac{\hbar^2}{2m} \frac{d^2\varphi(x)}{dx^2} \frac{1}{\varphi(x)} +V(x)=E\\
\Longrightarrow \psi(x,t) \equiv \varphi(x)
\underbrace{f(t)}_{f(0) e^{-i\frac{Et}{\hbar}}}= \varphi(x) \underbrace{f(0)}_{constant} 
e^{-i\frac{Et}{\hbar}}.$$

As the constant $f(0)$ can easily be absorbed in the wavefunction normalization, 
we can express $\psi(x,t) = \varphi(x) e^{-i\frac{E t}{\hbar}}$, up to a constant.

Effectively, the states $\psi(x,t)$ are *not-stationary but are evolving in time.*

However, any measurable quantities, such as the *probability density* 
$|\psi(x,t)|^2$ or the *expectation values of an observable*, 
$\langle A\rangle = \int \psi^*(x,t)A\psi(x,t)\ dx$ remain *time-independent*,
since $\left (e^{-i\frac{Et}{\hbar}}\right )^*e^{-i\frac{Et}{\hbar}}\equiv 1$,
$\forall\ t\in\mathbb{R}^+$. Therefore, all *stationary measurable quantities* 
do not evolve with time

$$\psi^*(x,t)\ \psi(x,t) \equiv \underbrace{\overbrace{e^{-i\frac{E t}{\hbar}}}^{scalar}
e^{+i\frac{E t}{\hbar}}}_{1}
\underbrace{\varphi^*(x) \varphi(x)}_{time\ independent}\ .$$

Of course, the *separable wavefunction solution* built from one energy eigenfunction, 
$\psi (x,t)= \varphi(x)f(t)$, represent only one particular class of solutions
to the Schrodinger equation. There are many other potential solutions involving
more entangled (*non-separable*) spacetime functions whose shapes depend
on the particular form of the *potential energy* $V(x)$. 

To describe more general TISE solutions, we can use a basis given by the 
eigenfunctions of the Hamiltonian, $\{\varphi(x)\}$. The eigenstate of the
TISE Hamiltonian do not evolve. However we can write any wavefunction as superposition
of the $\{\varphi(x)\}$ base eigenstates given by the energy eigenfunctions

$$\psi(x,t) = \sum_k \underbrace{c_k(t)\varphi_k(x)}_{space-time\\ separable}\ .$$
The coefficients $c_k(t)$ can be obtained at any instant $t$ by taking the
*inner product* $c_k(t)=\langle \varphi_k (x) |\psi(x,t)\rangle$. 
Plugging these in the Schrodinger equation 
$i\hbar \frac{\partial \psi(x,t)}{\partial t} = H \psi(x,t)$, we get


$$i\hbar \frac{\partial}{\partial t} \sum_k c_k(t)\varphi_k(x)= 
\sum_k c_k(t) H\varphi_k(x)$$

$$\Longrightarrow i\hbar  \sum_k \frac{\partial}{\partial t}c_k(t)\varphi_k(x)= 
\sum_k c_k(t) E_k\varphi_k(x)\ .$$

For each *base function* $\varphi_k(x)$ we then have a corresponding *coefficient equation* 

$$i\hbar  \frac{\partial}{\partial t}c_k(t)= c_k(t) E_k
\Longrightarrow c_k(t)= c_k(0)e^{-i\frac{E_k t}{\hbar}}\ .$$

Therefore, a general solution of the Schrodinger equation is 

$$\psi(x,t) = \sum_k {c_k(0)e^{-i\frac{E_k t}{\hbar}}\varphi_k(x)}\ .$$

**Remarks**

 - The energy eigenvalues (eigen-frequencies) $E_k=\hbar \omega_k$ indicate that the wavefunction is a superposition of waves $\varphi_k(x)$ each with a different frequency $\omega_k$ propagating in time. The behavior of quantum systems resembles wave propagation. Diffraction patterns formed by electrons scattering from a slit represent one instance of this wave superposition.

 - A special property only for the *energy eigenvalues*, not for other observables, is that the *probability of measuring a certain energy* $E_k$ at time $t$ is given by the squared amplitude of the coefficient of $\varphi_k(x)$, $|c_k(t)|^2=\left |c_k(0)e^{-i\frac{E_k t}{\hbar}}\right |^2=|c_k(0)|^2 .$ Hence, the *probability for each measurable energy is constant*, i.e., the probability of an energy state does not depend on time. In other words, 
 *energy is constant of the motion.* 

 - *Example*: Suppose we are interested in estimating the probability of finding the system at a 
certain position, $p(x) = |\psi(x,t)|^2$, which varies with time. Let $\psi(x,0) = c_1(0)\varphi_1(x) + c_2(0)\varphi_2(x)$, with $|c_1(0)|^2 + |c_2(0)|^2 = |c_1|^2 + |c_2|^2 = 1$, and $\varphi_1,\varphi_2$ normalized energy eigenfunctions. At time $t$, we have $\psi(x,t) = c_1(0)e^{-i\omega_1 t}\varphi_1(x) + c_2(0)e^{-i\omega_2 t}\varphi_2(x)$ and we can compute $p(x,t)$

$$p(x,t)= \left | c_1(0)e^{-i\omega_1 t}\varphi_1(x) + c_2(0)e^{-i\omega_2 t}\varphi_2(x)\right |^2\\
=\cdots = |c_1|^2+|c_2|^2 + \underbrace{2Re\left [c_1^*c_2\ \varphi_1^*\varphi_2 e^{-i(\omega_2-\omega_1)t}\right ]}_{wave\ interference}\ .$$

The *wave interference* term, $2Re\left [c_1^*c_2\ \varphi_1^*\varphi_2 e^{-i(\omega_2-\omega_1)t}\right ]$, describes interactions between different 
components of the initial wavefunction.

 - These expressions for the *time-dependent wavefunction* are only valid if the *potential energy is itself time-independent*. Otherwise, deriving closed-form solutions of the TISE are more difficult to obtain.

## Unitary Evolution

### TDSE - Time-Dependent Schrodinger Equation

Time-dependent Schrodinger equation describes the evolution of a closed system, 
which is *unitary* (reversible) and the evolution is given by the equation

$$i\hbar \frac{\partial |\psi(x,t)\rangle}{dt} = H|\psi(x,t)\rangle\ ,$$
where $H$ is the Hamiltonian (energy operator) of the system represents an observable, and $|\psi(x,t)\rangle$ is the *state vector* of the quantum system. 
The position-space wavefunction $\psi (x,t)$ can be written as the inner product
of a *time-dependent state vector* $|\psi (x, t) \rangle$ with convenient 
*position eigenstates* $| x\rangle$

$$\psi (x,t)=\underbrace{\langle x|}_{position\\ eigenstate}
\underbrace{\psi (t)\rangle}_{\ \ time-dependent\\ \ \ state\ vector}\ .$$

Again, the time-dependent Schrodinger equation describes the longitudinal evolution
of a quantum state.

## Propagators

The *time-independent Schrodinger equation* is used to find the *stationary states* 
of a quantum system, which are unchanged over time. Stationary state solutions
are associated with specific *energy levels* of the system. 

We can express the evolution of a state $|\psi(x,t)\rangle$ in terms of a 
*unitary propagator operator*

$$\psi(x,t) = U(t)\psi(x,0)\ ,$$
where the operator $U$ is unitary, $U^\dagger U = \mathbb{I}$. In general,
the unitary operator $U=U(x,t)$ could also be a function of space-time (not just time).
Let's confirm that this state $\psi(x,t) = U(t)\psi(x,0)$ is a solution of 
the to the Schrodinger equation 
$i\hbar \frac{\partial |\psi(x,t)\rangle}{dt} = H|\psi(x,t)\rangle$ by plugging in
$\psi(x,t) = U(t)\psi(x,0)$

$$i\hbar \frac{\partial}{dt}(U(t))\underbrace{\psi(x,0)}_{} = HU(t)
\underbrace{\psi(x,0)}_{}\ \ \Longrightarrow 
i\hbar \frac{\partial U(t)}{dt}= HU(t)\ .$$

The second step implication is valid since the equation holds
for $\forall \psi$ wavefunctions, and therefore, it holds for the unitary
operator $U$. When the Hamiltonian is *time independent*, the solution of
the second TISE equation is

$$i\hbar \frac{\partial U(t)}{dt}= HU(t) \ \Longrightarrow
U(t)=e^{-i\frac{Ht}{\hbar}}\ ,$$

where $U(t=0)=\mathbb{I}$ and $U$ is unitary, since
$U^\dagger(t) U(t)=e^{i\frac{Ht}{\hbar}}\ e^{-i\frac{Ht}{\hbar}}=\mathbb{I}$.

## Schrodinger vs. Heisenberg Picture

The [synergies between Schrodinger and Heisenberg pictures of quantum field theory (QFT) are clearly explicated in this article on operator-valued distributions in QFT](https://doi.org/10.48550/arXiv.1508.05908). 

The Schrodinger and Heisenberg pictures only differ by a Hilbert space basis change 
with respect to time-dependency. The Heisenberg picture utilizes an algebraic
matrix  mechanics formulation in an arbitrary basis where the Hamiltonian is not necessarily diagonal. The Schrodinger picture is useful with time-independent Hamiltonians where
$\partial_{t} H=0$ and the state of a system evolves with time, 
$|\psi(t_0)\rangle\to |\psi(t)\rangle$ by the *time-evolution operator* 
$|\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle$. 

The [Stone–von Neumann theorem](https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem) implies the *uniqueness of the canonical commutation relations between position and momentum operators*. That is, the theorem suggests *unitary equivalence* between the Heisenberg and 
the Schrodinger pictures, i.e., the two formulations are related by a basis change
in the Hilbert space.

### Schrodinger Picture

In the [Schrodinger picture](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture)
formulation of quantum mechanics, *operators are constant*
and the *states evolve in time.* At time $t$, the Schrodinger equation described a quantum state
by $|\psi(t)\rangle =U(t)|\psi(0)\rangle$, where 

 - the time-evolution operator, $U(t)=T\ e^{-{\frac{i}{\hbar}} \int_{0}^t dsH_{S}(s)}$, is induced by a (potentially) time-dependent Hamiltonian $H_{S}(t)$, 
 - the initial state of the system is $|\psi (0)\rangle$, and
 - $T$ and $\hbar$ are the time-ordering and the reduced Planck constant.

In the Schrodinger picture, the *expectation value* of any observable, $A_{S}(t)$,
in the state $|\psi (t)\rangle$ is a Hermitian linear operator, $\langle A\rangle _{t}$, 
also potentially time-dependent

$$\langle A\rangle _{t}=\langle \psi (t)|A_{{S}}(t)|\psi (t)\rangle .$$

### Heisenberg Picture

The [Heisenberg picture](https://en.wikipedia.org/wiki/Heisenberg_picture) of QM represents
*operators and observables as time-dependent*, whereas the *state vectors are time-independent*.

In the Heisenberg picture, the observable operators evolve with time 
$A_{H}(t) = U^{\dagger }(t)A_{S}(t)U(t)$, whereas quantum states remain constant 
with time. Hence, the expectation value can be computed in either picture
$\langle A\rangle_{t}=\langle \psi (0)|A_{H}(t)|\psi (0)\rangle$. 

The time-dependent Schrodinger equation (TDSE) 
$\frac{d}{dt}U(t)=-{\frac {i}{\hbar }}H_{S}(t)U(t)$ and using the product rule 
for differentiation we get the *Heisenberg's equation of motion*

$$\begin{aligned}{\frac{d}{dt}}A_{H}(t)&=\left({\frac {d}{dt}}U^{\dagger }(t)
\right)A_{S}(t)U(t)+U^{\dagger }(t)A_{S}(t)\left(\frac{d}{dt}U(t)\right)+U^{\dagger }(t)
\left({\frac {\partial A_{S}}{\partial t}}\right)U(t)\\&=
{\frac {i}{\hbar }}U^{\dagger }(t)H_{S}(t)A_{S}(t)U(t)-\frac{i}{\hbar}U^{\dagger }(t)
A_{S}(t)H_{S}(t)U(t)+U^{\dagger }(t)\left({\frac {\partial A_{S}}
{\partial t}}\right)U(t)\\&={\frac {i}{\hbar }}U^{\dagger }(t)H_{S}(t)U(t)U^{\dagger }(t)
A_{S}(t)U(t)-\frac {i}{\hbar}U^{\dagger }(t)A_{S}(t)U(t)U^{\dagger }(t)
H_{S}(t)U(t)+\left(\frac {\partial A_{S}}{\partial t}\right)_{H}\\&=
\frac {i}{\hbar}[H_{H}(t),A_{H}(t)]+\left(\frac {\partial A_{S}}{\partial t}
\right)_{H}\end{aligned}\ .$$

Note the synergies between the *Heisenberg* and the *Schrodinger Hamiltonians*
$H_{S}(t)$ and $H_{H}(t)$. 

### Heisenberg Equation

Examine the *time-derivative of expectation* ...

The [Heisenberg equation](https://en.wikipedia.org/wiki/Heisenberg_picture), 
describes the time evolution of operators and their relation to the 
time-derivative of expectation values. In the Heisenberg picture, operators evolve 
with time while states remain constant. 

When the Hamiltonian $H_{S}$ is time-independent, the time-evolution operator 
$U(t)=e^{-\frac {i}{\hbar }\ t\ H_{S}}$ yields the equivalence of the
Schrodinger and the Heisenberg Hamiltonians, $H_{H}\equiv H_{S}\equiv H$, since 
in this case, $U( t )$ commutes with $H$. Thus,

$$\langle A\rangle _{t}=\langle \psi (0)|e^{{\frac {i}{\hbar }}tH}A_{S}(t)
e^{-{\frac {i}{\hbar }}tH}|\psi (0)\rangle $$
and 

$$\begin{aligned}{\frac {d}{dt}}A_{H}(t) &= \frac {i}{\hbar }
[H,A_{H}(t)] + 
\underbrace{e^{\frac {i}{\hbar }tH}\left(
{\frac {\partial A_{S}}{\partial t}}\right)e^{-{\frac {i}{\hbar }}tH}}_
{trivial,\ 0}\end{aligned}\ ,$$

since $A_{S}\equiv A$ is also time-independent. So, we get
$$\frac{d}{dt}A_{H}(t) = \frac{i}{\hbar } [H,A_{H}(t)]\ ,$$

where $A_{H}(t)\equiv A(t)=e^{\frac{i}{\hbar }\ t\ H}\ A\ e^{-\frac {i}{\hbar}tH}$. 


# Wavefunctions, Kime-Functions, and Inference Functions

Recall from the [TCIU distributions section](https://www.socr.umich.edu/TCIU/HTMLs/Chapter3_DistributionTheory.html) that *test functions* $\phi(\theta)$ are *infinitely differentiable functions*
that either have *compact support* or their *tails are rapidly decreasing* (super-exponentially). We will extend the notion of *actions* of generalized functions $\psi$ on test functions. This action is defined via *duality pairing*, 
$\langle\psi, \phi\rangle =\int_{\mathbb{R}}{\psi^*(\theta)\phi(\theta)}\ d\theta$.

The extension to *actions of separable kime-functions* 
$$\mathfrak{P}(x,y,z,t,\ \theta)\equiv \underbrace{\Psi(x,y,z,t)}_{classical\ wavefunction}\times \underbrace{\ell(\theta)}_{kime-phase\\ distribution}:\overbrace{\mathbb{R}^3}^{{\bf{x}}=(x,y,z)}\times
\overbrace{\mathbb{C}}^{\kappa=t e^{i\theta}} \longrightarrow \mathbb{C}$$

on *test functions* $\phi$ is defined by

$$\underbrace{\langle\mathfrak{P}, \phi\rangle}_{kime-function\\ action} = 
\overbrace{\Psi(x,y,z,t)}^{spacetime\\ wavefunction}
\underbrace{\int_{\mathbb{R}}{\ell^*(\theta) \phi(\theta)}\ 
d\theta}_{\underbrace{\langle\ell, \phi\rangle}_{kime-phase\\ action}\in\mathbb{C}}\ .$$

The integral of the product of the *kime-phase distribution* $\ell$ and the test function $\phi$ represents the action of the kime-phase generalized function on the test function. This integral is an *aggregation function* weight-averaging all values of the test function against the corresponding kime-phase probability density function.

**Note**: Whereas $\theta\in \mathbb{R}$ and $\ell$ is a univariate
*kime-phase distribution*, its values are complex, as $\ell$ is a function of $e^{i\theta}=\cos(\theta)+ i \sin(\theta)$.

This suggests an *explicit relation between wavefunctions and kime-functions*, 
where actions of kime-functions on test-functions are just complex multiples of the wavefunctions. Since, wavefunctions are typically normalized to induce probabilistic interpretation of their magnitudes, this suggest the equivalence

$$\underbrace{\langle\mathfrak{P}, \phi\rangle}_{kime-function\ action\\ on\ test-function} = 
\underbrace{c\  \Psi(x,y,z,t)}_{scaled\ wavefunction}\ ,\ c \in\mathbb{C}\ .$$

Extending the classical notion of [generalized functions (distributions)](https://www.socr.umich.edu/TCIU/HTMLs/Chapter3_DistributionTheory.html), 
*kime-functions* can be considered as *generalized wavefunctions*, $\Psi$, 
which describe the state of a QM system *by their action* on test-functions 
$\varphi$ (outputting complex values) 

$$\left \langle \overbrace{\mathfrak{P}}^{kime-function} \bigg\rvert
\underbrace{\varphi}_{test-function} \right \rangle 
\underbrace {\left (\overbrace{x,y,z}^{\bf{x}\in\mathbb{R}^3},
\overbrace{t,\theta}^{\kappa\in\mathbb{C}}
\right )}_{spacekime}: \mathbb{R}^3\times \mathbb{C} \longrightarrow \mathbb{C}\ .$$

Since *kime-functions* are [linear functionals acting on test-functions](https://www.socr.umich.edu/TCIU/HTMLs/Chapter3_DistributionTheory.html),
they are elements of the *dual space* of the space of test functions, $D(\Omega)$. 

 - For any test functions $\{\varphi_1, \varphi_2\} \in D(\Omega)$ and 
$\forall \alpha , \beta\in\mathbb{C}$, the mapping $T: D(\Omega)\to \mathbb{C}$ should be *linear*

$$T_{\mathfrak{P}}(a\varphi_1 + b\varphi_2 )\equiv \langle \mathfrak{P}, a\varphi_1 +
b\varphi_2 \rangle = \alpha\langle \mathfrak{P}, \varphi_1\rangle + \beta\langle
\mathfrak{P},\varphi_2\rangle\ .$$

 - The mapping $T_{\mathfrak{P}}$, should also be *continuous*. For any test function $\varphi \in D(\Omega)$ and a corresponding sequence of test functions $\{ \varphi_{n} \} \in D(\Omega)$ with 
 
$$\varphi_n\underset{n\to\infty}{\stackrel{D}\longrightarrow}\varphi \ \ 
\Longrightarrow T_{\mathfrak{P}}(\varphi_n)\underset{n\to\infty}{\longrightarrow} T_{\mathfrak{P}}(\varphi)\ .$$

Denote $T_{\mathfrak{P}} = \langle \mathfrak{P}, \cdot \rangle$, which is a linear and continuous mapping from $D$ to the real or complex numbers. 

The following argument *does not really make sense*, as it does not account for kime, 
$\kappa=te^{i\theta}$,

$$\underbrace{\mathfrak{P}}_{kime-function} (x,y,z, t,\theta): 
\underbrace{\alpha\phi+ \beta\psi}_{superposition\ of\\ test\ functions} 
\longrightarrow 
\alpha \langle \mathfrak{P}({\bf{x}})\ ,\ \phi ({\bf{x}}) \rangle +
\beta \langle \mathfrak{P}({\bf{x}})\ ,\ \psi ({\bf{x}}) \rangle\\ =
\underbrace{\alpha \int_{\mathbb{R}^3} {\mathfrak{P}({\bf{x}})\ \phi ({\bf{x}})\ d{\bf{x}}}+
\beta \int_{\mathbb{R}^3} {\mathfrak{P}({\bf{x}})\ \psi 
({\bf{x}})\ d{\bf{x}}}}_{\in\ \mathbb{C}}\ ,\ \forall\ \alpha,\beta\in \mathbb{C}\ .$$

**Definition**: *Inference-functions*, or *kime-inference-functions*, 
denoted by Fraktur-format Latin letters
$$\mathfrak{A,B,C,D,E,F,G,\cdots, P, Q, R, S, T, U, V, W, X, Y,Z}$$
are a special kind of *kime-functions*, which are separable as products of 
*wavefunctions* over spacetime, denoted by *capital* Greek symbols, 
$\{\Psi, \Phi, \Xi, \cdots\}$, and 
*generalized-functions (kime-phase distributions)*, $\ell(e^{i\theta})$. 

Indeed, kime *inference-functions* are in the dual-space of the Hilbert space of
test functions, i.e., kime *inference-functions* act on test functions as follows:

$$\underbrace{\mathfrak{P}}_{inference-function\\ (Fraktur-P)} \left (\underbrace{x,y,z}_{{\bf{x}},\ space}, 
\underbrace{t,\cdot}_{kime,\ \kappa=te^{i\theta}}\right )\underbrace{\equiv}_{separability}
\overbrace{\Psi({\bf{x}},t)}^{wavefunction}\underbrace{\ell(\cdot)}_{phase\\ distrib.} 
\ \ {\text{  acting on  }}\ \ 
\underbrace{\alpha\phi+ \beta\psi}_{superposition\ of\\ test\ functions}, \\ 
\\ 
\Longrightarrow \mathfrak{P}({\bf{x}},\kappa)\ (\alpha\phi+ \beta\psi) \equiv \Psi({\bf{x}},t) \left [\ell(\cdot)\ (\alpha\phi+ \beta\psi)\right ]\ =\  
\underbrace{\alpha\Psi({\bf{x}},t)}_{scalar\ \in\ \mathbb{C}} 
\left\langle \ell(\cdot)\ ,\ \phi  \right\rangle +
\underbrace{\beta\Psi({\bf{x}},t)}_{scalar\ \in\ \mathbb{C}}  
\left\langle \ell(\cdot)\ ,\ \psi  \right\rangle\\ =
\underbrace{\alpha\Psi({\bf{x}},t) \int_{\mathbb{R}} {\ell^*(e^{i\theta})\ \phi (\theta)\ d\theta}+
\beta \Psi({\bf{x}},t) \int_{\mathbb{R}} {\ell^*(e^{i\theta})\ \psi (\theta)\ d\theta}}_{\in
\ \mathbb{C}}\\
= \left (\underbrace{\alpha \int_{\mathbb{R}} {\ell^*(e^{i\theta})\ \phi (\theta)\ d\theta}}_{\in
\ \mathbb{C}} +
\underbrace{\beta \int_{\mathbb{R}} {\ell^*(e^{i\theta})\ \psi (\theta)\ d\theta}}_{\in
\ \mathbb{C}}\right ) \ \Psi({\bf{x}},t) \\
= \left (\alpha \left\langle \ell(\cdot) \ ,\ \phi  \right\rangle +
\beta \left\langle \ell(\cdot) \ ,\ \psi  \right\rangle
\right )\ \Psi({\bf{x}},t) \\
= \left\langle \ell(\cdot) \ ,\ \alpha \psi +\beta\psi \right\rangle
\ \Psi({\bf{x}},t) =\Psi({\bf{x}},t) \left\langle \ell(\cdot) \ ,\ \alpha \psi +
\beta\psi \right\rangle,\ \forall\ \alpha,\beta\in \mathbb{C}\ .$$

To simplify the notation, we can suppress the subscript of the wavefunction $\Psi$, 
since technically speaking the *kime inference-function* and the corresponding
*wavefunction* are linked via the *phase-distribution*, $\ell(\cdot)$,

$$\mathfrak{P}\underbrace{(\cdot,\cdot)}_{spacekime}= \Psi\underbrace{(\cdot,\cdot)}_{spacetime}
\ell\underbrace{(\cdot)}_{kime-phase}\ , \ \ 
\Psi({\bf{x}},t)\equiv \Psi_{\mathfrak{P}}({\bf{x}},t)\ .$$

Recall from the [TCIU Distribution Theory Section](https://www.socr.umich.edu/TCIU/HTMLs/Chapter3_DistributionTheory.html) that 
the *class of test functions* operated on by distributions can vary. Examples include:

 - A class of test functions representing the *space of infinitely differentiable functions with compact support*, $\mathcal{C}_c^{\infty}(\mathbb{R})$. 
 - In Fourier analysis, the set of *Schwartz functions* 

$$\mathcal{S}(\mathbb{R^n})=\{f\in C^{\infty}(\mathbb{R^n}):
||f||_{\alpha,\beta}<\infty \,, \forall \alpha, \beta\} \tag{1}$$

&nbsp; represents another useful class of natural test functions consisting of all *infinitely differentiable functions of rapid decay*, i.e., functions $\phi$ such that $x^n \phi(x)\underset{x\to\pm\infty}{\longrightarrow} 0,\ \forall\ n\in\mathbb{N}$. This rapid decay conditions guarantees that the Fourier transforms of Schwartz functions are also Schwartz functions. 

## Examples

### Example 1: Kime-Phase Distribution Multiplication by a Function

Consider the *product* of a generalized function $\ell(e^{i\theta})$ 
(kime-phase distribution) and a given function $f(\cdot)$. This product $f\ell$ is
defined in terms of its action on any test function $\phi\in\mathcal{D}(\Omega)$: 

$$\left \langle \underbrace{f\ \ell}_{product}(e^{i\theta}),\ \phi\right \rangle \equiv \langle \ell(e^{i\theta}),f\ \phi\rangle,
\ \ \ \forall \phi\in\mathcal{D}(\Omega)\Longrightarrow 
\underbrace{f\ \phi}_{product}\in\mathcal{D}(\Omega).$$

This implies that the action of the (product) *distribution* $f\ \ell(e^{i\theta})$ on a test function $\phi$ is identical to the action of the distribution $\ell(e^{i\theta})$ on 
(the modified product) test function $f\ \phi$. For instance, suppose
$f(x)$ is a *smooth function* and the phase distribution is just the Dirac delta
$\ell(e^{i\theta}):=\delta(\theta)$. Then, the product 
$f\ \ell\equiv f(\theta)\delta(\theta)$ acts on any test function $\phi\in\mathcal{D}(\Omega)$ 
by evaluating the product of $f$ and $\phi$ at the origin ($\theta=0$)

$$\langle f(\theta)\ \ell(e^{i\theta}),\ \phi(\theta) \rangle =
\langle \delta(\theta),\ f(\theta)\ \phi(\theta)\rangle = f(0)\ \phi(0).$$

### Example 2: Principal Value Distribution

A more nuanced example of a *generalized phase function* is the 
[Cauchy principal value distribution](https://en.wikipedia.org/wiki/Cauchy_principal_value),
$\ell(\theta)\equiv\mathcal{P}\left (\frac{1}{\theta}\right )$, defined as the Cauchy principal 
value of the function $\frac{1}{\theta}$, which is not Lebesgue/Riemann integrable over
$\theta\in\mathbb{R}$. The *Cauchy principal value phase-distribution* acts on
a test function $\phi(\theta)$ by

$$\langle \ell(e^{i\theta}),\ \phi(\theta) \rangle =
\left \langle \mathcal{P}\left (\frac{1}{\theta}\right ),\ \phi(\theta)
\right \rangle = \lim_{\epsilon\to 0}{\int_{|\theta|\gt\epsilon}{\frac{\phi(\theta)}{\theta}\ 
\ d\theta}}\ .$$

One application of the Cauchy principal value phase-distribution is to compute or estimate
definite integrals that would otherwise not be well-defined, e.g., in this case
due to a singularity at the origin $\theta=0$. Recall the
[L'Hopital's rule](https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule),
which allows us to estimate the value of the integral near the origin, 
$$\int_{|\theta|\lt\epsilon}{\frac{\phi(\theta)}{\theta}\ 
\ d\theta},\ \ \ \lim_{\epsilon\to 0}{\frac{\phi(\theta)}{\theta}}
\underbrace{=}_{\text{L'Hopital}} \lim_{\epsilon\to 0}{\frac{\phi'(\theta)}{\theta'}}=
\lim_{\epsilon\to 0}{\frac{\phi'(\theta)}{1}}=\lim_{\epsilon\to 0}\phi'(\theta)\ .$$

### Example 3: Sign Function

The *sign generalized function (distribution)* 
$$sgn(\theta)\equiv \begin{cases} 
      -1 &,\ \theta\leq 0 \\
      0 &,\ \theta=0 \\
      +1 &,\ \theta\ge 0 
   \end{cases}$$

acts on test functions $\phi(\theta)$ by weight-averaging (or convolving) the 
test function by the sign function

$$\langle \ell(e^{i\theta}),\ \phi(\theta) \rangle =
\left \langle sgn (\theta),\ \phi(\theta) \right \rangle =
\int_{\mathbb{R}}{sgn(\theta)\ \phi(\theta) \ d\theta}\\
= \int_{0}^{\infty}{\phi(\theta) \ d\theta}-\int_{-\infty}^{0}{\phi(\theta) \ d\theta} \ .$$

Clearly, for *symmetric* test functions, $\phi(-\theta)=\phi(\theta)$, this action is trivial, 
$\langle \ell,\ \phi \rangle =0$.


### Example 4: Laplace Distribution

Let's consider the action of the [Laplace distribution, aka, double exponential distribution](https://en.wikipedia.org/wiki/Laplace_distribution) with 
location ($\mu\in\mathbb{R}$) and scale $b\in\mathbb{R}^+$ parameters

$$\ell_{\mathcal{L}}(\theta \mid \mu, b) = \frac{1}{2b} e^{\left( -\frac{|\theta - \mu|}{b} \right)}$$

on test functions $\phi$:

$$\langle \ell_{\mathcal{L}},\ \phi \rangle =
\frac{1}{2b} \int_{\mathbb{R}}{e^{-\frac{|\theta - \mu|}{b}}\ \phi(\theta) \ d\theta} \ .$$

**Problem**: Work out the details for some simple test-functions ....


## Connection to Quantum Field Theory (QFT) and Operator-Valued Distributions

Recall that *probability distributions* are mathematical abstractions (models) that
facilitate *generative (statistical) inference* and *random drawing* (sampling) of measurable observations from various processes. A probability distribution 
themselves cannot be (holistically) observed, but samples from each distribution 
are observable. 

Similarly, *quantum fields* are mathematical abstractions (models) providing 
mechanisms for *organizing*, *tracking the evolution*, and explicating 
*generators of the algebra of observables*. Quantum fields are 
*not themselves directly observable* - the corresponding 
*actual observables* are their actions on *test functions*, e.g.,
$\varphi(f)\in\mathbb{C}$, rather than $\varphi$.

Suppose $\mathcal{H}$ is a [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space), a Euclidean vector space over $\mathbb{R}^3$ representing the states of a quantum particle. The particle 
spatiotemporal dynamics are in $\mathbb{R}^3\times\mathbb{R}^+$ and 
$\mathcal{H}=L_2(\mathbb{R}^3)$. For quantum systems with multiple identical particles, this extends to [Fock spaces](https://en.wikipedia.org/wiki/Fock_space). 
Often for simplification, we consider single-particle systems with 

$$\mathcal{H}=L_2(\mathcal{X}=\mathbb{R}^3, measure=\mu=d^{3}{\bf{x}}))$$ 

representing the space of *square-integrable functions*. Of course, this is a
measure-theoretic representation where objects in $\mathcal{H}$ are actually
*equivalence classes* of square integrable functions, not functions themselves. 
Recall that many different functions in $L_2$ are *equivalent*, e.g., when they 
differ on a set of measure zero. The more general
Fock spaces ($n$ particles) include symmetric or antisymmetric square integrable functions, commonly denoted by $Sym(\mathcal{H})$, is a quotient space of 
equivalence classes of functions in the tensor algebra 
$\bigoplus_{n \in \mathbb{N}} \mathcal{H}^{\otimes n}.$

For *single particle* systems, states are unit vectors $\psi\in \mathcal{H}$,
*observables* are *self-adjoint bounded operators* $T$ on $\mathcal{H}$
whose expectation values in the state $\psi$ are equal to 
$\langle T\rangle\equiv \langle \psi|T|\psi\rangle\in \mathbb{C}.$
More specifically, *pure states* are points in the [projective space](https://en.wikipedia.org/wiki/Projective_space) 
$\mathbb{P}\mathcal{H}$ representing *hyper-lines* in $\mathcal{H}$ as equivalence classes of unit vectors modulo a phase factor (random kime-phase). 
Consider a $C^*$ algebra $\mathcal{A}$ of the bounded operators on $\mathcal{H}$.
Then, states are linear functionals $\langle\cdot\rangle:\mathcal{A}\to\mathbb{C}$
satisfying two conditions: $\langle \mathbb{I} \rangle=\mathbb{I}$ and 
$\langle AA^* \rangle\in\mathbb{R},\ \forall A\in \mathcal{A}$. 
This definition ensures that a linear superposition of states is another state
in the Hilbert space and all pure states are of the form 
$\langle \psi|\cdot| \psi\rangle .$

For systems with *multiple particles*, *states* are elements of $Sym(\mathcal{H})$
and *observables* are self-adjoint (Hermitian) bounded operators on the
Hilbert space $Sym(\mathcal{H})$ of the states. Given a single particle state
$\psi$, an example of a two-particle state is the element $\frac{1}{\sqrt{2}}\psi\otimes \psi \in Sym(\mathcal{H})$.

Whereas in *quantum mechanics* we compute using unit vectors and self-adjoint operators,
in *quantum field theory* the calculations are done on *operator-valued distributions*,
i.e., the Hilbert space $Sym(\mathcal{H})$ of the multi-vector states. The 
operator-valued distribution's action of a quantum field $\Phi$ on a
test function $f({\bf{x}}),\ {\bf{x}}\in\mathbb{R}^3$ is a Hermitian
operator $\Phi$ on $Sym(\mathcal{H}).$ Thus, the quantum fields are *operator-valued distributions*, not just states on $Sym(\mathcal{H}).$

The *Hilbert space itself a mathematical abstraction*, not a physical object. On
the other hand, the *algebra of observables is physical* as all real-measurements
are either (atomic) pure states, or superpositions of pure states and all
*AI/ML/statistical inference* and *predictions* are based on such physical 
measurements (data-driven inference). The Hilbert spaces are the abstract 
mathematical models of more tangible physical systems. 
Also, the Hilbert space models are unique up to isomorphisms 
reflecting kime-phase uncertainty, e.g., reflecting quantum fluctuation effects
on measurable observations.

In QFT, a multi-particle Hilbert space is a Fock space, 
$\mathcal{H}_{{many}\choose{particles}} = Sym\left(\mathcal{H}_{{single}\choose{particles}}\right),$ where the natural physical class of Hermitian operators
preserve some *locality.* For instance, such operators shrink or expand
the support by $\varepsilon$ or temper the tails of the test functions at
infinity (e.g., ensure super-exponential decay). These 
$\mathcal{H}_{{many}\choose{particles}}$ operators are defined at any spacetime location 
resembling the definition of classical quantum mechanics operators that can
be measured at any spacetime location.

In a [Gedankenexperiment](https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments), assume we can track/observe a system state
characteristic at any spacetime location. Operators evaluate the 
values of the measurable characteristics of the quantum field. Consider
the full Hilbert space example with a pair of *creating* and *annihilating* 
quantum field operators. *Functional operators* on $\mathcal{H}_{{many}\choose{particles}}$ 
transform given spacetime locations, $\mathbb{R}^3$, into classical
quantum mechanical operators, i.e., bounded Hermitian operators on 
$\mathcal{H}_{{many}\choose{particles}}$ complex-value (scalar) outputs. 

We have to utilize *distributions*, rather than *functions*, to ensure that
uncertainty and commutator relations can be expressed in terms of *Dirac* $\delta$ *distribution*, i.e., in $\mathbb{R}^3$, $\delta(\cdot)=\delta^3(\cdot).$ 
For instance, if $p,q\in\mathbb{R}^3$ are two canonically conjugate quantum fields, 
the corresponding *commutator*, $[\cdot,\cdot]$, of the *creation* $a$ and 
*annihilation* $a^{\dagger}$ operators is

$$\underbrace{[a(p), a^{\dagger}(q)]}_{commutator}=
\underbrace{(2\pi)^3\delta^3(p - q)\ \mathbb{I}}_{distribution}\ ,$$
and

$$[a(p), a(q)] = 0\ \ , \ \  [a^{\dagger}(p), a^{\dagger}(q)] = 0\ ,$$
where for a pair of operators $A,B\in\mathcal{H}_{{many}\choose{particles}}$, $[A,B]:=AB-BA$, and $\mathbb{I}$ is the *identity operator* on the Hilbert space
$\mathcal{H}_{{many}\choose{particles}}$.

This implies that the fields $\{p,q\}$ must also be *distributions.* Similarly 
for spacetime fields, representing Fourier transformations
of the *creation* and *annihilation* operators, whose time-ordered 
[Green's function, i.e., correlation function, or correlator](https://en.wikipedia.org/wiki/Correlation_function_(quantum_field_theory)),
will also be a Dirac $\delta$ distribution. Therefore, the spacetime fields 
are *operator-valued distributions*, rather than functions.

[An introduction to operator-valued distributions and the relevant Hilbert spaces is provided in this QFT book](https://souravchatterjee.su.domains/qft-lectures-combined.pdf).

## Kime Unitary Operators

**Definition 1**: A *unitary linear operator* $U :\mathcal{H}\to\mathcal{H}$ has the property
$||U\psi|| = ||\psi||,\ \forall\ \psi\in\mathcal{H}$.

**Definition 2**: A *strongly continuous unitary group* $\{U_t\}_{t\in\mathbb{R}}$
is a collection of unitary operators such that

 - $U_{t_1+t_2} = U_{t_1}\ U_{t_2},\ \forall\ t_1,t_2\in\mathbb{R}$, and
 - $\forall\ \psi \in\mathcal{H}$, the $\mathbb{R}\to\mathbb{C}$ mapping $t \mapsto U_t(\psi)$ is continuous.

**Definition 3**: A *kime continuous unitary group* 
$\{U_{\kappa}\}_{\kappa\in\mathbb{C}}$ is a collection of unitary operators such that

 - $U_{\kappa_1 \cdot \kappa_2} = U_{\kappa_1}\cdot U_{\kappa_2},\ \forall\ \kappa_1,\kappa_2\in\mathbb{C}$, and
 - $\forall\ \psi \in\mathcal{H}$, the $\mathbb{C}\to\mathbb{C}$ mapping $\kappa \mapsto U_{\kappa}(\psi)$ is continuous.
 

### Stone - von Neumann Theorem

The [Stone - von Neumann theorem](https://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups) establishes a one-to-one correspondence between 
*self-adjoint operators on a Hilbert space* $\mathcal {H}$ and 
*one-parameter families of continuous unitary operators* $\{U_{t}\}_{t\in \mathbb {R}}$:

$$(1)\ \ \forall t_{o}\in \mathbb {R} ,\ \psi \in {\mathcal {H}}:\ \lim _{t\to t_o}U_{t}(\psi )=U_{t_o}(\psi),$$
$$(2)\ \ \forall s,t\in \mathbb {R} :\ U_{t+s}=U_{t}U_{s}\ .$$

A [historical perspective of the Stone-von Neumann theorem is presented in this article, *A Selective History of the Stone-von Neumann Theorem*](https://www.math.umd.edu/~jmr/StoneVNart.pdf).

This theorem supports a constructive definition of the *derivative of continuous unitary operators* 
$t\mapsto U_t$, see the $5$th QM postulate below.

The one-to-one correspondence between *one parameter strongly continuous unitary groups of operators*
$U^A :\mathcal{H}\to\mathcal{H}$ and *self-adjoint operators* 
$A^U :\mathcal{H}\to\mathcal{H}$ is given by

$$A^U\ \psi = \lim_{t\to 0}{\frac{U^A_t(\psi) - \psi}{it}}\\
U^A_t=e^{itA^U}\ .$$

Denote the *domain* of $A$ by 

$$D(A) = \left \{\psi\in\mathcal{H}: \lim_{t\to 0}{\frac{U_t(\psi)-\psi}{it}}\ {\text{exists}}\right \}\ .$$
Conversely, for any self-adjoint operator $A^o$, there exists a strongly
continuous unitary group $\{U^{A^o}_t\}_{t\in\mathbb{R}}$, such that this relation
between the operator $A^o$ and the unitary group $U^{A^o}$ is satisfied on the domain of $A^o$.

[See this Physics Stack-Exchange discussion about a generalization of the Stone-von Neumann Theorem over kime](https://physics.stackexchange.com/questions/802005/complex-extension-of-the-stone-von-neumann-theorem),
as well as [this Math-Physics Stack-Exchange example of defining strongly-continuous projective unitary representations of $\mathbb{C}$ on $L^2(\mathbb{R},dx)$](https://math.stackexchange.com/questions/4865565/complex-extension-of-the-stone-von-neumann-theorem) by

<!-- https://physics.stackexchange.com/questions/780863/relation-between-stone-von-neumann-theorem-and-bargmanns-theorem?rq=1 -->

$$\mathbb{C}\cong \mathbb{R}^2\ni (a,b)\mapsto U(a,b) := e^{i(\overline{aX - i b \frac{d}{dx}})},$$

$$\mathbb{C}\cong \mathbb{R}^2\ni (a,b)\mapsto V(a,b) := e^{i(\overline{aX - i 2b \frac{d}{dx}})}.$$

### Kime generalization

**Problem**: Explore transforming the *additive group* over $t\in\mathbb{R}$ to a 
*multiplicative group* $\kappa=te^{i\theta}\in\mathbb{C}$, where $\forall\ \psi\in\mathcal{H}$, the $1:1$ correspondence between 
*one parameter kime continuous unitary groups of operators*, i.e., 
*operator-valued distributions*, $U^A :\mathcal{H}\to\mathcal{H}$ and 
*self-adjoint operators* 
$A^U :\mathcal{H}\to\mathcal{H}$ is given by

$$\overbrace{A^U \psi}^{operator-valued\\ distribution} = \lim_{t\equiv|\kappa|\to 0}{\frac{U^{A^U}_{\kappa}(\psi) - \psi}{it}}\\
\overbrace{U^{A^U}_{\kappa}\underbrace{(\varphi)}_{test\\ function}}^{kime\ unitary\ operator}=
\underbrace{e^{-itA^U}}_{operator} \ \ \underbrace{\ell\left(e^{i\theta}\right)}_{distribution} \varphi\ ,$$

where the action of the *kime-inference function on test functions*,
$\langle\mathfrak{P}, \phi\rangle,$ is defined by the kime-phase action on 
test functions, $\langle\ell, \phi\rangle = \int_{\mathbb{R}}{\ell^*(\theta) \phi(\theta)}\ d\theta \in\mathbb{C}$,

$$\underbrace{\langle\mathfrak{P},
\phi\rangle}_{kime-function\\ action} =
\overbrace{\Psi(x,y,z,t)}^{spacetime\\ wavefunction}
\underbrace{\int_{\mathbb{R}}{\ell^*(\theta) \phi(\theta)}\
d\theta}_{\underbrace{\langle\ell, \phi\rangle}_{kime-phase\\
action}\in\mathbb{C}}\ .$$

Technically, $\ell(\theta)\equiv \ell(e^{i\theta})$.

These one parameter kime unitary operators $\{U_{\kappa}\ |\ \kappa\in\mathbb{C} \}$ are *continuous*
$$\forall \kappa_{o}\in \mathbb {C} ,\ \psi \in {\mathcal {H}}:\ \lim _{\kappa\to \kappa_o}U_{\kappa}(\psi )=U_{\kappa_o}(\psi),$$

To simplify all notation, we will be suppressing the extra (unnecessary) superscripts
signifying the $U\equiv U^{A^U} \longleftrightarrow A^U\equiv A$ correspondence.

$$\forall \kappa_1=t_1e^{i\theta_1},\ \kappa_2=t_2e^{i\theta_2}\in \mathbb {C} ,\\ U_{\kappa_1\cdot \kappa_2}=
U_{t_1e^{i\theta_1}\cdot t_2e^{i\theta_2}} = U_{t_1 t_2 e^{i(\theta_1+\theta_2)}} =
\underbrace{e^{-i(t_1 t_2)A}}_{operator} \ \ \underbrace{\ell\left(e^{i(\theta_1+\theta_2)}\right)}_{distribution}\\
\overbrace{=}^{\theta_1\perp \theta_2\\ \ell,\ separable} \underbrace{e^{-i(t_1)A}\ \ell_1\left(e^{i\theta_1}\right)}_{U_{\kappa_1}} \ \ \underbrace{e^{-i(t_2)A}\ \ell_2\left(e^{i\theta_2}\right )}_{U_{\kappa_2}}=
U_{\kappa_1}U_{\kappa_2}.$$

Recall that $\forall\ \kappa=te^{i\theta}\in\mathbb{C},$ $U_{\kappa}$ 
is an *operator-valued distribution* acting on *test functions*
$\varphi\in\mathcal{H}$ and producing complex scalars

$$\underbrace{U_{\kappa_1\cdot \kappa_2}(\varphi)}_{\in\mathbb{C}}=
U_{\kappa_1}(\varphi) \cdot U_{\kappa_2}(\varphi)\ .$$

To explicate the kime-dynamics of states at any kime $\kappa\in\mathbb{C}$,
consider an initial state $|\varphi_{\kappa_o}\rangle$. Without loss of generality, we can assume that $\kappa_o=te^{i\theta}=0$, i.e., $t=0$. So, the starting
initial state is $|\varphi_{\kappa_o}\rangle\equiv |\varphi_{o}\rangle$.

As the state at kime $\kappa\in\mathbb{C}$ is measurable, the temporal dynamics
of the system can be expressed in terms of the kime unitary operator group action

$$|\varphi_{\kappa}\rangle=U_{\kappa}(|\varphi_{o}\rangle)=
\underbrace{e^{-i t A}}_{operator} \ \ \underbrace{\ell\left(e^{i\theta}\right)}_{distribution} (|\varphi_{o}\rangle)\ ,$$

where $\ell\left(e^{i\theta}\right)$ is a (prior) model of the kime-phase distribution, which can be *sampled once* for single observations, or 
*sapled multiple times* corresponding to multiple *repeated measurements.*

Taking the partial derivative of $|\varphi_{\kappa}\rangle$ with respect to
the *kime-magnitude* (*time*, $t$) yields the *kime-Schrodinger equation*

$$\frac{\partial |\varphi_{\kappa}\rangle}{\partial t}=-iA
\underbrace{\ \ |\varphi_{\kappa}\rangle\ \ }_{\left (e^{-i t A}\right )
\ell\left(e^{i\theta}\right) (|\varphi_{o}\rangle)} \ .$$

For instance, consider a *free evolution* (no external forces), 
where we are modeling the *energy (Hamiltonian) kime-evolution* of the 
state of a particle of mass $m$ in $1D$. Assume only *kinetic energy* $K$ 
is at play, without *potential energy*, $V=0$. Then, the energy 
*Hamiltonian operator* $H=A$ is self-adjoint

$$A\equiv H=-\frac{1}{2m}\frac{d^2}{dx^2}\ .$$

In this case, the explicit form of the *kime-independent Schrodinger equation*
describing the *physical state of a quantum-mechanical system* is

$$\frac{\partial |\varphi_{\kappa}\rangle}{\underbrace{\partial t}_{t=|\kappa|}}=-iA
\underbrace{\ \ |\varphi_{\kappa}\rangle\ \ }_{\left (e^{-i t A}\right )
\ell\left(e^{i\theta}\right) (|\varphi_{o}\rangle)} =
+i\frac{1}{2m}\frac{\partial^2}{\partial x^2} |\varphi_{\kappa}\rangle\ .$$

For simplicity, here we are working with normalized units, $c=\hbar = 1$. In the more general $3D$ kinetic energy (free potential) case, the system Hamiltonian is

$$A\equiv H=-\frac{h{^2}}{8\pi{^2}m}\left(\underbrace{\dfrac{\partial{^2}}
{\partial{x^2}}+\dfrac{\partial{^2}}{\partial{y^2}}+\dfrac{\partial{^2}}
{\partial{z^2}}}_{Laplacian,\ \nabla^2}\right)\ .$$

This *kime-independent Schrodinger equation* has an explicit solution
$$|\varphi(\kappa)\rangle = \left (e^{-i t E}\right )
\ell\left(e^{i\theta}\right) (|\varphi_{o}\rangle)\ ,$$

where $E$ represent observable energies. Since $\ell\left(e^{i\theta}\right)$ 
is a kime-phase distribution on $[-\pi,+\pi)$ and the observable energies
(eigenvalues of the Hamiltonian) are finite, this suggests that $\forall\ \psi_o\in L^2(\mathbb{C})$

$$||\psi_{\kappa}||_{L^{\infty}} \underset{t\to\infty}{\ \longrightarrow 0}\ .$$

Therefore, as $t\to\infty$ the *PDF of the position* vanishes and there is 
no limiting probability distribution for the position of the particle in $1D$
as it can spreads out across the entire real line.

This was the solution of the *kime-independent Schrodinger equation* in 
*position coordinates*. Let's consider the same *free evolution* (kinetic energy only)
in *momentum coordinates.* Recall that the Fourier transform provides a 
linear bijective mapping between *position coordinates*, spacetime representation,
$\varphi(x),$ and *momentum coordinates*, k-space frequency representation, ${\hat{\varphi}}(p)$. 

Again for simplicity, we'll consider the momentum of a free particle in $1D$
and set $c=\hbar=1$. The solution of the Schrodinger equation in momentum coordinates is

$${\hat{\varphi}}_{\kappa}(p)=\langle p |{\hat{\varphi}}_{\kappa}\rangle = 
\left (e^{-i t p^2}\right ) \ell\left(e^{i\theta}\right) 
(\langle p |{\hat{\varphi}}_{\kappa}\rangle)=
\left (e^{-i t p^2}\right ) \ell\left(e^{i\theta}\right) 
({\hat{\varphi}}_{o}(p))\ .$$

Clearly, 
$||{\hat{\varphi}}_{\kappa}(p)||^2 \equiv ||{\hat{\varphi}}_{o}(p)||^2,$
suggesting that the *momentum PDF is static in kime*.
For a free-evolution (no potential energy) this finding is not surprising,
since the momentum of a free particle should be preserved. However, 
in real observations, quantum fluctuations (intrinsic randomness) will affect
repeated measurements of a given observation (e.g., energy). This intrinsically
stochastic behavior is modeled by the kime-phase distribution
$\ell\left(e^{i\theta}\right)\sim \Phi_{[-\pi,+\pi)]}$.

**Problem**: Extend this $1D$ free-particle example to a *Hamiltonian including both kinetic and potential energy* components 
$$H(\psi) =\underbrace{-\frac{1}{2m}\frac{d^2}{dx^2}}_{kinetic}\psi(x) +
\overbrace{(V\psi)(x)}^{\overbrace{V(x)}^{potential}\ \psi(x)}\ .$$

Work out examples like a *harmonic oscillator* with potential energy
$V(x)=\frac{1}{2}m \omega^2 x^2$, where $\omega$ is the *oscillation frequency*.

...

### QM Postulates

The firsts *four quantum mechanics (QM) postulates* include:

 - `P1`: The state of a physical system is described by a vector in a separable
complex Hilbert space $\mathcal{H}$.
 - `P2`: To each (real-valued) observable $O$ corresponds a Hermitian operator
$\hat{O}$ on $\mathcal{H}$. Note that $\hat{O}$ can be a bounded linear operator such that $\hat{O} = \hat{O}^{\dagger}$, but not all operators $\hat{O}$ are bounded.
 - `P3`: If $\hat{O}$ is the operator for an observable $O$, then any experimentally observed value of $O$ must be an eigenvalue of $\hat{O}$.
 - `P4`: If the observable $O$  corresponds to the operator $\hat{O}$ and suppose $\hat{O}$ has an orthonormal sequence of eigenvectors $\{|\psi_n\rangle\}_n$ paired with eigenvalues $\{\lambda_n\}_n$. Then if the quantum system is in state $\psi\in \mathcal{H}$, the probability that the observed value of $O = \lambda$ is 

$$\frac{\sum_{i:\lambda_i=\lambda}|\langle\psi_i|\psi\rangle|^2}{||\psi||^2}\ .$$

 - `P5`: If the system is in inertial state (only kinetic energy without potential energy) not affected by external influences, then its *state evolves in time* as  $|\psi(t)\rangle = U_t|\psi(0)\rangle$, for some strongly continuous unitary group $\{U_t\}_t$ that only depends on the system (and not on the state).

The *Stone - von Neumann* theorem allows us to compute the derivative of the time evolution (i.e., explicate the time-dynamics). More specifically, there *exists a unique self-adjoint operator* $\hat{H}$, the system *Hamiltonian*, such that 
$U_t = e^{-it\hat{H}}$ satisfies

$$\frac{d}{dt}U_t=-i\hat{H}\ U_t=-i\hat{H} e^{-it\hat{H}}=-iU_t\hat{H}=-ie^{-it\hat{H}}\hat{H}\ .$$

**Kime-Evolution Problem**: Extend these formulations from positive *real time*, $t\in\mathbb{R}^+$, to *complex time (kime)*, $\kappa\in\mathbb{C}$.

# Kime Relation to Reality of Quantum States and PBR Theorem

In 2012, [Matthew Pusey, Jonathan Barrett and Terry Rudolph proved the PBR Theorem](https://doi.org/10.1038/nphys2309), which a quantum foundations 
*no-go theorem* for interpreting the *nature of quantum states*. In regard to 
realistic *hidden variable theories* explaining predictions of quantum mechanics,
the PBR theorem rules that pure quantum states must correspond directly to 
states of reality. This correspondence can't be *epistemic*, i.e., it can't
represent *probabilistic or incomplete states of knowledge about reality*. 
Subsequently, in 2017, a 
[follow up paper *The PBR theorem: Whose side is it on?*](https://doi.org/10.1016/j.shpsb.2016.11.004) discussed implications of the PBR theorem
in relation to reality of quantum states. Specifically this study discriminates
between *epistemic interpretations* of quantum states and *instrumentalism.*
The [PBR theorem also has connections to de Finetti’s probability interpretation as rational degrees of belief, the Einstein-Podolsky-Rosen (EPR) experiment, and Bell’s theorem](https://doi.org/10.1007/978-3-642-21329-8_16).

 - (*Objective* or *physical*) $\psi$-*ontic models* of the wavefunction correspond to the physical states of the system. More specifically, $\psi$-ontic models the wavefunction correspond directly to the physical state of the system;
 - (*Knowledge* ot *belief*) $\psi$-*epistemic models* of the wavefunction represent *(partial?) knowledge about the physical state of the system*, not the actual physical reality. In other words, $\psi$-epistemic models only represent virtual proxy knowledge about the physical state of the
system. 

There are two possible modes of proxy knowledge in *incomplete epistemic models*:
 - The wavefunction $\psi$ could just yield a *partial description* of the actual physical state of the system, or
 - The wavefunction $\psi$ is just a *partial representation of our knowledge* about that state. 

In the *case of incomplete* $\psi$-*ontic models*, it is conceivable that $\psi$ 
could be supplemented with further (yet-unknown) parameters – *hidden variables*
- to *complete* the real-representaiton of the system. In this case, a fixed 
wavefunction $\psi$ could correspond to various physical
states of the system, which can potentially be disambiguated by additional knowledge
about the extra hidden variables.

In the case of *incomplete* $\psi$-*epistemic models* the completion of the
knowledge to achieve a perfect reconstruction of physical reality cannot be
accomplished by hidden variables!

The *representation incompleteness* takes two forms: 

 - $\Psi$ just gives a partial description of the physical state, or 
 - a partial representation of our knowledge about that state. 
 
An incomplete $\Psi$-*ontic model* could potentially be augmented with further parameters,
e.g., *hidden variables*, suggesting that the wavefunction could correspond to 
various physical states of the system that are distinguishable by knowing the 
values of the additional hidden variables. 

$\Psi$-*epistemic models* can also be either *complete* or *incomplete*, however,
completing them cannot be accomplished by adding hidden variables.

At face value, it appears that PBR theorem supports other no-go theorems, 
[Bell's theorem](https://en.wikipedia.org/wiki/Bell%27s_theorem), 
that rule out the possibility of explaining the QM predictions 
using *local hidden variable theories*. Specifically, the PBR theorem 
may rule out physical reality interpretation via independent hidden variables,
i.e., quantum states prepared independently can't have independent 
hidden variable descriptions.

A *statistical description of an ensemble* of similar systems makes no definite
claims about individual members of the ensemble. This makes it challenging 
to reconcile *probabilistic descriptions of an ensemble* and the precise physical
description of each individual system comprising the ensemble. 
A major point of division is whether quantum states tell us anything about the
individual system. Specifically, the question of how to explicate the
*collapse of the wavefunction* due to measurement (i.e., random sampling from the 
underlying probability distribution). If wavefunctions directly represent the 
physical states of individual systems, upon measuring a physical property,
their *instantaneous collapse (to specific "critical" values)* still needs to be
explained. If quantum probabilities govern the distributions of ensembles of 
similar systems, then measurements on individual systems should yield definite
values; In a statistical distribution sense, critical values that correspond.

In essence, the measurement-induced *wavefunction collapse* is analogous the 
*collapse of a roll of a fair hexagonal die* from ensemble states 
$\{|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle, |6\rangle\}$ of equal
probabilities $p(|i\rangle)=\frac{1}{6}, \forall\ i\in\{1, 2, \cdots, 6\}$.
At each roll of the die, the probabilities characterize an ensemble to either 
of the $6$ states. In the ensemble interpretation, the wavefunction $\psi$
*does not represent a physical state of an individual system*. The notions of
quantum phenomena, e.g., superposition, interference and entanglement, become more
enigmatic under an ensemble interpretation of quantum mechanics. 

**Problem**: Explore the dichotomy of the Heisenberg uncertainty principle 
in the ensemble interpretation, spacekime interpretation, and 5DSTM consortium 
interpretation of an extra force in 5D that does not have a 4D analogue. 
Can the uncertainty principle reflect a lower-dimensional (4D) statistical 
projection law unlishing dispersion (variability) of sampled (critical) values 
in an ensemble of systems? Can this MInkowski 4D spacetime uncertainty manifest
the loss of $1$ degree-of-freedom (kime-phase distribution),
which transforms 5D definiteness of spacekime events into stochastic values 
representing measurable properties of individual systems?




## PBR Theorem Assumptions

Two of the critical assumptions of the PBR theorem are:

 - The an isolated (not entangled) *quantum system has a real physical state*, not necessarily completely described by quantum theory, but a physical state that is objective and independent of the observer.
 - Quantum systems *prepared independently* have *independent physical states*.

### Classical Mechanics Iterpretation

For instance, given a point particle moving in 1D, at time $t$. Classical mechanics
models the physical state of the particle as a function of its position $x$ and 
momentum $p=mv$, scalar mass $m$ and vector velocity $v$. Thus, the 
physical state of a particle is reflected as a point $(x,p)$ in the 2D phase
space. Of course, the other physical properties of the particle (e.g, charge) 
are assumed to be either *fixed* or *functions* of the state, e.g.,
the energy $H=H(x,p)$. In a way, other particle physical properties can be considered as
functions constant in time. The exact physical state of the particle at a given point in the phase space is stochastic, subject to some well-defined probability distribution 
$\xi(x,p)$. 

**Problem 1**: Can we consider the *kime-phase distribution* 
$\ell_{(x,p)}(\cdot)\sim \Phi[-\pi,\pi)$ as this enigmatic exact physical state 
probability distribution $\xi(x,p)$, which does not directly represent
reality and evolves over time according to the 
[Liouville's equation](https://en.wikipedia.org/wiki/Liouville%27s_equation)? 
The state probability distribution function $\xi(x,p)$ describes the time evolution 
of the state, but instead of reality, it represents a state of knowledge
about an experimenter's uncertainty about the physical state of the particle.

**Problem 2**: Explore the [proof of the quantum de Finetti representation theorem](https://doi.org/10.1063/1.1494475) in relation to the *spacekime interpretation*,
and the SOCR *DNN Exchangeability, Probability Symmetry, Invariance, and Equivariance* paper
by Yueyang Shen *et al.*.


### Quantum Mechanics Interpretation

**Definition**: A *physical property* of a system is some function of the physical state.
For each state of the physical system, $\lambda$, consider a one-parameter family of probability distributions $\{\xi_{\gamma}(\lambda)\}_{\gamma}$. If each pair of distributions in this family are disjoint, 

$$\mathrm{supp}\left (\xi_{\gamma}(\lambda)\right)\cap \mathrm{supp}\left (\xi_{\delta}(\lambda)\right)=\emptyset,\ \forall\ \gamma\not=\delta\ ,$$

then the system’s physical state (label) $\gamma$ is uniquely fixed by $\lambda$ 
and $\gamma$ is a *physical property*.

However, $\gamma$ would *not be a physical property* when $\exists\ \gamma_1\not= \gamma_2$, such 
that the corresponding distributions $\xi_{\gamma_1}(\lambda)$ and $\xi_{\gamma_2}(\lambda)$
both assign positive probability to some overlapping region of positive measure ($\nu$), i.e., 
$$\nu\left (\mathrm{supp} (\xi_{\gamma_1}(\lambda)\right)\cap \mathrm{supp}\left (\xi_{\gamma_2}(\lambda))\right)\gt 0\ .$$
In this situation, a scalar value 
$$\underbrace{\lambda}_{system’s\\ physical\ state} \in  \underbrace{\mathrm{supp} (\xi_{\gamma_1}(\lambda))\cap \mathrm{supp} ( (\xi_{\gamma_2}(\lambda))}_{\not= \emptyset}\ .$$
would be consistent with either label, which is a contradiction.

The quantum state is a state of knowledge reflecting the uncertainty
about the real physical state of the system. Assume there is a mathematical-statistics
model associated with the physical state $\Xi$ of the system. Recording an experimental
observation by sampling the process yields probability likelihoods of different 
outcome measurement follow the distribution $\xi_{\alpha}\sim \Xi$. When a 
controlled quantum experiment is carefully prepared to ensure consistent IID outcomes,
quantum theory associates a *pure* ($|\psi_{\xi_{\alpha}}\rangle$) or a *mixed* 
($\sum_{\alpha}\omega_{\alpha}|\psi_{\alpha}\rangle$) quantum state, yet the 
the physical state $\Xi$ is not fixed uniquely by the system preparation process.
Rather $\xi_{\alpha}\sim \Xi$ is a draw (sample) from the *probability distribution*
of the physical outcome.

The main question is **whether quantum state corresponds directly to reality or merely represents partial information about the state**. Consider the following two scenarios in the case of classical mechanics:

 - *Case 1*: The experimenter only knows the system energy $E$, and completely uncertain, about other characteristics. This (limited) knowledge corresponds to a distribution $\xi_E(x,p)$ which is uniformly distributed over all the 2D phase space and the system Hamiltonian $H(x,p)=E$. Different energy values $E'\not= E''$ correspond to disjoint regions of phase space, since energy is a physical property of the system. Therefore, the *supports* of distributions are *disjoint*, $\mathrm{supp}\left (\xi_{E'}(x,p)\right)\cap \mathrm{supp}\left (\xi_{E''}(x,p)\right)=\emptyset$.
 - *Case 2*: Assume that a pair of probability distributions have overlapping supports, i.e., $\mathrm{supp}\left (\xi_{E'}(x,p)\right)\cap \mathrm{supp}\left (\xi_{E''}(x,p)\right)\not=\emptyset$. Then, the observable quantities $E'$ and $E''$ cannot refer to a single physical property of the system.

### Pure, Superposition & Mixed Quantum States

**Definitions**:

 - A *pure quantum state* is a state that can not be written as a probabilistic mixture,
i.e., a linear combination, of other quantum states. 
 - Quantum *density matrix* is a quantum tensor generalization of the classical concept of probability distribution to provide the complete description of a quantum state of a physical system and any observable quantities that can be extracted from from the state. While *state vectors* and *wavefunctions* can only represent *pure states*, density matrices can also represent *mixed states* that represent quantum systems with unknown preparation, i.e., statistical quantum ensembles of possible preparations, or physical systems with entanglements or with system-environment interactions (e.g., *decoherence*).
 - [Density operators](https://en.wikipedia.org/wiki/Density_matrix) are linear projection operators, paired with coresponding *density matrices*. In a 2D Hilbert space scenario, we start with a basis of two states $|0\rangle $ amd $|1\rangle $ where the corresponding density operator is represented by the (density) matrix $(\rho _{ij})=\left({\begin{matrix}\rho_{00}&\rho _{01}\\ \rho _{10}&\rho _{11}\end{matrix}}\right)=\left({\begin{matrix}p_{0} & \rho _{01}\\ \rho _{01}^* & p_{1} \end{matrix}}\right)$. The main diagonal elements are real numbers summing up to one. The off-diagonal elements (coherences) are complex conjugates of each other satisfying the positive semi-definite requirement $(\rho _{ij}) \gt 0$.

Density operators facilitate multi-particle quantum mechanics calculations.
More generally, in a quantum system with a $d$-dimensional Hilbert space $\mathcal{H}$, take
an arbitrary state $|\psi\rangle \in\mathcal{H}$, and define $\hat{\rho} = |\psi\rangle\, \langle\psi|$. This is the desity projection operator for $|\psi\rangle$. 
As finite dimensional linear operators can be represented as matrices, this *density operator*
is represented by a *density matrix* having the following properties:

 - It is *Hermitian*
 - If $\hat{Q}$ is an observable with *eigenvalues* $\{q_{\mu}\}$ corresponding to the *eigenstates* $|\mu\rangle$, then $\hat{Q}$ measurements on $|\psi\rangle$ will yield values $q_{\mu}$ with probability $p_{\mu} = \big|\langle \mu | \psi\rangle\big|^2 = \langle \mu |\, \hat{\rho}\, | \mu \rangle$.
 - Let $Q$ be an *observable* of a physical system, and suppose an *ensemble is in a mixed state*, where each of the individual *pure states* $|\psi_{j}\rangle$ occurs with a corresponding probability $p_{j}$. Then the *mixed-state density operator* is $\rho=\rho_{mixed} =\sum_{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|$ and the *mixed-state expectation value* of the measurement is also expressed in terms of the  of *pure states*:

$$\langle Q\rangle =\sum _{j}p_{j}\langle \psi _{j}|Q|\psi _{j}\rangle =\sum _{j}p_{j}\operatorname {tr} \left(|\psi _{j}\rangle \langle \psi _{j}|Q\right)=\operatorname {tr} \left(\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|Q\right)=\operatorname {tr} (\rho\ Q)\ .$$
The matrix operator $\operatorname {tr}$ denotes the trace. For *pure states*, 
the expectation value is $\langle Q\rangle =\langle \psi |Q|\psi \rangle$, 
whereas for *mixed states* the expectation value is
$\langle Q\rangle =\operatorname {tr} (\rho Q).$

Recall that the trace $\mathrm{tr}$ of a matrix is the sum of the diagonal elements,
which is a scalar invariant of the basis, i.e., $\mathrm{tr}$ is basis-independent.

Density operators representing pure states have the following properties:
 - The density operator can be written as an outer product of a state vector $|\psi \rangle$ with itself, $\rho =|\psi \rangle \langle \psi |$;
 - The operator is an idempotent projection of rank one $\rho =\rho^{2}$;
 - The operator purity is unitary $\operatorname {tr} (\rho ^{2})=1.$

There is an important difference between a *probabilistic (statistical) mixture* of
quantum states and *quantum superpositions* of pure states. 
In a simple 2D scenario, a physical system prepared with *equal probability* in either one state 
$|\psi _{1}\rangle ={1\choose 0}$ or another state $|\psi _{2}\rangle={0\choose 1}$
can be described by the (rank $2$) *density operator of the mixed state*

$$\rho ={\overbrace{\frac {1}{2}}^{equal\\ prob}}
{\begin{pmatrix} \overbrace{1}^{|\psi_{1}\rangle} & \overbrace{0}^{|\psi_{2}\rangle} \\
0 & 1 \end{pmatrix}} \ .$$

with $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ assumed to be orthogonal pure states. 

On the other hand, a *quantum superposition* of these two pure states with equal probability amplitudes yields another *pure state* 
$|\psi \rangle =\frac{1}{\sqrt {2}}\left (|\psi _{1}\rangle +|\psi _{2}\rangle \right)$ represented by the (rank $1$, not $2$) density matrix

$$ |\psi \rangle \langle \psi |={\frac {1}{2}}{\begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}}.$$

Whereas probabilistic statistical *mixture* reflects independent factors, *quantum superposition* reflects potential *quantum interference*. For a more direct statistical analogy, the *mixture vs. superposition* duality is analogous to the duality between *linear models with and without interaction* effects

$${\text{(independence, no interaction model)}}\\
m_1 = \operatorname {lm} (y \sim A\ +\ B,\ data = input)\\ \ \\
{\text{(model with interaction)}}\\
m_2 = \operatorname {lm} (y \sim A + B + A:B,\ data = input)$$

For a given a pure state, the uncertainties of statistical *mixed-states* 
and a *quantum superpositions* are quantified and interpreted differently. 
Quantum amplitudes reflect interference measured by preparing many copies
of the same state (drawing multiple IID samples from a fixed statistical distribution)
and measuring incompatible observables. This interference is in terms of the double-slit experiment where quantum superpositions correspond to particles going through both 
slits at once, which produces interference at the detection screen in the back.

A *mixed state* represents a statistical mixture of two or more pure states, 
whereas a *superposition state* refers to a joint state composed simultaneously 
of some other states.

Consider a quantum system that has $50\%$ chance to be observed in state 
$|\psi_1\rangle$ and $50\%$ to be in another state $|\psi_2\rangle$. An example 
of a *superposition state* is
$$|\psi \rangle = \frac{1}{\sqrt{2}}\left(|\psi_1\rangle +|\psi_2\rangle \right)\ ,$$
since its a *pure state*, there is a $0\%$ chance that the system is in either of those states $|\psi_1\rangle,|\psi_2\rangle$, and a $100\%$ chance that the system is in the superposition state $|\psi \rangle$. 

This probabilistic quantification of state likelihoods is made before making any measurements. 
Once we observe a (*physical*) measurement, i.e, (*statistically*) draw from the state probability distribution corresponding to the observable characteristic property
denoted by $|\psi\rangle$, this process corresponds to a collapse the system wavefunction, which results in a $50\%$ chance that the actual measurement (sample) is in *either of the two states* $|\psi_i\rangle,\ i\in\{1,2\}$.

### Light Polarization Example of Mixed and Superposition States

Consider a light bulb source emitting completely random polarized photons with the
following *mixed state density matrix*

$$\rho_{mixed}={\begin{bmatrix}\frac{1}{2} & 0\\ 0 & \frac{1}{2}\end{bmatrix}}\ .$$
All photons passing through a *vertical plane polarizer* will be all vertically polarized
with a *pure state density matrix*

$$\rho_{pure}={\begin{bmatrix} 1& 0 \\ 0 & 0 \end{bmatrix}}\ .$$

Each individual photon will either have a right or left circular polarization
described by either orthogonal quantum states $|\mathrm {r} \rangle$ or
$|\mathrm {l} \rangle$ }, or by a superposition of the two pure states
$\alpha |\mathrm {r} \rangle + \beta |\mathrm {l} \rangle$, where 
$|\alpha |^{2} + |\beta |^{2}=1$. The observed polarization (detected measurement) 
will exhibit *linear*, *circular*, or *elliptic* pattern. 

For instance, a single *vertically polarized photon* described by the state 
$|\mathrm {v} \rangle =\frac{1}{\sqrt {2}}(|\mathrm {r} \rangle +|\mathrm {l} \rangle )$
that passes through a *circular polarizer* will only show a pure state polarized light
$|\mathrm {r} \rangle$ or $|\mathrm {l} \rangle$, but not a mixture of both. While
a barrage of photons will have (approximately) half of the photons absorbed by the polarization filter. It's not true *half of the photons are in one state* $|\mathrm {r} \rangle$
and the other half in the state $|\mathrm {l} \rangle$. Passing  
$\frac{1}{\sqrt {2}}(|\mathrm {r} \rangle + |\mathrm {l} \rangle )$ through a 
linear polarizing filter will not absorb any of the photons, however passing
pure states $|\mathrm {r} \rangle$ or $|\mathrm {l} \rangle$ will indeed result in
absorption of half the photons.

On the other hand, *non-polarized light* (e.g., from an incandescent light bulbs)
cannot be described as $\alpha |\mathrm {r} \rangle + \beta |\mathrm {l} \rangle$ 
linear, circular, or elliptical polarization state. *Polarized light* passes through
a polarizer with a $50\%$ intensity loss irrespective of the orientation of the 
polarization filter. *Unpolarized light* can be described as a *statistical ensemble* 
where each photon has either $|\mathrm {r} \rangle$ or $|\mathrm {l} \rangle$ polarization 
with equal probability $\frac{1}{2}$. Similarly for photons with equal probability of 
either vertical $|\mathrm {v} \rangle$ or horizontal $|\mathrm {H} \rangle$ polarization.
The density operator for unpolarized light is

$$\rho ={\frac{1}{2}}|\mathrm {r} \rangle \langle \mathrm {r} |+{\frac{1}{2}}|\mathrm {l} \rangle
\langle \mathrm {l} |=\frac{1}{2} |\mathrm {h} \rangle \langle \mathrm {h} |+
\frac {1}{2}|\mathrm {v} \rangle \langle \mathrm {v} | = 
\frac {1}{2}{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}}.$$

In radioactive decay, non-polarized light may be generated by using *entangled states*
that emit two entangles photons traveling in opposite directions, representing
the entangles quantum state 
$\frac{1}{\sqrt {2}} (|\mathrm {r} ,\mathrm {l} \rangle + |\mathrm {l} ,\mathrm {r} \rangle )$. 
The joint state of the two photons represents a *pure state*, whereas
the density matrix for each individual photon computed as the partial trace of
the joint density matrix represents a *mixed state.*


# Observables (Quantum Measurements) and Linear Transforations as Quantum Operators

Following the notation used in [this article, "*An introduction to quantum computing for statisticians and data scientists*"](https://doi.org/10.3934/fods.2024013), 
we will explicate the pair of *quantum operator* representations of *observables* and 
*linear operators*.

Classical probability theory intuition collides with quantum theoretic interpretations
due to quantum interference. At first glance, quantum theory may be counter-intuitive
in describing precisely (deterministically) and completely the behavior of tiny atomic particles.

Linear algebra over complex (finite/infinite dimensional Hilbert) vector spaces provides 
the foundation of quantum theory. All closed quantum systems are described as vectors
over Hilbert spaces, where vectors are pure quantum states, $|\psi\rangle$ and 
the *ket* operator $|\cdot\rangle$ is

$$|\psi\rangle \equiv \begin{pmatrix}
\psi_1 \\ \psi_2 \\ \vdots \\ \psi_n \\ \vdots \end{pmatrix}\ .$$

The *coordinate values*, $\psi_m\in\mathbb{C}$, aka *amplitudes*, are used
to express the quantum state $|\psi\rangle$ as a *linear combination* (*superposition*)
of *quantum base states*, vectors in the Hilbert space. Suppose the standard 
base states (basis vectors) are $\{|e_m\rangle\}_m$, then

$$|\psi\rangle = \psi_1 |e_1\rangle + \psi_2 |e_2\rangle + \ldots + 
\psi_n |e_n\rangle + \ldots\ ,$$

which can be finite or infinite depending on the situation.

In the Hilbert (vector) space, the *inner product* of two vectors $|\phi\rangle$ 
and $|\psi\rangle$ is defined similarly to the inner product in the 
classical Euclidean space. However, since the scalar base-field is $\mathbb{C}$,
the inner product uses *complex conjugation* and *transposition* of the left vector,
$|\phi\rangle$, expressed using the *bra* notation, i.e., the conjugate-transpose 
is a row-vector $\langle\phi| = (\phi_1^*\ \phi_2^*\ \ldots\ \phi_n^*\ \ldots)$. 
The asterisk $^*$ indicates complex conjugation. The inner product 
between two states $|\phi\rangle$ and $|\psi\rangle$ is 

$$\langle\phi|\psi\rangle = \overbrace{\phi^* \cdot \psi=\sum_m \phi_m^*\psi_m}^{discrete\ case}
\ \ \ \langle\phi|\psi\rangle = \overbrace{\int_{\mathbb{R}} 
\phi^*(x)\psi (x)\ dx}^{continuous\ case}\ .$$
Quantum normalization of the states facilitates the probabilistic interpretation
the squared *amplitudes* $|\psi_m|^2$ as *likelihoods*, 
i.e., $\forall\ |\psi\rangle$, we have $\langle\psi|\psi\rangle = \sum_m |\psi_m|^2 = 1$.
Hence, the squared amplitudes $|\psi_m|^2$ quantify the probabilities of future 
outcomes of prospective quantum measurement.

*Observables* (*quantum measurements*) characterizes (quantize) unknown quantum states. 
For an arbitrary quantum state, $|\psi\rangle$, and a given reference basis, $|e_m\rangle$,
the outcomes of a quantum measurement
are always stochastic. For instance, if an instrument measures certain *observable* $|\psi\rangle$
in the standard basis, the actual *measurement value* (*outcome*) $m$ corresponding to the state $|e_m\rangle$ with probability $|\psi_m|^2$. The Copenhagen interpretation of quantum 
mechanics suggests that immediately after the measurement, the quantum state
*collapses to the basis vector* $|e_m\rangle$ corresponding to the outcome $m$. 

In a classical probabilistic setting, imagine a closed container containing balls 
of different colors. Prior to randomly drawing a ball from the container, 
we don't know it's color. Yet, taking many repeated samples (drawing a random 
ball and recording it's color, before replacing it and mixing the balls in 
the container) allows us to estimate (quantify) the *color probability distribution*
over the set of all possible ball colors. Prior to the measuremetn observation,
the color of the next ball is a *random variable.* Drawing a ball at random, 
according to the ball color probability distribution, corresponds to instantiating
a random ball color, i.e., at the time of observing the ball, the experimental
measurement (sampling) collapses the probability distribution into a specific color. 
Analogously, we can characterize (quantize) quantum states by repeatedly preparing 
the controlled experiment, making quantum measurements, and extrapolating (modeling)
the sampling distribution.

Quantum theory may be viewed as both *counter-intuitive* (stochastic interpretability)
and *powerful* (reliable computing, estimation, and prediction). Quantum states are 
observed according to their *squared amplitudes*, which are the corresponding 
probabilities quantify the likelihoods of each outcome. 
As the amplitudes are complex values, their corresponding squared amplitudes
probability masses (or density values) cancel out under *linear quantum states transformations.* 

A quantum state $|\psi_A\rangle$ transforms into another state in the same Hilbert space $|\psi_B\rangle$ by $|\psi_B\rangle = U |\psi_A\rangle$, where $U$ is a *linear unitary operator.*
*Quantum interference* refers to the *cancellation of (complex) amplitudes* under 
linear transformations. We will see that in general, a *quantum measurement*, 
or a *quantum algorithm*, $A$ is a *collection of linear operators* $A\equiv\{M_m\}$  
corresponding to a series of outcomes (or indices of outcomes) $\{m\}$ 
occurring with corresponding series of probabilities $\{p_m\}$. Hence, quantum
measurements are *series of linear transformations* mapping input states
$|\psi_{\text{input}}\rangle$ into corresponding output states 
$|\psi_{\text{output}}\rangle$

$$|\psi_{\text{output}}\rangle = A |\psi_{\text{input}}\rangle\ .$$

More specifically, a measurement applied to a quantum state $|\psi\rangle$ 
results in a specific outcome $m$ with probability 
$p_m = \langle \psi | M_m^\dagger M_m | \psi \rangle.$ Furthermore,
the normalized state of the system after the measurement is 
$${\text{Observation}}=\frac{\overbrace{M_m |\psi\rangle}^{measurement\ output}} 
{\underbrace{\sqrt{\langle \psi | M_m^\dagger M_m | \psi \rangle}}_{normalization\ const.}}.$$

## Qubit - an Example of a Simple Quantum System

The basic unit of information in modern computers is a $0/1$ *bit*, whereas
the basic information-storage unit of a quantum computer is a *qubit*, $|q\rangle$,
which is a 2D system $|q\rangle = a |0\rangle + b |1\rangle$, 
where $|0\rangle$ and $|1\rangle$ form an *orthonormal basis* of the corresponding
2D Hilbert space. The *basis states* are $|0\rangle=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ 
and $|1\rangle=\begin{pmatrix} 0 \\ 1 \end{pmatrix}$. 
The superposition (composite) *qubit state* $|q\rangle=\begin{pmatrix} a \\ b \end{pmatrix}$
is a normalized vector in the 2D Hilbert space describing the state of the 
single-qubit quantum system. The complex-scalar value *amplitudes* are the coefficients 
$a, b \in \mathbb{C}$, constrained so that $|a|^2 + |b|^2 = 1$. Hence, subject to 
a normalization constraint (llosing one degree of freedom, the superposition state
$|q\rangle$ is a linear combination of the basis states $|0\rangle$ and $|1\rangle$.

A quantum measurement of $|q\rangle$ in the basis of $\{|0\rangle, |1\rangle\}$ yields $0$ with probability $|a|^2$ and $1$ with probability $|b|^2$. But prior to making a *physical measurement* (*random sampling*, in statistical terms), the quantum state $|q\rangle$
is in a superposition of the two base states. In a way, the state $|q\rangle$ is 
analogous to a random variable taking the classical bit values $0$ and $1$ with
corresponding probabilities $|a|^2$ and $|b|^2$. Yet, there is a crucial difference: the coefficients $a,b\in\mathbb{C}$ do not need to be positive reals! This complexification
property is crucial to the power of quantum computers.

This 2D superposition property extends to collections of *multiple qubits*, 
referred to as *quantum registers*, e.g., a register of $3$ qubits can support 
quantum states expressed in terms of the $2^3=8$ base states $\{|b_1 b_2 b_3\rangle\}$
encoding the base-2 (binary) representations of the numerical outcomes 
$\{m\}=\{0,1,2,3,4,5,6,7\}$

$$|\psi\rangle = \sum_m \psi_m|e_m\rangle \equiv \psi_{000} |000\rangle + \psi_{001} 
|001\rangle + \psi_{010} |010\rangle + \psi_{011} |011\rangle + \psi_{100} |100\rangle 
+ \psi_{101} |101\rangle + \psi_{110} |110\rangle + \psi_{111} |111\rangle .$$

For $b_k = \{1, 0\},\ k\in\{1,2,3\}$, the *base states* $|b_1 b_2 b_3\rangle$, 
form an orthonormal basis of the Hilbert space, and the *3-qubit register* supports $2^3 = 8$ dimensional quantum states, i.e., superpositions of $8$ basis states. 
The shorthand notation $|b_1 b_2 b_3\rangle=|b_1\rangle \otimes |b_2\rangle \otimes |b_3\rangle$,
where $\otimes$ represents the *tensor (outer) product*, i.e., $|b_1 b_2 b_3\rangle$ 
is an 8-dimensional vector. The tensor form shows that the state 
$|b_1 b_2 b_3\rangle$ is *separable*, i.e., each qubit in the state can be manipulated *independently* of the other qubits. In general, quantum states are *entangled*, that is
formed by multiple qubits and *cannot* be expressed as a tensor product.
Generally, entangled states $|\psi\rangle$ are *not separable*.

The superposition property and entanglement enable efficient encoding of information 
supporting enormous computational efficiencies and parallelism. There is nothing special
with the *binary* (2D) and *tertiary* (3D) examples above. This composite state superposition
representaiton generalizes to using $n$-qubit registers can encode $2^n$D vectors
with $2^n - 1$ *independent amplitudes*, one less to account for the normalization of quantum states condition, which introduces a interdependence (loss of one degree of freedom). 
Note that the special case of a *separable state* on $n$ qubits can only encode $n$ 
independent amplitudes. Quantum entanglement provides this massive complexification of
the state space, where $n$ qubits can encode $2^n-1$ independent amplitudes.

## Example - Quantum Algorithms

Quantum operations are series of *linear transformations* and *quantum measurements* 
on a set of quantum registers. As an example, just like classical algorithms are
often represented as a series of functions, quantum algorithms (operations) maybe 
implemented via a series of simple quantum gates, 
analogous to the traditional logic gates - *AND*, *XOR*, or *NOT.* Quantum gates are 
elementary quantum transformations that act on one, two, three, or more qubits.

Quantum algorithms take an *input quantum state* and transform it into a corresponding
*output state* that encodes the desired result. The superposition property imply
the efficiency of quantum algorithms offering *exponential speedup* over classical
algorithms. For instance, a *classical algorithm* that checks $100$-bit strings, i.e., 
$2^{100} \approx 10^{30}$ possibilities, has to check each of the $2^{100}$ strings. 
On the other hand, using *quantum parallelism*, a *quantum algorithm* could 
potentially perform the same task in massive parallel fashion labeling the correct 
and incorrect strings using only $100$ operations, one for each of the $100$ qubits 
holding the $2^{100}$ strings in a quantum superposition. 

The (probabilistic) output of a naive quantum computation will be a superposition 
of all the $2^{100}$  labeled results (unsettling). If all results are approximately
equally probable, then it would take $\sim 10^{30}$ measurements to extract the 
correct answer, which negates any quantum computing benefits. To solve this issue,
quantum algorithms leverage *quantum interference* - the property that quantum amplitudes
are complex numbers that can cancel out during the computation (just like in a simple additive operation mode, adding positive and negative real numbers tend to cancel out). Quantum algorithms
rely on quantum interference to *suppress the amplitudes of the wrong answers*
while *amplifying the amplitudes of the correct answers*. Hence, output states
of efficient quantum algorithms are superpositions of the desired answers, where
just a few measurements are sufficient to estimate the final answer within a
predefined precision (controlled error-rate).


## Quantum States, Measurability and Quantum Operators

In our prior example of a quantum state created on a 3-qubit registers, the expansion of
$|\psi\rangle$ explicates the bit strings of individual qubit basis states as integer numbers,
e.g., $\underbrace{|101\rangle}_{3-qubit\\ register}=\underbrace{|5\rangle}_{integer}$,
where the $n$-qubit quantum state $|\psi\rangle = \sum_{m=0}^{N-1} \psi_m |m\rangle$, 
$N=2^3$, and the integer representation of the bit string is $m=|m\rangle$. 
Hence, the 3-qubit quantum state $|\psi\rangle$ is described by a normalized $8$D
vector over complex numbers, $(\psi_0, \psi_1, \cdots, \psi_7)^T$. 
The *orthonormal basis states* 
$$\underbrace{|m\rangle}_{integer\ rep.} \equiv 
\underbrace{|b_1 b_2 \cdots b_n\rangle}_{binary\ rep.}\ ,$$ 
where $\forall\ m\in\{1, 2, \cdots, n\},\ b_m = \{0, 1\}$ is the computational basis
of the Hilbert space.

Much like in the classical case of Euclidean spaces, the *inner product* between two
quantum states $|\psi\rangle$ and $|\phi\rangle$ in an $N$D Hilbert space
can be expressed in terms of the product of the paired vector coordinates 

$$\langle\phi|\psi\rangle = \underbrace{\sum_{m=0}^{N-1} \phi_m^*\psi_m}_{finite\ dim\ space},
\ \ \ 
\langle\phi|\psi\rangle = \underbrace{\int_{\mathbb{R}} \phi^* (x)  \psi(x) \ dx }_{infinite\ dim\ space}.$$

In quantum notation, the conjugate-transpose of a *ket* is a *bra*, and vice-versa

$$\left (\overbrace{|\phi\rangle}^{ket} \right )^*=
\overbrace{\langle\phi|}^{bra}= \sum_{m=0}^{N-1} \phi_m^* \langle m|\ ,
\ \ {\text{and  }} \ \left (\overbrace{\langle\phi|}^{bra} \right )^*=
\overbrace{|\phi\rangle}^{ket}= \sum_{m=0}^{N-1} \phi_m | m\rangle\ ,$$

and the pure (base) *bra* $\langle m|$ is the conjugate transpose of the *ket* $|m\rangle$.
For example, in a 2D Hilbert space, if $|0\rangle=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$, 
then $\langle 0|=\begin{pmatrix} 1 & 0 \end{pmatrix}$. As basis states are orthonormal,
the inner product is expressed as $\langle\cdot|\cdot\rangle$, e.g., for a pair of basis states $|i\rangle$ and $|j\rangle$ is 

$$\langle i|j\rangle = \delta_{ij} \equiv \begin{cases} 
1, & \text{if } i = j \\
0, & \text{if } i \neq j 
\end{cases} \ ,$$

where $\delta_{ij}$ is the Kronecker delta function. More generally,
for $N$D states, the inner product is

$$\langle \phi|\psi\rangle = \left( \sum_{j=0}^{N-1} \phi_j^* \langle j| \right) \left( \sum_{i=0}^{N-1} \psi_i |i\rangle \right) = \sum_{j=0}^{N-1} \sum_{i=0}^{N-1} \phi_j^* \psi_i \langle j|i\rangle = \sum_{i=0}^{N-1} \phi_i^* \psi_i. $$

## Quantum Operators

*Quantum operators* are linear and are represented by complex matrices *acting* on vectors 
in the $N$D Hilbert space. For instance, an operator $A=A_{N \times N}=(a_{ij})_{i,j=0}^{N-1}$
can be expressed in *bra-ket* (matrix/tensor) form

$$A = \sum_{i,j=0}^{N-1} a_{ij} |i\rangle \langle j| \ ,$$

which acts on quantum states $|\psi\rangle = \sum_{k=0}^{N-1} \psi_k |k\rangle$ by 
standard matrix multiplication

$$A |\psi\rangle = \left( \sum_{i,j=0}^{N-1} a_{ij} |i\rangle \langle j| \right) \left( \sum_{k=0}^{N-1} \psi_k |k\rangle \right) \overbrace{=}^{\langle j|k\rangle = \delta_{jk}}\\
\sum_{i,j=0}^{N-1} a_{ij} \psi_j |i\rangle = \overbrace{\sum_{i=0}^{N-1} 
\underbrace{\left( \sum_{j=0}^{N-1} a_{ij} \psi_j 
\right)}_{scalar\ coef\ \in\mathbb{C}}\overbrace{|i\rangle}^{basis}}^{linear\ superposition}\ .$$

Note that vector$\times$scalar multiplication associative property and the 
orthonormality of the basis vectors, $\langle j|k\rangle = \delta_{jk}$, 
leads to simplifications of the linear combinations of outer products of 
basis states $|i\rangle \langle j|$. 


**Definition**: Not all linear operators are quantum operators, but all quantum operators
are linear. There are *two types of quantum operators*: *unitary transformations* and *observables.* 

 - *Unitary transformations* map one quantum state into another.
 - *Observables* are related to quantum measurement and probabilistically characterize quantum states.
 - A linear operator $U$ is *unitary* if $U^{-1} = U^\dagger$, where $\dagger$ denotes the *conjugate transpose*. Unitary operators preserve the unit norm of quantum states. 
 - The *identity operator* $I$ is a unitary operator expressing quantum states in alternative bases. Let $\{|a_i\rangle\}_{i=0}^{N-1}$ be an orthonormal set of basis states of an $N$D Hilbert space and let $\{|b_j\rangle\}_{j=0}^{N-1}$ be an alternative set of orthonormal basis states of the same vector space. A state $|\psi\rangle=\{|a_i\rangle\}$ in expressed in terms of first basis states can also be expressed in terms of second basis states by using the identity operator $I = \sum_{j=0}^{N-1} |b_j\rangle \langle b_j|$
 
$$|\psi\rangle = \sum_{i=0}^{N-1} \psi_i |a_i\rangle = \left( \sum_{j=0}^{N-1} |b_j\rangle \langle b_j| \right) \left( \sum_{i=0}^{N-1} \psi_i |a_i\rangle \right) = \sum_{j=0}^{N-1} \left( \sum_{i=0}^{N-1} \psi_i \langle b_j | a_i \rangle \right) |b_j\rangle = \sum_{j=0}^{N-1} \psi_j |b_j\rangle\ ,$$
where $\psi_j = \sum_{i=0}^{N-1} \psi_i \langle b_j | a_i \rangle\in \mathbb{C}$ are the (scalar) coordinates of $|\psi\rangle$ in the second basis $\{|b_j\rangle\}$.

## Observables and Quantum Measurements

Quantum measurements characterize probabilistically quantum states. Quantum operations
transform input quantum states into output measurements of the resulting state that
reflect the desired result with high probability. Symbolically, *quantum measurements*
are collections of operators $\{M_m\}$ that correspond to outcomes $m$ (as
before, $m$ can represent the actual outcomes or the index of the outcomes), which
occur with corresponding probabilities $p_m$. A measurement applied to a 
quantum state $|\psi\rangle$ yields the outcome $m$ with probability 
$$p_m = \langle \psi | M_m^\dagger M_m | \psi \rangle\ .$$

The (normalized) state of the system after the measurement is 
$$\frac{M_m |\psi\rangle }{\sqrt{\langle \psi | M_m^\dagger M_m | \psi \rangle}}.$$

Because the probability of all possible outcomes adds up to $1$, $\sum_m p_m = 1$
and for all quantum states, the measurement operators satisfy the *completeness relation*
$\sum_m M_m^\dagger M_m = I$.

For example, quantum measurement of a qubit $|q\rangle$  in the binary computational basis 
is a collection of a pair of measurement operators, *projection measurement operators* onto
the two base states. Symbolically, $P_0 = |0\rangle \langle 0|$ and 
$P_1 = |1\rangle \langle 1|$ correspond to outcomes $0$ and $1$, respectively. 
These quantum operators satisfy the completeness relation 
$$\sum_m M_m^\dagger M_m \equiv P_0^\dagger P_0 + P_1^\dagger P_1 = |0\rangle \langle 0| 0\rangle \langle 0| + |1\rangle \langle 1| 1\rangle \langle 1| = |0\rangle \langle 0| + |1\rangle \langle 1| = I.$$

The chance that $0$ is the measured value is quantified by
$p_0 = \langle q | P_0^\dagger P_0 | q \rangle = \langle q | 0\rangle \langle 0 | q \rangle = | \langle 0 | q \rangle |^2 = |a|^2.$ 
Similarly, the chance of observing $1$ is $p_1 = | \langle 1 | q \rangle |^2 = |b|^2$. Because $|a|^2 + |b|^2 = 1$, $p_0 + p_1 = 1$.

More generally, the measurement of an $N$D quantum state $|\psi\rangle$ in the 
orthonormal basis $\{|a_i\rangle\}_{i=1}^N$ is the *set of projection operators* 
$\{P_m = |a_m\rangle \langle a_m|\}$. The probability $p_m$ that the measurement of
state $|\psi\rangle$ yields basis state $|a_m\rangle$ is 
$p_m = | \langle a_m | \psi \rangle |^2$.

This property is called the **Born rule** and suggests that 
**quantum states are determined up to a complex phase pre-factor** 
$e^{i\theta}$, where the *phase* $\theta \in \mathbb{R}$, and $i=\sqrt{-1}$ is
the imaginary unit. Hence, the state $|\psi\rangle$ is equivalent to all states 
in this form $e^{i\delta} |\psi\rangle,\ \forall\theta\in\mathbb{R}$. 
The *global phase pre-factor* $e^{i\delta}$ is directly related to the 
*kime-phase* and its physical meaning beyond the reflection of repeated
measurement distribution fluctuations is still being investigated.

**Observables** are linear self-adjoint (Hermitian) operators, $K^\dagger = K$,
whose eigenvalues are all real, but are not necessarily norm-preserving. 

Let $|\kappa_j\rangle$ be eigenvectors of the observable $K$ with (real) eigenvalues 
$\kappa_j$, so that $K |\kappa_j\rangle = \kappa_j |\kappa_j\rangle$. 
Then, the observable $K = \sum_j \kappa_j |\kappa_j\rangle \langle \kappa_j|$,
where the states $|\kappa_j\rangle$ form an orthonormal basis. If 
$P_j = |\kappa_j\rangle \langle \kappa_j|$ is the *projection operator* onto the 
subspace spanned by $|\kappa_j\rangle$, the observable $K$ is linked to the quantum
measurement by the complete set of projection operators $\{P_j\}$ with outcomes $\{\kappa_j\}$. This quantum measurement is called the measurement of the observable $K$.

For a quantum state $|\psi\rangle$, the *expectation value* of the result 
of the measurement of $K$ is
$$\mathbb{E}_\psi[K] \equiv \langle K \rangle = \sum_j p_j \kappa_j = \sum_j | \langle \kappa_j | \psi \rangle |^2 \kappa_j,$$

where $\langle \cdot \rangle$ denotes the expectation value of an observable and 
$p_j = | \langle \kappa_j | \psi \rangle |^2$ is the probability that the measurement
of $K$ in the state $|\psi\rangle$ yields the value $\kappa_j$. 
Since $| \langle \kappa_j | \psi \rangle |^2 = \langle \psi | \kappa_j \rangle \langle \kappa_j | \psi \rangle$, the expectation value is

$$\langle K \rangle = \sum_j \langle \psi | \kappa_j \rangle \langle \kappa_j | \psi \rangle \kappa_j = \langle \psi | \left( \sum_j \kappa_j |\kappa_j\rangle \langle \
\kappa_j| \right) | \psi \rangle = \langle \psi | K | \psi \rangle\ .$$

Thus, the expectation value of an observable $K$ in a quantum state 
$|\psi\rangle$ is $\langle \psi | K | \psi \rangle$. Hence, we can
compute the expectation values of observables without explicit knowledge of the
measurement probabilities.


...

# Notes

 - Minkowski 4D spacetime appears to be a very special Euclidean manifold. The reason for this mathematical-uniqueness is the topological characterization of 4D. For any dimension other than $4$, $n\in\mathbb{Z}\setminus\{4\}$, there are *no exotic smooth structures* on $\mathbb {R}^{n}$. That is, when $n\not= 4$, any smooth manifold
$\mathcal{M}$ *homeomorphic* to $\mathbb {R}^{n}$ is also *diffeomorphic* to 
$\mathbb {R}^{n}$. The existence of [exotic $\mathbb{R}^4$'s](https://en.wikipedia.org/wiki/Exotic_R4) is only possible in $4$D, e.g., Minkowski spacetime. Exotic $\mathbb{R}^4$'s are differentiable manifolds that are *homeomorphic* to $\mathbb{R}^4$, i.e., *shape preserving*, but not *diffeomorphic* to $\mathbb{R}^4$, i.e., *non smooth manifolds*. These strange constructs only exist in $4$D.
 - The uniqueness of the existence of exotic $\mathbb{R}^4$ manifolds is directly related to the [Clay Mathematics Institute *Millennium Prize Problems*](https://en.wikipedia.org/wiki/Millennium_Prize_Problems) and the [GTR, Yang–Mills Theory, Gauge Theory, Quantum electrodynamics (QED), Symmetry Groups, and the physics Standard Model](https://en.wikipedia.org/wiki/Gauge_theory). In  gauge theory terms, the *Standard Model* unifies the description of $3$ of the fundamental forces - electromagnetism, weak nuclear interactions, and strong nuclear interactions. The [Standard Model](https://en.wikipedia.org/wiki/Standard_Model) can be thought of as a *non-Abelian gauge theory* with the symmetry group $U(1) \times SU(2) \times SU(3)$ and a total of $12$ gauge bosons - $1$ photon, $3$ weak bosons, and $8$ gluons.
 - This also relates to the extension of [quantum field theory (QFT) from *operator-valued functions* to *operator-valued distributions*](https://plato.stanford.edu/entries/quantum-field-theory/). An *operator-valued distribution* is a functional $\Psi$ from the Schwartz space of test-functions to bounded linear operators on the Hilbert space $\mathcal{H}$ having the following property, $\forall\ \eta, \xi \in\mathcal{H}\ \Longrightarrow\ \langle\eta|\Psi|\xi\rangle$ is a *tempered distribution*. A tempered distribution $f:\mathbb{R}\to\mathbb{C}$ is a functional assigning a complex-number to each test function $\phi$, i.e., $\langle f,\cdot\rangle(\phi)\equiv \langle f,\phi\rangle=\int_{\mathbb{R}}f^*(\theta)\ \phi(\theta)\ d\theta$.
 
 

# References

 - [Introduction to Applied Nuclear Physics (OCW MIT)](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/).

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