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This TCIU section 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).
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 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:
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')
pThe 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 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
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.
In Chapter 3 (Time Complexity), we show examples of 1D and 2D matter waves. Below is an example of a complex wave function of constant amplitude.
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
pIn 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
\[i\hbar {\frac {\partial }{\partial t}}\psi (x,t)=\left[ \underbrace{-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}}_{kinetic}+\underbrace{V(x,t)}_{potential}\right]\psi (x,t)\ ,\]
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 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 \(h\) 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. 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 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.
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 \((x,t)\).
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, \(p=1 - 0.125 = 0.875\).
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 \([0,L/4]\)
\[\begin{align} P(x\in [0, L/4]) &= \frac{2}{L} \int_0^{L/4} \sin^2\left (\frac{\pi}{L} x\right ) dx = \frac{\pi -2}{4\pi}\approx 0.090845\ . \end{align} \]
# 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 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.
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 \(x\) 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.
# 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
\[P([x,x+\Delta x], t)=|\psi(x,t)|^2 \Delta x\ .\]
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 \(C=\sqrt{\frac{2}{L}}\).
The classical mechanics solutions to the 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
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 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 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.
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 \(x=0\).
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 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 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.
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 \(|\psi(t)\rangle = \hat{U}( t ) |\psi(0)\rangle\).
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.
\[\begin{align*} 1 =\langle \psi |\psi \rangle &= \int_0^L \psi^* (x) \psi(x)\ dx = \int_0^L \left(C e^{+i\omega t} \sin \, \dfrac{\pi x}{L}\right) \left(C e^{-i\omega t} \sin \, \dfrac{\pi x}{L}\right) dx \\[4pt] &= C^2 \int_0^L \sin^2 \, \dfrac{\pi x}{L}\ dx = C^2 \dfrac{L}{2}\ \ \Longrightarrow\ \ C = \sqrt{\dfrac{2}{L}}. \end{align*} \]
\[\begin{align*}\langle x \rangle &= \int_0^L \psi^* (x) x \psi(x) \ dx = \int_0^L \left(C e^{+i\omega t} \sin \, \dfrac{\pi x}{L}\right) x \left(C e^{-i\omega t} \sin \, \dfrac{\pi x}{L}\right) \ dx \\[4pt] &= C^2 \int_0^L x \, \sin^2 \, \dfrac{\pi x}{L}\ dx = C^2 \dfrac{L^2}{4} \ \ \Longrightarrow \ \ \langle x \rangle = \dfrac{L}{2}. \nonumber \end{align*} \nonumber\]
\[\underbrace{\hat{p}}_{operator}|\psi(x)\rangle \equiv -i\hbar\frac{d}{dx} \psi(x) = -i\hbar \dfrac{d}{dx} \left (C e^{-i\omega t}\sin \frac{\pi x}{L} \right ) = - i\frac{C\ \hbar}{2L} e^{-i\omega t} \cos\frac{\pi x}{L}.\]
\[\begin{align*} \Longrightarrow \ \ \langle p \rangle &= \int_0^L \underbrace{\left(C e^{+i\omega t}\sin \frac{\pi x}{L}\right)}_{\underbrace{\langle\psi(x)|}_{bra}} \underbrace{\left(-i \frac{Ch}{2L} e^{-i\omega t} \cos\frac{\pi x}{L}\right)}_{\hat{p}\underbrace{|\psi(x)\rangle}_{ket}} \ dx \\[4pt] &= -i \dfrac{C^2 \hbar}{4L} \int_0^L \underbrace{\sin \frac{2\pi x}{L}}_{odd\ function\\ about\ x=\frac{L}{2}}\ dx =0\ . \end{align*}\]
\[\begin{align} \underbrace{-\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}}_{\hat{K}} \psi (x) &= - \dfrac{\hbar^2}{2m} \dfrac{d^2}{dx^2} \left (C e^{-i\omega t} \, \sin \, \dfrac{\pi x}{L}\right ) \\[4pt] &= - \dfrac{\hbar^2}{2m} C\ e^{-i\omega t} \frac{d^2}{dx^2} \, \sin \, \dfrac{\pi x}{L} = \frac{C\ \hbar ^2}{8mL^2} e^{-i\omega t} \, \sin \, \dfrac{\pi x}{L}. \end{align}\]
\[\begin{align*} \Longrightarrow \langle K \rangle &= \int_0^L \left( C\ e^{+i\omega t} \sin \frac{\pi x}{L}\right) \left(\dfrac{C\ \hbar^2}{8mL^2} e^{-i\omega t} \sin \frac{\pi x}{L}\right) \ dx \\[4pt] &= \dfrac{C^2\ \hbar^2}{8mL^2} \int_0^L \sin^2 \, \dfrac{\pi x}{L}\ dx = \dfrac{C^2\ \hbar^2}{8mL^2} \dfrac{L}{2} = \dfrac{\hbar^2}{8mL^2}. \end{align*}\]
Summarizing:
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.
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.
The fourth postulate of QM states:
\[i\hbar \frac{\partial |\psi\rangle}{\partial t} = H(t)| \psi\rangle\ ,\] 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 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 \(\psi(x,0)\).
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, the 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.
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.
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 \(U^\dagger(t) U(t)=e^{i\frac{Ht}{\hbar}}\ e^{-i\frac{Ht}{\hbar}}=\mathbb{I}\).
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 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.
In the Schrodinger 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
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
\[\langle A\rangle _{t}=\langle \psi (t)|A_{{S}}(t)|\psi (t)\rangle .\]
The 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 \(H_{H}(t)\).
Examine the time-derivative of expectation …
The Heisenberg equation, 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 \(A_{H}(t)\equiv A(t)=e^{\frac{i}{\hbar }\ t\ H}\ A\ e^{-\frac {i}{\hbar}tH}\).
Recall from the TCIU distributions section 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), 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, they are elements of the dual space of the space of test functions, \(D(\Omega)\).
\[T_{\mathfrak{P}}(a\varphi_1 + b\varphi_2 )\equiv \langle \mathfrak{P}, a\varphi_1 + b\varphi_2 \rangle = \alpha\langle \mathfrak{P}, \varphi_1\rangle + \beta\langle \mathfrak{P},\varphi_2\rangle\ .\]
\[\varphi_n\underset{n\to\infty}{\stackrel{D}\longrightarrow}\varphi \ \ \Longrightarrow T_{\mathfrak{P}}(\varphi_n)\underset{n\to\infty}{\longrightarrow} T_{\mathfrak{P}}(\varphi)\ .\]
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 that the class of test functions operated on by distributions can vary. Examples include:
\[\mathcal{S}(\mathbb{R^n})=\{f\in C^{\infty}(\mathbb{R^n}): ||f||_{\alpha,\beta}<\infty \,, \forall \alpha, \beta\} \tag{1}\]
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.
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 (\(\theta=0\))
\[\langle f(\theta)\ \ell(e^{i\theta}),\ \phi(\theta) \rangle = \langle \delta(\theta),\ f(\theta)\ \phi(\theta)\rangle = f(0)\ \phi(0).\]
A more nuanced example of a generalized phase function is the Cauchy principal value distribution, \(\ell(\theta)\equiv\mathcal{P}\left (\frac{1}{\theta}\right )\), 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, which allows us to estimate the value of the integral near the origin, \[\int_{|\theta|\lt\epsilon}{\frac{\phi(\theta)}{\theta}\ \ d\theta},\ \ \ \lim_{\epsilon\to 0}{\frac{\phi(\theta)}{\theta}} \underbrace{=}_{\text{L'Hopital}} \lim_{\epsilon\to 0}{\frac{\phi'(\theta)}{\theta'}}= \lim_{\epsilon\to 0}{\frac{\phi'(\theta)}{1}}=\lim_{\epsilon\to 0}\phi'(\theta)\ .\]
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, \(\langle \ell,\ \phi \rangle =0\).
Let’s consider the action of the Laplace distribution, aka, double exponential 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 ….
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, 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. 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 \(\mathbb{P}\mathcal{H}\) 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, 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, will also be a Dirac \(\delta\) distribution. Therefore, the spacetime fields are operator-valued distributions, rather than functions.
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
Definition 3: A kime continuous unitary group \(\{U_{\kappa}\}_{\kappa\in\mathbb{C}}\) is a collection of unitary operators such that
The Stone - von Neumann theorem establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \(\mathcal {H}\) and one-parameter families of continuous unitary operators \(\{U_{t}\}_{t\in \mathbb {R}}\):
\[(1)\ \ \forall t_{o}\in \mathbb {R} ,\ \psi \in {\mathcal {H}}:\ \lim _{t\to t_o}U_{t}(\psi )=U_{t_o}(\psi),\] \[(2)\ \ \forall s,t\in \mathbb {R} :\ U_{t+s}=U_{t}U_{s}\ .\]
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, as well as this Math-Physics Stack-Exchange example of defining strongly-continuous projective unitary representations of \(\mathbb{C}\) on \(L^2(\mathbb{R},dx)\) by
\[\mathbb{C}\cong \mathbb{R}^2\ni (a,b)\mapsto U(a,b) := e^{i(\overline{aX - i b \frac{d}{dx}})},\]
\[\mathbb{C}\cong \mathbb{R}^2\ni (a,b)\mapsto V(a,b) := e^{i(\overline{aX - i 2b \frac{d}{dx}})}.\]
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.
…
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), \(\kappa\in\mathbb{C}\).
In 2012, Matthew Pusey, Jonathan Barrett and Terry Rudolph proved the PBR Theorem, 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? 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.
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:
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, 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?
Two of the critical assumptions of the PBR theorem are:
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? 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 in relation to the spacekime interpretation, and the SOCR DNN Exchangeability, Probability Symmetry, Invariance, and Equivariance paper by Yueyang Shen et al..
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:
Definitions:
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:
\[\langle Q\rangle =\sum _{j}p_{j}\langle \psi _{j}|Q|\psi _{j}\rangle =\sum _{j}p_{j}\operatorname {tr} \left(|\psi _{j}\rangle \langle \psi _{j}|Q\right)=\operatorname {tr} \left(\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|Q\right)=\operatorname {tr} (\rho\ Q)\ .\] 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 \(|\psi_i\rangle,\ i\in\{1,2\}\).
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.
Following the notation used in this article, “An introduction to quantum computing for statisticians and data scientists”, 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 \[{\text{Observation}}=\frac{\overbrace{M_m |\psi\rangle}^{measurement\ output}} {\underbrace{\sqrt{\langle \psi | M_m^\dagger M_m | \psi \rangle}}_{normalization\ const.}}.\]
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.
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).
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 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.
\[|\psi\rangle = \sum_{i=0}^{N-1} \psi_i |a_i\rangle = \left( \sum_{j=0}^{N-1} |b_j\rangle \langle b_j| \right) \left( \sum_{i=0}^{N-1} \psi_i |a_i\rangle \right) = \sum_{j=0}^{N-1} \left( \sum_{i=0}^{N-1} \psi_i \langle b_j | a_i \rangle \right) |b_j\rangle = \sum_{j=0}^{N-1} \psi_j |b_j\rangle\ ,\] 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 \(\{|b_j\rangle\}\).
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.
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