#  Measurement of a quantum system with a classical apparatus using ensembles
on configuration space

Marcel<EMAIL_ADDRESS>and Sebastian Ulbricht1

(1 _Physikalisch–Technische Bundesanstalt, D–38116 Braunschweig, Germany_
(Dated: ) )

###### Abstract

Finding a physically consistent approach to modelling interactions between
classical and quantum systems is a highly nontrivial task. While many
proposals based on various mathematical formalisms have been made, most of
these efforts run into difficulties of one sort or another. One of the first
detailed descriptions was given by Sudarshan and his collaborators who,
motivated by the measurement problem in quantum mechanics, proposed a Hilbert
space formulation of classical-quantum interactions which made use of the
Koopman-von Neumann description of classical systems. Here we use the approach
of ensembles on configurations space to give a detailed account of a classical
apparatus measuring the position of a quantum particle that is prepared in a
superposition of two localized states. We show that the probability of the
pointer of the classical apparatus is left in a state that corresponds to the
probability of the quantum particle. A subsequent observation of the pointer
leads to an update of its probability density. From this we can obtain
information about the position of the quantum particle, leading to an update
of its wave function. Since this formalism incorporates uncertainties and
finite measurement precision, it is well suited for metrological applications.
Furthermore, it resolves fundamental issues that appear in the case of a
quantum description of the apparatus.

##  1 Introduction

There are a number of good reasons for developing theoretical models that
allow for interactions between quantum and classical systems.

For many calculations, it is convenient or even necessary to describe parts of
a complex system classically for physical or computational reasons, while the
rest of the system is described quantum mechanically. Ideally, such models
should be free of inconsistencies and should allow for well defined
approximation schemes.

Mixed classical-quantum models also arise when questions of fundamental
physics are considered. For some systems, e.g., such that involve gravity,
there is no full quantum theory yet. A detailed description of classical
gravity (within the framework of general relativity) interacting with quantum
matter (described by quantum field theory) can provide insight that helps in
the search for a quantum theory of gravity. However, if gravity does not have
to be quantized [1, 2, 3, 4, 5], it becomes essential to find a consistent
theory that couples quantum fields to a classical gravitational field.

Another example from foundations of physics comes from measurement theory: in
the Copenhagen interpretation of quantum mechanics, a measuring apparatus is
described in classical terms [6, 7]. This implies a coupling (typically left
undefined) between the quantum system and the classical apparatus.

Due to the many applications in applied and fundamental physics, various
approaches describing mixed classical-quantum systems have been developed. The
first general model was worked out by E. C. G. Sudarshan [8], who proposed a
Hilbert space formulation incorporating the Koopman-von Neumann description of
classical systems. In this paper, we use instead the formalism of ensembles on
configuration space [9, 10, 11] to study the measurement of a quantum system
with a classical apparatus. This approach brings to the fore a number of
interesting conceptual issues, such as the question of how “classicality” may
be defined, the role of the environment, and the connection between the state
of the measuring apparatus and the collapse of the wave function [9].

The model discussed in this paper describes a simple type of measurement.
However, the techniques that we use can also be applied to the description of
complex measurement processes and provide a blueprint for modeling
measurements carried out with more realistic devices. In particular, they
allow us to consider the finite precision in the preparation of an initial
quantum state and the limitations on the calibration of the classical
apparatus. Moreover, it is possible to keep track on the evolution of
measurement uncertainties. This makes our approach highly suitable for the
analytical discussion of non-idealized quantum particle position measurements
and, thus, for modern high precision quantum metrology.

Our paper is organized as follows. In order to present a consistent
description of mixed classical-quantum systems, we first give an introduction
to the formalism of ensembles on configuration space in Sec. 3. After that, in
Sec. 4, we describe the measurement procedure in three steps: the initial
condition, the system during the interaction of the classical apparatus and
the quantum particle, and the evolution of the classical pointer after the
interaction. These steps are described in detail in the Secs. 5 and 6. Finally
we consider the measurement of the classical pointer, from which the
information about the quantum particle position is obtained in Sec. 7. A
discussion of the implications of our analysis for the discussion of
classical-quantum systems and for metrological application can be found in
Sec. 8.

Before we start with the main task of this publication, to describe the
measurement of a quantum particle by a classical apparatus, we discuss briefly
some of the difficulties of quantum measurement theory in the next section.

##  2 Some aspects of the quantum measurement problem

The earliest formal model of measurement goes back to von Neumann, who started
with a fully quantum mechanical description of the measurement process and of
both the quantum particle and the apparatus. He then went on to describe the
unitary evolution of the quantum system that results when the coupling to the
measuring apparatus is strong [12]. The apparatus is left in a state that can
be described by a reduced density matrix. In this formalism, however,
difficulties arise when a macroscopic measurement apparatus is used: In this
case, the device ends up with properties that are not compatible with
observations [13, 14, 7]. As Roland Omnès points out [7], this issue arises
due to the presence of off-diagonal elements of the density matrix: “A linear
superposition of macroscopically distinct states of the measuring apparatus is
obtained” and “The theory predicts possible interferences between
macroscopically different states of the apparatus that are associated with
different data”.

While there are a number of different proposals for dealing with this issue,
there is no generally accepted resolution [13]. In von Neumann’s original
formulation, it is assumed that the unitary evolution of the quantum system in
the moment of measurement is replaced by a different type of evolution, the
collapse of the wave function , that is discontinuous, noncausal and
nonunitary.

In this paper, we consider a description of quantum measurement in which a
classical measuring apparatus is coupled to a quantum system using the
formalism of ensembles on configurations space [9]. In this approach, the
measurement of the quantum observable can be modeled via an interaction with a
strictly classical measuring apparatus, followed by a direct measurement of
the classical configuration. Our model of measurement is not subject to the
problem discussed above, since the final state of the apparatus is described
in terms of a classical mixture rather than a quantum superposition of
macroscopic states. The measuring apparatus ends up in a particular state and
is subject to a given uncertainty. This model supports the assumption of the
Copenhagen interpretation that the pointer of a measurement device should be
regarded as a truly ‘classical’ object.

By extending the description of the measurement process to include the
inevitable interaction of the classical measuring device with the environment,
the ensemble that describes the apparatus and quantum system effectively
decoheres into a proper mixture of noninteracting independent ensembles. Thus,
each element of the mixture after measurement satisfies the condition of
_strong separability_ , ruling out any possibility of nonlocal energy flow and
nonlocal signaling [9]. It is important to point out that the role of the
environment in this context differs from the one that we are accustomed to in
decoherence theory [15]. Instead, it is analogous to the role that the
environment plays in the description of a cloud chamber track; that is, the
environment selects a particular trajectory out of all the possible ones,
which are consistent with the initial conditions [16, 17, 18].

##  3 Interacting classical-quantum systems: the approach of ensembles on
configuration space

Finding a physically consistent approach that models interactions between
classical and quantum systems is a highly nontrivial task. While many
proposals based on various mathematical formalisms have been made, most of
these efforts run into difficulties of one sort or another. One of the first
detailed descriptions of mixed classical-quantum systems was given by
Sudarshan and collaborators. Motivated by the measurement problem in quantum
mechanics, they proposed a Hilbert space formulation of classical-quantum
interactions, which made use of the Koopman-von Neumann description of
classical systems [8, 19, 20, 21]. In recent years, this work has been further
developed and extended [22, 23, 24, 25, 26].

Other common descriptions of classical-quantum interactions include the mean
field approach, in which classical observables appear as parameters in a
quantum Hamiltonian operator [27, 28, 29], and the trajectory approach, which
relies on modifying the de Broglie–Bohm formulation of quantum mechanics [30,
31, 32]. In this paper, we will focus on an alternative description, the
configuration ensemble approach, which is based on a general formalism for
describing physical systems in terms of ensembles evolving on a configuration
space [9]. While each of these approaches have proven useful for given
applications (e.g., to provide an approximate description of a system where it
is desirable to model some components classically while the rest of the system
is modeled quantum mechanically), the question of their applicability in a
more general sense still needs to be evaluated. In particular, there is a lack
of models that consider experiments in a realistic fashion.

Our description of interacting classical and quantum systems is formulated
using ensembles on configuration space [9, 10, 11]. The basic idea of the
approach is simple: physical systems are described by ensembles on
configuration space, with equations of motion derived from an action
principle. The system is then described by an ensemble of configurations with
probability density $P(x,t)$, where $P(x,t)\geq 0$ and $\int dx\,P(x,t)=1$. To
derive equations of motion, we introduce an _ensemble Hamiltonian_
$\tilde{H}[P,S]$, where $S$ is canonically conjugate to $P$. In this
Hamiltonian formulation, the evolution of the system is given by the equations

$\frac{\partial P}{\partial
t}=\left\\{P,\tilde{H}\right\\}_{PB}=\frac{\delta\tilde{H}}{\delta
S},~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\partial S}{\partial
t}=\left\\{S,\tilde{H}\right\\}_{PB}=-\frac{\delta\tilde{H}}{\delta P},$ (1)

where $\\{A,B\\}_{PB}$ is the Poisson bracket of two functionals of the
canonical variables $P$ and $S$.

For the non-relativistic description of classical particle, the ensemble
Hamiltonian is given by

$\displaystyle{H}_{C}[P,S]$ $\displaystyle=$ $\displaystyle\int
dx\,P\left[\frac{|\nabla S|^{2}}{2m}+V(x)\right].$ (2)

In order to describe the dynamics of a quantum particle, an additional term is
added to the classical ensemble Hamiltonian $H_{C}$:

$\displaystyle{H}_{Q}[P,S]$ $\displaystyle=$
$\displaystyle{H}_{C}[P,S]+\frac{\hbar^{2}}{4}\int
dx\,P\,\frac{1}{P^{2}}\frac{(\nabla P)^{2}}{2m}.$ (3)

This additional term is proportional to the Fisher information of the
probability density $P$ [33]. It gives rise to the term proportional to
$\hbar^{2}$ that appears in Eq. (4), commonly known as the quantum potential
due to the role assigned to it in Bohmian mechanics [14].

Evaluating the equations of motion of Eq. (1) using the quantum ensemble
Hamiltonian ${H}_{Q}[P,S]$, we obtain

$\frac{\partial P}{\partial t}+\nabla\cdot\left(P\frac{\nabla
S}{m}\right)=0,~{}~{}~{}\frac{\partial S}{\partial t}+\frac{|\nabla
S|^{2}}{2m}+V+\frac{\hbar^{2}}{2m}\frac{\nabla^{2}P^{1/2}}{P^{1/2}}=0,$ (4)

which were first derived by Madelung in 1926 [34]. In the case of $\hbar=0$,
these equations reduce to the classical equations of motion calculated from
${H}_{C}[P,S]$. The first equation in Eq. (4) is a continuity equation, the
second equation is the classical Hamilton-Jacobi equation when $\hbar=0$ and a
modified Hamilton-Jacobi equation when $\hbar\neq 0$. Introducing the complex
canonical transformation $\psi:=\sqrt{P}~{}e^{iS/\hbar}$,
$\psi^{*}:=\sqrt{P}~{}e^{-iS/\hbar}$, Eq. (4) takes the form of the
Schrödinger equation

$i\hbar\frac{\partial\psi}{\partial
t}=\frac{-\hbar^{2}}{2m}\nabla^{2}\psi+V\psi.$ (5)

An advantage of the formalism introduced above is that it treats quantum and
classical particles on an equal footing, only distinguished by the choice of
ensemble Hamiltonian. This feature allows the formalism to be extended in a
natural and consistent way to mixed quantum-classical systems.

A hybrid ensemble Hamiltonian where the classical and quantum particles
interact via a potential term $V(q,x,t)$ is given by

${H}_{QC}[P,S]=\int
dq\,dx\,P\,\left[\frac{|\nabla_{x}S|^{2}}{2M}+\frac{|\nabla_{q}S|^{2}}{2m}+\frac{\hbar^{2}}{4P^{2}}\frac{|\nabla_{q}P|^{2}}{2m}+V(q,x,t)\right].$
(6)

Here $q$ denotes the configuration space coordinate of a quantum particle of
mass $m$ and $x$ that of a classical particle of mass $M$. The equations of
motion for $P$ and $S$ derived from ${H}_{QC}$ are

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle-\nabla_{q}.\left(P\frac{\nabla_{q}S}{m}\right)-\nabla_{x}.\left(P\frac{\nabla_{x}S}{M}\right),$
(7a) $\displaystyle\frac{\partial S}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{|\nabla_{q}S|^{2}}{2m}-\frac{|\nabla_{x}S|^{2}}{2M}-V+\frac{\hbar^{2}}{2m}\frac{\nabla_{q}^{2}P^{1/2}}{P^{1/2}}.$
(7b)

In this way, mixed quantum-classical states are described in a unified way in
terms of ensembles in configuration space, ensuring mathematical compatibility
of the quantum and classical sectors. Furthermore, in this unified approach
interactions between quantum and classical systems can be easily incorporated.

As one would expect, an ensemble for a system that consists of a classical and
a quantum particle is defined to be independent if the conditions

$P(x,q)=P_{C}(x)P_{Q}(q),\qquad\quad S(x,q)=S_{C}(x)+S_{Q}(q)$ (8)

are satisfied [9]. We will assume this property in Sec. 4 when defining
initial conditions for a measurement process.

While the interpretation of $P$ as a probability density is straightforward,
there are some subtle issues concerning its canonical conjugate $S$. The
physical interpretation of $S$ can be derived from its role in defining local
energy and momentum densities. To make this clear, we notice that the ensemble
Hamiltonians satisfy the relation ${H}[\lambda P,S]=\lambda{H}[P,S]$. Taking
the derivative of this relation with respect to $\lambda$, using the Leibniz
rule and evaluating the result at $\lambda=1$ leads to

${H}=\int dx\,P\frac{\delta{H}}{\delta P}=-\int dx\,P\frac{\partial
S}{\partial t}=-\langle\partial S/\partial t\rangle,$ (9)

which shows that $-\partial S/\partial t$ is a local energy density.

Furthermore, one can check that $\int dx\,P\nabla S$ is the canonical
infinitesimal generator of translations, since spatial variations of $P$ and
$S$ can be expressed as

$\displaystyle\delta P(x)$ $\displaystyle=$
$\displaystyle\delta\textbf{x}\cdot\left\\{P,\int dx\,P\nabla
S\right\\}_{PB}=-\delta\textbf{x}\cdot\nabla P,$ (10a) $\displaystyle\delta
S(x)$ $\displaystyle=$ $\displaystyle\delta\textbf{x}\cdot\left\\{S,\int
dx\,P\nabla S\right\\}_{PB}=-\delta\textbf{x}\cdot\nabla S,$ (10b)

under action of the generator. Therefore $P\nabla S$ can be considered a local
momentum density. These results are generally valid; i.e. they hold true for
classical, quantum and mixed systems.

Having introduced the formalism to describe classical-quantum systems with
ensembles in configuration space, we apply it in the next sections to the
specific scenario of a classical apparatus measuring a quantum particle’s
position.

##  4 Model of a quantum measurement with a classical apparatus

Our aim is to describe a quantum measurement where the measurement apparatus
is described classically using the approach of ensembles in configuration
space, introduced in Sec. 3. There we showed that the dynamics of the system
and their interaction are given by ensemble Hamiltonians. Thus in this
specific case we will be concerned with a Hamiltonian that has three
contributions: the one describing the classical pointer, the one describing
the quantum particle, and the one describing the interaction between the
measuring apparatus and the quantum particle,

$\displaystyle H_{C}$ $\displaystyle=$ $\displaystyle\int
dqdx\,P\left\\{\frac{(\partial_{x}S)^{2}}{2M}\right\\},$ (11a) $\displaystyle
H_{Q}$ $\displaystyle=$ $\displaystyle\int
dqdx\,P\left\\{\frac{(\partial_{q}S)^{2}}{2m}+\frac{\hbar^{2}}{8m}\frac{(\partial_{q}P)^{2}}{P^{2}}\right\\},$
(11b) $\displaystyle H_{CQ}$ $\displaystyle=$ $\displaystyle\alpha(t)\int
dqdx\,P\left\\{\partial_{x}S\,q\right\\},$ (11c)

where $\alpha(t)$ is the time dependent interaction strength. This approach,
already introduced in Ref. [9], follows the basic idea of von Neumann for the
description of a strong measurement with the difference that the apparatus is
classical. The interaction term, Eq. (11c), couples the momentum of the
classical pointer to the position of the quantum particle and only plays a
role while the measuring device is turned on222In the Sudarshan model of mixed
classical-quantum systems [8], the interaction is given by the operator
$\hat{H}_{CQ}=-i\alpha(t)q\frac{\partial}{\partial x}$ that is non-observable,
since it contains the hidden variable
$\hat{\lambda}_{x}=-i\frac{\partial}{\partial x}$. The functional $H_{CQ}$ of
Eq. (11c), however, is an observable in the approach of ensembles on
configuration space.. During this short time it is assumed to be strong enough
that we can ignore the other contributions. Thus we distinguish between three
different phases of the measurement process: before, during, and after the
interaction,

$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle H_{C}+H_{Q}\hskip
100.00015ptt=0$ (12a) $\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle
H_{C}+H_{Q}+H_{CQ}\,\,\approx\,\,H_{CQ}\hskip 25.00003pt0<t\leq\varepsilon$
(12b) $\displaystyle H_{3}$ $\displaystyle=$ $\displaystyle H_{C}+H_{Q}\hskip
100.00015pt\varepsilon<t\leq t_{m}\,.$ (12c)

The temporal sequence of this phases also resembles the sequence of the next
sections: In Sec. 5 we will first introduce the initial condition at $t=0$ and
then describe the system during the interaction phase ($t\leq\varepsilon$).
The evolution of the classical pointer after the interaction ($t>\varepsilon$)
is the subject of Sec. 6. Finally we consider the measurement of the classical
pointer at the time $t_{m}$ and the information that we obtain about the
quantum system in Sec. 7.

##  5 Interaction between the classical measuring apparatus and the quantum
system

We make the standard assumption that the state of the measuring apparatus does
not depend on the state of the quantum system before the measurement takes
place. This means that the correlations between the quantum system and the
measuring apparatus only appear through the interaction during the measurement
process.

We assume furthermore that the quantum system is isolated from the environment
while the measurement takes place. This necessarily brings in an asymmetry
with respect to the measuring apparatus, which we assume satisfies some basic
requirements of classicality [9]. Classicality implies not only classical
equations of motion but also that an object that is described classically must
be _macroscopic_. This is a reasonable assumption, since in the opposite case
of _microscopic_ scales, objects are subject to quantum effects and any
classical description must be given up in favor of a quantum mechanical
treatment.

One important property of macroscopic objects is that they cannot be isolated
from the environment: a macroscopic object cannot avoid scattering photons and
other particles. Thus, for example, the pointer of a measuring device will be
continuously undergoing scattering processes, implying there are a large
number of external degrees of freedom. Another important property of
macroscopic objects is that they have a very large number of internal degrees
of freedom. When describing a classical measuring apparatus, we commonly
discuss only _one_ relevant degree of freedom, say the position of a pointer.
In reality, however, there is an enormous number of “irrelevant” degrees of
freedom, say $10^{23}$, which must be treated statistically if they are
modeled at all (usually they are simply neglected). Thus, classicality is not
just an assumption about mathematical equations of motion, it is also an
assumption about the physical model used for the description of a physical
object.

###  5.1 Initial conditions

Figure 1: Initial condition for the probability density of the mixed quantum-
classical system. The quantum particle is most likely localized around one of
the two positions $q=\pm q_{0}$. The classical pointer is located in the
center of the coordinate system. Both the classical pointer and the quantum
system are initially prepared with finite precision, represented by the
standard deviations $\sigma_{C}$ and $\sigma_{Q}$, respectively.

We assume that the two subsystems (classical and quantum) are independent at
$t=0$. The classical pointer’s location is described by a Gaussian centered at
$x=0$ and a standard deviation $\sigma_{C}$. This allows for a description of
measurements in which there is an initial uncertainty associated with the
pointer of the classical measuring apparatus, which is unavoidable in a
realistic description of a measurement. The quantum system is in a
superposition of two Gaussian wavefunctions localized at $q=\pm q_{0}$, each
with a standard deviation $\sigma_{Q}$. This describes a single quantum
particle that is most likely localized around one of two positions. We
consider the case where $\sigma_{C}<q_{0}$ so that the two peaks of the
initial quantum probability density are well separated, see Figure 1. Thus,
the initial probability density of the mixed classical-quantum system is given
by

$\displaystyle P(x,q,0)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\,\sigma_{C}}\,\exp\left(-\frac{x^{2}}{2\sigma_{C}^{2}}\right)$
(13)
$\displaystyle\times\;\frac{1}{2\sqrt{2\pi}\,\sigma_{Q}}\left[\exp\left(-\frac{(q-q_{0})^{2}}{2\sigma_{Q}^{2}}\right)+\exp\left(-\frac{(q+q_{0})^{2}}{2\sigma_{Q}^{2}}\right)\right]$
$\displaystyle=:$ $\displaystyle
P_{C}(x,0;\sigma_{C})\,P_{Q}(q,0;\sigma_{Q},q_{0}),$

where we have chosen the prefactors, such that $P(x,q,0)$ is normalized to
one.

A complete description of the initial conditions would also require a
specification of the initial condition for the variable $S$. However, it turns
out that an explicit expression of $S(x,q,0)$ will not be needed for the rest
of our analysis. We will omit it here and refer to Appendix B for those
interested in more details concerning the initial state and the time
evolution.

###  5.2 The hybrid system during interaction ($0<t\leq\varepsilon$)

In our model we assume a strong measurement, where during the brief
interaction time we can neglect the kinetic energy terms in the Hamiltonian.
During this time we only consider the changes in the state caused by the
interaction term $H_{CQ}$. Under this assumptions the equations of motion for
$P$ and $S$ become

$\frac{\partial P}{\partial t}=\frac{\delta H_{CQ}}{\delta
S}\quad,\quad\frac{\partial S}{\partial t}=-\frac{\delta H_{CQ}}{\delta P},$
(14)

which, given the explicit form of the interaction Hamiltonian Eq. (11c), lead
to the partial differential equations

$\frac{\partial P}{\partial
t}=-\alpha(t)\,q\,\partial_{x}P\quad,\quad\frac{\partial S}{\partial
t}=-\alpha(t)\,q\,\partial_{x}S\,.$ (15)

The general solution to this equations can be found by direct calculation and
is given by

$P(x,q,t)=P(x-qk({t}),q,0)\quad S(x,q,t)=S(x-qk({t}),q,0)\,,$ (16)

where the dimensionless parameter
$k(t)=\int_{0}^{t}dt^{\prime}\,\alpha(t^{\prime})$ is the integrated
interaction strength.

As can be seen from Eq. (16), the interaction does not change the shapes of
the functions $P$ and $S$ and only leads to the displacement $x\rightarrow
x-qk({t})$. This is consistent with the observation that $\int
dx\,P\left\\{\partial_{x}S\right\\}$ is the generator of translations of $x$,
as shown by Eqs. (10). Also notice that, as a result of the interaction, the
initially independent classical and quantum subsystems develop correlations
and can not be treated independently anymore.

In what follows, we focus on the probability density that describes the
location of the pointer at times $t=0$ and at $t=\varepsilon$; i.e., at the
beginning and at the end of the interaction. Initially, the probability
density of the pointer is given by

$P_{C}(x,0)=\int
dq\,P(x,q,0)=\frac{1}{\sqrt{2\pi}\,\sigma_{C}}\,\exp\left(-\frac{x^{2}}{2\sigma_{C}^{2}}\right)\,.$
(17)

At the end of the interaction with the quantum system, however, the statistics
of the pointer will be given by the integral of $P(x,q,t)$ from Eq. (16) over
the quantum coordinate $q$, evaluated at $t=\varepsilon$. A direct
calculation, which is presented in Appendix B, leads to the probability
density of the pointer in the form

$\displaystyle P_{C}(x,\varepsilon)$ $\displaystyle=$ $\displaystyle\int
dq\,P(x,q,\varepsilon)$ (18) $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2\pi}\left(\sigma_{C}^{2}+\sigma_{Q}^{2}k^{2}(\varepsilon)\right)^{1/2}}\;\left\\{\exp\left[-\frac{\left(x-q_{0}k(\varepsilon)\right)^{2}}{2\left(\sigma_{C}^{2}+\sigma_{Q}^{2}k^{2}(\varepsilon)\right)}\right]\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left.\exp\left[-\frac{\left(x+q_{0}k(\varepsilon)\right)^{2}}{2\left(\sigma_{C}^{2}+\sigma_{Q}^{2}k^{2}(\varepsilon)\right)}\right]\right\\}\,.$

As expected, this expression depends on the uncertainties of both the pointer
and the quantum system.

To interpret Eq. (18) more easily, it is useful to consider the case where the
initial uncertainty of the pointer $\sigma_{C}$ is small, such that the
relation $\sigma_{C}<<\sigma_{Q}\alpha(\varepsilon)$ holds. This simplifies
Eq. (18) to

$\displaystyle P_{C}(x,\varepsilon;\sigma_{C})\\!\\!\\!$
$\displaystyle\rightarrow$
$\displaystyle\\!\\!\\!\frac{1}{2\sqrt{2\pi}k({\varepsilon})\sigma_{Q}}\\!\left\\{\\!\exp\\!\left[-\frac{(x-k({\varepsilon})q_{0})^{2}}{2(k({\varepsilon})\sigma_{Q})^{2}}\right]\\!\\!+\\!\exp\\!\left[-\frac{(x+k({\varepsilon})q_{0})^{2}}{2(k({\varepsilon})\sigma_{Q})^{2}}\right]\\!\right\\}\,\,\,$
(19) $\displaystyle=$
$\displaystyle\\!\\!\\!P_{Q}(x,0;k({\varepsilon})\sigma_{Q},k({\varepsilon})q_{0})\,,$

where the expression for $P_{Q}$ is given in Eq. (13). Thus we see that after
interaction with the quantum system, the pointer ends up in a state with the
same probability density as the quantum system at $t=0$, but with the location
and standard deviation parameters scaled according to $\sigma_{Q}\rightarrow
k(\varepsilon)\sigma_{Q}$ and $q_{0}\rightarrow k(\varepsilon)q_{0}$. This is
precisely the behaviour that allows us to use the classical pointer as a
measurement device for the quantum particle position.

##  6 The evolution of the classical pointer after the interaction ends
($t>\varepsilon$)

We are now interested in the evolution of the pointer after the interaction
with the quantum system has come to an end. To give an analytical description
of the pointer at a time $t>\varepsilon$, we need to assume that the classical
measuring device has been calibrated and that we know its response to the
preceding interaction. This response of the apparatus is encoded in the
function $\alpha(t)$ which we introduced in Eq. (11c) as the time-dependent
strength of interaction, and now needs to be determined.

In what follows, we will consider an apparatus with a pointer that quickly
responds to the interaction with the quantum system at the very beginning of
the measuring process and moves with a constant velocity, afterwards. This
assumption is reasonable, since it resembles the behaviour of detectors in
real experiments.333To simplify the analysis, we will make one minor departure
from the operating principles of measuring apparatuses: Measuring devices
include damping forces that ensure that the pointer stops quickly once the
apparatus is turned off. Without damping or friction, the pointer would
continue at a constant velocity even after the interaction stops. We will
assume here that there is no damping, so that the pointer does indeed continue
moving at a constant velocity after time $t=\varepsilon$. We are aware that
this is an unrealistic assumption but it does simplify the analysis
considerably. We ask the reader to consider then that this how our measuring
apparatus works! (It would be rather straightforward to add a damping force to
the analysis, but little would be gained by doing that.) In our model, this
means that the strength of the interaction $\alpha(t)=\lambda$ for
$0<t<\varepsilon$ is a non-vanishing constant that depends on the
characteristics of the detector and presumably needs to be determined by
calibration. This is equivalent to setting $k({t})=\lambda t$. As can be seen
from Eq. (16), this implies that the classical pointer moves with a constant
velocity $\lambda q$ during the interaction time.

###  6.1 Statistics of the classical pointer as the interaction comes to an
end

Let us now consider the case where the initial uncertainty of the pointer
$\sigma_{C}$ is small enough so that $\sigma_{C}<<\sigma_{Q}\lambda t$. Then,
as the system approaches the time when the interaction is turned off, the
statistics of the classical pointer is given by the probability density

$\displaystyle P_{C}(x,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2\pi}\sigma_{Q}\lambda
t}\exp\left[-\frac{\left(x-\lambda
q_{0}t\right)^{2}}{2\sigma_{Q}^{2}\lambda^{2}t^{2}}-\frac{\left(x+\lambda
q_{0}t\right)^{2}}{2\sigma_{Q}^{2}\lambda^{2}t^{2}}\right].\qquad$ (20)

This expression shows nicely how the initially Gaussian distribution for the
classical pointer at $t=0$ splits up into two Gaussians that move with
constant velocities $\pm\lambda q_{0}$ while they broaden with a standard
deviation $\sigma_{Q}\lambda t$, as can be seen in Fig. 2. However, the
probability density Eq. (20) only contains partial information: While it gives
the correct statistics for measurements carried out on the pointer, it
contains no information about correlations between $x$ and $q$. For a complete
description of these relations we would need to evolve the joint probability
distribution $P(x,p,t)$ using the Hamiltonian equations of motion after
choosing an appropriate function $S(x,q,t)$. Such an complete description,
however, is not required, since after the interaction ceases, only the
classical pointer is accessible. For times $t>\varepsilon$, both the classical
pointer and the quantum particle independently obey the equations of motion of
free particles. Thus, for our purposes it is enough to focus on the time
development of the probability density $P_{C}(x,t)$ and how to extract
information about the quantum particle position from a measurement of the
pointer’s motion.

Figure 2: Time evolution of the classical (red) and the quantum subsystem
(blue) after interaction. Both systems propagate freely, after the interaction
strength $\alpha(t)$, that determines the velocity of the pointer, is set to
zero. One can see, that the initial Gaussian probability distribution for the
pointer splits up and after some time resembles the bimodal distribution of
the quantum system. This phase lasts until the pointer is measured and both
subsystems are updated as described in Sec. 7.

###  6.2 The motion of the pointer for $t>\varepsilon$: Two complementary
descriptions

In this section we will give two equivalent but complementary descriptions of
the state of the pointer after the time of interaction, each associated with a
particular interpretation of measurement. The statistics of the pointer’s
position is defined in a unique way by the time dependent probability density
$P_{C}(x,t)$. We show, however, that it is possible to express this time
dependent probability density in different ways using the concept of classical
mixtures, which leads to different but equivalent descriptions of the state of
the pointer corresponding to different types of measurements.

A brief review on classical mixtures is given in Appendix C. In what follows,
we will first show how the state of the pointer can be described as a
classical mixture in terms of pairs of functions $P_{C|Q}(x,t;q)$,
$S_{C|Q}(x,t;q)$ and a density for the parameter $q$, where $q$ is both a
label for the elements of the classical mixture and the coordinate of the
quantum particle. Afterwards, we will discuss a second description, where the
state is given in terms of a pair of functions $P_{C}(x,t)$, $S_{C}(x,t)$
without introducing a mixture of different elements.

####  6.2.1 Description of the state of the pointer in terms of a classical
mixture

As we have seen above, the assumption that the detector has a response
$k(t)=\lambda t$ implies that the pointer moves with a constant velocity while
the interaction takes place. Since the position of the quantum particle is
uncertain, the velocity of the pointer, given by $q\lambda$, will also be
uncertain during the interaction and after the interaction ends. The pointer
then will continue moving with a constant velocity $q\lambda$ that depends on
the probability $P_{Q}(q,t)$ of finding the quantum particle particle at the
location $q$. Thus, the statistics of the pointer after interaction can be
formally described in terms of a _classical mixture_. To introduce these
mixtures, we use the product rule of probability theory
$P(x,q,t)=P_{C|Q}(x|q,t)P_{Q}(q,t)$, which allows us to write a probability as
the product of a conditional probability $P_{C|Q}(x|q,t)$ and a prior
probability $P_{Q}(q,t)$. With this in hand, we can express the classical
probability $P_{C}(x,t)$ by

$\displaystyle P_{C}(x,t)$ $\displaystyle=$ $\displaystyle\int
dq\,P_{C|Q}(x|q,t)\,P_{Q}(q,t),$ (21)

where each element of the mixture $P_{C|Q}(x|q,t)$ describes a classical
particle moving with velocity $\lambda q$. To make this clear, we can bring
$P_{C}(x,t)$ from Eq. (20) into the form

$\displaystyle P_{C}(x,t)$ $\displaystyle=$ $\displaystyle\int
dq\,\delta(x-\lambda qt)$ (22)
$\displaystyle\times\frac{1}{2\sqrt{2\pi}\sigma_{Q}}\;\left[\exp\left(-\frac{(q+q_{0})^{2}}{2\sigma_{Q}^{2}}\right)+\exp\left(-\frac{(q-q_{0})^{2}}{2\sigma_{Q}^{2}}\right)\right],$

where we find $P_{C|Q}(x|q,t)=\delta(x-\lambda qt)$ in direct comparison with
Eq. (21).444The probability that the particle has velocity $\lambda q$ is
given by the second line of Eq. (22). By comparison with Eq. (21) one could
expect the appearance of a time dependent probability density for the
propagating quantum particle. The given expression, however, corresponds to
the initial condition $P_{Q}(q,0;\sigma_{Q},q_{0})$ for the quantum particle
from Sec. 5.1. This is a consequence of the assumption that during the short
time of interaction $0<t<\varepsilon$ the quantum system is “frozen in time”
and, therefore, is described by its initial condition. In our model of a
strong measurement this is valid, since the time $\varepsilon$ is considered
to be smaller than any typical time scale of the dynamics of the quantum
particle.

To complete the description of the state of the pointer, we now consider the
appropriate solutions of the Hamilton-Jacobi equation. To each element of the
mixture $P_{C|Q}(x|q,t)$ we can assign a corresponding function

$S_{C|Q}(x,t;q)=-\frac{M\lambda^{2}q^{2}}{2}t+M\lambda qx$ (23)

that is labeled by $q$. This representation will be useful later when we
consider a single measurement of the position of the pointer.

Notice that $S_{C|Q}(x,t;q)$ cannot be any arbitrary solution of the Hamilton-
Jacobi equation, as the equations of motion of the ensemble have to be
satisfied with the given choice of $P_{C|Q}(x|q,t)$. In Appendix C we check
that the functions $P_{C|Q}(x|q,t)$, $S_{C|Q}(x,t;\lambda q)$ satisfy the
Hamilton-Jacobi equation and continuity equations

$\frac{\partial S_{C|Q}}{\partial t}+\frac{1}{2M}\left(\frac{\partial
S_{C|Q}}{\partial x}\right)^{2}=0,\quad\frac{\partial P_{C|Q}}{\partial
t}+\frac{\partial}{\partial x}\left(P_{C|Q}\frac{1}{M}\frac{\partial
S_{C|Q}}{\partial x}\right)=0,\qquad$ (24)

for an arbitrary choice of the parameter $q$.

In Sec. 7 we will use the description of the state by classical mixtures to
describe a measurement of the pointer that selects one of the elements of the
mixture and leaves the pointer in that state. Before we do that, however, we
will briefly discuss the interpretation of the pointer’s probability density,
when no classical mixtures are introduced.

####  6.2.2 Description of the state of the pointer without introducing a
classical mixture

The statistics of the pointer as given by Eq. (20) provides a second,
equivalent description of the dynamics of the pointer after the interaction
period. This probability density $P_{C}(x,t)$ then describes two Gaussians
travelling at constant velocity away from each other. Each of the Gaussians
has a width that changes linearly in time. This is only possible if the
velocity $v(x)=\frac{1}{M}\frac{\partial S}{\partial x}$ of the classical
ensemble for a given time $t$ is proportional to $x$. The corresponding
function $S_{C}(x,t)$ that completes the description of the state of the
pointer is given by

$S_{C}(x,t)=\frac{Mx^{2}}{2t}.$ (25)

The functions $P_{C}(x,t)$ and $S_{C}(x,t)$ then satisfy both the Hamilton-
Jacobi equation and the corresponding continuity equation,

$\frac{\partial P_{C}}{\partial t}+\frac{\partial}{\partial
x}\left(P_{C}\frac{1}{M}\frac{\partial S_{C}}{\partial
x}\right)=0,\qquad\frac{\partial S_{C}}{\partial
t}+\frac{1}{2M}\left(\frac{\partial S_{C}}{\partial x}\right)^{2}=0,$ (26)

as can be checked by direct calculation. This equivalent representation is
useful if we would consider an experiment to check the predicted probability
density of the pointer by a large number of measurements of identically
prepared systems, without being concerned with a subsequent update of the
pointer’s state.

##  7 The measurement

The next and final step of our model is an observation of the pointer followed
by an update of the probability densities of the pointer and the quantum
particle. We will also show that, unlike the usual description of a
measurement, which assumes a quantum apparatus interacting with a quantum
system, our approach does not rely on the linearity of quantum mechanics in
order to compute the outcome of the measurement [12, 13]. In this section,
moreover, we will make use of the possibility of giving a description of the
state of the pointer in terms of a classical mixture.

###  7.1 Measurements and update of the state of the pointer

If we carry out a large number of measurements of the pointer for an ensemble
of identically prepared systems, we would expect the data to converge to the
probability density of Eq. (20). However, this probability density does not
describe the results we would expect when _the same_ pointer is measured
repeatedly. The different statistics of the pointer in both cases were
discussed in Sec. 6.2.

In what follows, we will consider a single measurement of the pointer _at a
particular time $t$_. To simplify the analysis, we furthermore neglect
possible measurement uncertainties555The generalization to a non-ideal
measurement is straightforward: A measurement with a given uncertainty will
select a range of elements of the mixture, where the spread is determined by
its uncertainty.. Such a single, ideal measurement selects one particular
element $q^{\prime}$ of the mixture in Eq. (21) from all the possible values
$q$ that label the mixture. This necessarily leads to an update of the state
of the pointer according to

$P_{C}(x,t)~{}\rightarrow~{}P_{C|Q}(x|q^{\prime},t)=\delta(x-\lambda
q^{\prime}t),\quad S_{C}(x,t)~{}\rightarrow~{}S_{C|Q}(x,t;q^{\prime})$ (27)

where $S_{C|Q}(x,t;q^{\prime})$ is given by Eq. (23) and $q^{\prime}$ is a
parameter that has now been determined by a measurement. In this way, the
measurement determines the velocity of the pointer to be $v=\lambda
q^{\prime}$. The model realistically predicts that the measured pointer (as a
free particle) will stay in free motion, as it would be observed in such an
experimental situation.

From the moment of the first measurement on, subsequent measurements will have
to be consistent with the new probability density from Eq. (27). Thus, we will
never see the pointer randomly jumping to an other position, but moving
smoothly in one direction along its trajectory $x(t)=\lambda q^{\prime}t$.

As a result of the update of Eq. (27), the state of the pointer no longer
depends on the configuration space coordinate $q$ of the quantum particle.
Instead, the measured quantity $q^{\prime}$ enters the equations. Thus, we
find that the hybrid classical-quantum ensemble decomposes into _independent_
classical and quantum subsystems. This rules out any possibility of nonlocal
energy flow and non local signaling between classical and quantum subsystems
[35], as discussed in more detail in Ref. [9].

###  7.2 The role of the environment

Before we describe in the next section how a measurement of the classical
pointer leads to an update of the quantum system, it lends itself to briefly
discuss how the classical pointer is actually measured. Generally, there are
many conceivable ways of obtaining information about the pointer’s position.
One way to obtain this information is to shine light on the pointer. Then the
environment is filled with particles, particularly photons in this case. These
particles will be scattered by the pointer at its current position. This way,
an observer “looking at the pointer” or camera taking a picture, would obtain
the information about the pointer’s position from the scattered light. Under
normal conditions, a large number of photons will scatter from the pointer
while a measurement takes place, and each of these scattering events may be
considered a potential measurement.

While the environment plays an important rule in the measurement process, it
does not affect the equations that describe the motion of the pointer: Since
the photons are incident from all directions, the average transfer of momentum
will be null. Moreover, the difference in momenta between the macroscopic
pointer and a single photon is so large that one can neglect the recoil of the
pointer as it scatters the photons.

The consequences of the coupling between the classical pointer and the
environment can be described by an approach [9] similar to that by N. F. Mott
in 1929, explaining the appearance of straight tracks in a Wilson cloud
chamber when an atom undergoes radioactive decay with the emission of an
$\alpha$-particle [16, 36]: While the $\alpha$-particle is initially described
as a spherical matter wave that emerges from the atomic nucleus, the
continuous interaction with the environment picks out a particular direction
of motion.

The technical details of our analysis of the measurement of the classical
pointer are of course very different from the ones in Mott’s argument, but the
analogy is clear: in our case, an initial probability density that describes a
mixture of pointer states with different speeds and directions of motion is
updated due to the interaction with the environment. Consequentially, all
scattering events in which photons scatter from the pointer can only take
place along the trajectory of one particular element of the mixture.

###  7.3 Update of the state of the quantum system

Having discussed the update of the state of the classical pointer in the
previous sections, we can now discuss the update of the state of the quantum
system, which is the last step in the measurement process. We consider again
the case where the initial uncertainty of the pointer $\sigma_{C}$ is small
enough so that $\sigma_{C}<<\sigma_{Q}\lambda t$, as already assumed in Sec.
5.2. Under this assumption, the joint probability of the mixed clasical-
quantum system can be expressed in the form

$\displaystyle P(x,q,t)$ $\displaystyle=:$ $\displaystyle
P_{Q|C}(q|x,t)P_{C}(x,t)$ (28) $\displaystyle=$ $\displaystyle\frac{\lambda
t}{\sqrt{2\pi}\sigma_{C}}\exp\left[-\frac{\lambda^{2}t^{2}}{2\sigma_{C}^{2}}\left(q-\frac{x}{\lambda
t}\right)^{2}\right]$
$\displaystyle\times\frac{1}{2\sqrt{2\pi}\sigma_{Q}\lambda
t}\exp\left[-\frac{\left(x-\lambda
q_{0}t\right)^{2}}{2\sigma_{Q}^{2}\lambda^{2}t^{2}}-\frac{\left(x+\lambda
q_{0}t\right)^{2}}{2\sigma_{Q}^{2}\lambda^{2}t^{2}}\right]\,,$

where in the first equality we have used the product rule of probability
theory to express $P(x,q,t)$ in terms of the conditional probability
$P_{Q|C}(q|x,t)$ and the prior probability $P_{C}(x,t)$. A detailed
calculation of Eq. (28) can be found in Appendix B

We suppose now that at time $t=t_{m}$ we measure the position $x=x_{m}$ of the
pointer. For the sake of simplicity we will assume a measurement of $x_{m}$
and $t_{m}$ with negligible uncertainty666As already mentioned in Sec. 7.1 the
generalization to a non-idealized measurement would be straightforward.. Then
the probability density for the location of the pointer has to be updated to a
delta function, and the probability for $q$ has to be updated to

$P_{Q|C}(q|x_{m},t_{m})=\frac{1}{\sqrt{2\pi}\left(\frac{\sigma_{C}}{\lambda
t_{m}}\right)}\exp\left[-\frac{1}{2\left(\frac{\sigma_{C}}{\lambda
t_{m}}\right)^{2}}\left(q-\frac{x_{m}}{\lambda t_{m}}\right)^{2}\right]\,.$
(29)

This means that the measurement localizes $q\rightarrow q_{m}$ with an
uncertainty that depends on the initial uncertainty of the pointer according
to

$q_{m}\sim\frac{x_{m}}{\lambda
t_{m}},\qquad\sigma_{Q,m}=\frac{\sigma_{C}}{\lambda t_{m}}.$ (30)

This expressions are independent of the uncertainty of the initially prepared
quantum system, as we would expect from such a measurement. Since the
equations of motion for the quantum particle have to be fulfilled also after
measurement, the update will affect both variables $P$ and $S$. This leads to
what is described as the _collapse of the wave function_ in the literature on
quantum measurement theory.

Notice that the probability density for the quantum particle does not depend
on the configuration space coordinate $x$ of the classical pointer, but only
on the measured value $x_{m}$. As pointed out above, in the discussion on the
update of the state of the pointer, the hybrid classical-quantum ensemble
decomposes into independent classical and quantum subsystems.

##  8 Summary and Discussion

In this work we have used the approach of ensembles on configuration space to
construct a model where a classical apparatus (in the precise sense given
throughout the previous sections) is used to measure a quantum system. In Sec.
3 we start with the introduction of the the mathematical formalism of this
approach to mixed classical-quantum systems. The measurement process was then
described in Sec. 4 \- 6 in three chronological steps: the initial condition,
the system during the interaction of the classical apparatus and the quantum
particle, and the evolution of the classical pointer after the interaction.
Finally, in Sec. 7, we discussed the measurement of the pointer of the
classical apparatus, from which the information about the position of the
quantum particle was obtained.

We have shown that describing the apparatus classically within the formalism
of ensembles on configuration space solves the main difficulties, discussed in
Sec. 2, which would arise when also the apparatus is described quantum
mechanically. One important feature of the model is the asymmetry between the
measuring apparatus and the quantum system being measured: While we assume
that the quantum system can be isolated from the environment during the
measurement, the classical system inevitably interacts with the environment.
As a consequence, the environment forces an update of the probability density
of the pointer. This way, the classical mixture which mirrors the probability
density of the quantum state is updated to a single element of the mixture.
The motion of the pointer is then described by this very element, without the
possibility of being changed to another element by additional measurements. A
further, more technical, difference between the treatment of classical and
quantum systems is that one may describe the quantum system in terms of a
superposition of eigenstates (as per the linearity of quantum mechanics),
while this is not possible for the state of the pointer: Since after the
measurement one element is chosen out of the classical mixture, it is
impossible for the pointer to be left in a state where there is interferences
between macroscopically different states. This is of advantage in comparison
to the purely quantum description of the measuring process, as discussed in
Sec. 2. Furthermore, the interpretation of the probabilities that appear in
mixtures are straightforward as they are clearly epistemic in nature.

One can conclude that the notion of the “classicality” of the measuring
apparatus requires more than just classical equations of motion for the
pointer. In addition, it is essential that the system has to be macroscopic,
can not be decoupled from the environment, and has a large number of degrees
of freedom in order to describe a system as “classical”.

In our model, the collapse of the wavefunction of the quantum system
corresponds to an update which takes place when the classical pointer is
measured. As a result of this update, the mixed classical-quantum ensemble
decomposes into two independent subsystems, classical and quantum, which are
no longer correlated. This also prevents any superluminal communication
between the pointer and the quantum system [35], as discussed in more detail
in Ref. [9].

Since our model allows for a detailed description of the steps of the
measurement process, it is possible to keeps tracks of the uncertainties
associated with each step. In particular, they allow us to consider the finite
precision in the preparation of an initial quantum state and the limitations
on the calibration of the classical apparatus. Therefore, our approach enables
the analytical discussion of non-idealized quantum particle position
measurements and is applicable to the challenges of modern high precision
quantum metrology. The comparatively simple model we used to introduce the
basic formalism can be extended to describe even more complex measurement
processes.

## Acknowledgement

We want to thank Michael J. W. Hall for comments and valuable discussions. We
also want to thank the organizers of the conference Koopman Methods in
Classical and Classical-Quantum Mechanics for a very stimulating meeting. One
of the authors acknowledges support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germanys Excellence Strategy -
EXC-2123 QuantumFrontiers - 390837967.

## Appendices

##  Appendix A Validity of the approximation during the time of interaction

In Sec. 5.2 we have introduced an approximate solution of the modified
Hamilton-Jacobi equation and continuity equation during the time of
interaction under the assumption of a strong coupling constant. We made the
further assumption that the interaction is effectively instantaneous. In what
follows, we demonstrate the validity of this approximation under the
additional assumption that the pointer is originally at rest.

###  A.1 Initial state for the quantum particle

The wavefunction that we use as initial condition is the sum of two stationary
Gaussian wavefunctions which are located at $\pm q_{0}$. The full wave
function of the quantum system can be written as

$\displaystyle\psi(q,t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[\psi_{+}(q,t)+i\psi_{-}(q,t)],$ (31)
$\displaystyle\psi_{\pm}(q,t)$ $\displaystyle=$
$\displaystyle\sqrt{P_{\pm}(q,t)}\,\mathrm{e}^{iS_{\pm}(q,t)/\hbar}$ (32)

where [14]

$\displaystyle P_{Q}^{\pm}(q,t)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi\sigma^{2}(t))^{1/2}}\exp\left[-\frac{1}{2}\left(\frac{q\pm
q_{0}}{\sigma(t)}\right)^{2}\right],$ (33) $\displaystyle S_{Q}^{\pm}(q,t)$
$\displaystyle=$ $\displaystyle-\frac{\hbar}{2}\tan^{-1}\left(\frac{\hbar
t}{2m\sigma_{Q}^{2}}\right)+\frac{(q\pm
q_{0})^{2}\hbar^{2}t}{8m\sigma_{Q}^{2}\sigma^{2}(t)},$ (34)
$\displaystyle\sigma(t)$ $\displaystyle=$
$\displaystyle\sigma_{Q}\left[1+\left(\frac{\hbar
t}{2m\sigma_{Q}^{2}}\right)^{2}\right]^{1/2}.$ (35)

In particular, we need $P_{Q}$ and $S_{Q}$ at time $t=0$ to be able to
calculate the evolution of the classical pointer after interaction. Since
$S_{\pm}(q,0)=0$, the wavefunction reduces to
$\psi(q,0)=\sqrt{P_{Q}^{+}(q,0)}+i\sqrt{P_{Q}^{-}(q,0)}$, so that we have

$\displaystyle P_{Q}(q,0)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2\pi}\,\sigma_{Q}}\left[\exp\left(-\frac{(q+q_{0})^{2}}{2\sigma_{Q}^{2}}\right)+\exp\left(-\frac{(q-q_{0})^{2}}{2\sigma_{Q}^{2}}\right)\right],$
(36) $\displaystyle S_{Q}(q,0)$ $\displaystyle=$ $\displaystyle 0.$ (37)

###  A.2 Initial state for the classical pointer

We assume that the initial state of the pointer is described by a _classical
mixture_ (see Appendix C for a definition). The elements of this mixture
consist of states labeled by a parameter $v$, of the form

$\displaystyle S_{C|v}(x|v,t)$ $\displaystyle=$
$\displaystyle-\frac{Mv^{2}t}{2}+Mvx,$ (38) $\displaystyle P_{C|v}(x|v,t)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\,\sigma_{C}}\;\exp\left(-\frac{(x-vt)^{2}}{2\sigma_{C}^{2}}\right)\,.$
(39)

We have introduced the notation “$|v$” to emphasize that these quantities are
defined with respect to the parameter $v$, which corresponds to the velocity
of the pointer, as can be seen by evaluating $\frac{1}{M}\frac{\partial
S}{\partial x}=v$. The probability density $\rho(x,p,t)$ associated with this
classical mixture is given by

$\rho(x,p,t)=\int\,dv\,P_{v}(v)\,\delta\left(p-\frac{\partial
S_{C|v}(x|v,t)}{\partial x}\right)\,P_{C|v}(x|v,t)$ (40)

where $P_{v}(v)\geq 0$ and $\int\,dv\,P_{v}(v)=1$ so that $\int
dx\,dp\,\rho(x,p,v)=1$. One can show that such a classical mixture is strictly
equivalent to a classical phase-space distribution [9].

For the purpose of checking the approximation, we do not need to specify the
details of the mixture such as the functional form of $P_{v}(v)$. We will only
assume that such a distribution is symmetric and the mixture satisfies $<v>=0$
so that the pointer is stationary. At the end of the calculation, we will
consider the limit where the velocity of the pointer goes to zero.

###  A.3 The approximate solution

It is not necessary to use the full mixture when examining the validity of the
approximate solution. It is sufficient to consider any particular element
$S_{C|v}(x|v,t)$, $P_{C|v}(x|v,t)$. We consider the initial conditions

$\displaystyle S(x,q,0)$ $\displaystyle=$ $\displaystyle
S_{C|v}(x|v,t)+S_{Q}(q,0)=Mvx,$ (41) $\displaystyle P(x,q,0)$ $\displaystyle=$
$\displaystyle P_{C|v}(x|v,t)P_{Q}(q,0)$ (42) $\displaystyle=$
$\displaystyle\frac{1}{4\pi\sigma_{C}\sigma_{Q}}\;\exp\left(-\frac{(x-vt)^{2}}{2\sigma_{C}^{2}}\right)$
$\displaystyle\qquad\times\left[\exp\left(-\frac{(q+q_{0})^{2}}{2\sigma_{Q}^{2}}\right)+\exp\left(-\frac{(q-q_{0})^{2}}{2\sigma_{Q}^{2}}\right)\right]$

for the combined system of quantum particle and classical pointer.

As in Sec. 6, we assume an interaction strength of the form $\alpha(t)=\lambda
qt$ during the time $0<t<\varepsilon$. Under this assumptions, we obtain

$\displaystyle S(x,q,0<t<\varepsilon)$ $\displaystyle=$ $\displaystyle
S(x-\lambda qt,q,0)$ (43) $\displaystyle P(x,q,0<t<\varepsilon)$
$\displaystyle=$ $\displaystyle P(x-\lambda qt,q,0)$ (44)

Then, during $0<t<\varepsilon$, the explicit form of $P$ and $S$ is given by

$\displaystyle S(x,q,t)$ $\displaystyle=$ $\displaystyle Mv(x-\lambda qt)$
(45) $\displaystyle P(x,q,t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\sigma_{C}\sigma_{Q}}\;\exp\left(-\frac{(x-vt-\lambda
qt)^{2}}{2\sigma_{C}^{2}}\right)$ (46)
$\displaystyle\qquad\times\left[\exp\left(-\frac{(q+q_{0})^{2}}{2\sigma_{Q}^{2}}\right)+\exp\left(-\frac{(q-q_{0})^{2}}{2\sigma_{Q}^{2}}\right)\right]$

If we now introduce again the kinetic energy terms of the Hamiltonian that we
previously neglected because the interaction term was dominant, the modified
Hamilton-Jacobi equation is given by

$\displaystyle-\frac{\partial S}{\partial t}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{2M}\left(\frac{\partial S}{\partial
x}\right)^{2}+\frac{1}{2m}\left(\frac{\partial S}{\partial
q}\right)^{2}+\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{P}}\frac{\partial^{2}\sqrt{P}}{\partial
q^{2}}\right]+\lambda q\frac{\partial S}{\partial x}$
$\displaystyle\Rightarrow-Mv\lambda q$ $\displaystyle=$
$\displaystyle\left[\frac{Mv^{2}}{2}+\frac{M^{2}v^{2}\lambda^{2}t}{2m}+\frac{\hbar^{2}}{2m}\left(\frac{1}{\sqrt{P}}\frac{\partial^{2}\sqrt{P}}{\partial
q^{2}}\right)\right]-Mv\lambda q$ (47)

where the terms in square brackets are the ones that we previously omitted.

The third term in square brackets is the quantum potential term. One can show
that it remains finite provided both $\sigma_{C},\sigma_{Q}>0$. Furthermore,
since the quantum potential is the only term that is proportional to
$\hbar^{2}$, we can neglect it with respect to the other “classical” terms
(without $\hbar$) that contribute to the energy. Finally, in the limit
$v\rightarrow 0$, we can neglect all remaining terms of order $v^{2}$ when
compared to the terms linear in $v$. Thus all the terms in the square brackets
can be neglected. This shows that the approximation is indeed valid.

One must also check that the continuity equation,

$\frac{\partial P}{\partial t}+\frac{\partial}{\partial
x}\left(P\frac{1}{M}\frac{\partial S}{\partial
x}\right)+\frac{\partial}{\partial q}\left(P\frac{1}{m}\frac{\partial
S}{\partial q}\right)=0,$ (48)

is also valid during the time of interaction ($0<t<\varepsilon$) under the
assumptions of the approximation. A straightforward calculation using Eqs.
(45-46) shows that the equation is indeed satisfied provided all terms of
higher order in $v$ and $t$ are neglected with respect to the terms linear in
$v$; that is, in the limit $v\rightarrow 0$ and for the case of small
interaction times $\varepsilon$.

##  Appendix B Explicit expressions for probability densities

In what follows, for the sake of completeness, we give some explicit forms for
probability densities that have been used for the calculation. We begin with
the initial probability density, given by

$\displaystyle P(x,q,0)$ $\displaystyle=$ $\displaystyle P_{C}(x,0)P_{Q}(q,0)$
(49) $\displaystyle=$
$\displaystyle\frac{1}{4\pi\sigma_{C}\sigma_{Q}}\;\exp\left(-\frac{x^{2}}{2\sigma_{C}^{2}}\right)\left[\exp\left(-\frac{(q+q_{0})^{2}}{2\sigma_{Q}^{2}}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.+\exp\left(-\frac{(q-q_{0})^{2}}{2\sigma_{Q}^{2}}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\eta_{C}\eta_{Q}}}{4\pi}\;\exp\left(-\frac{\eta_{C}x^{2}}{2}\right)\left[\exp\left(-\frac{\eta_{Q}(q+q_{0})^{2}}{2}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.+\exp\left(-\frac{\eta_{Q}(q-q_{0})^{2}}{2}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\eta_{C}\eta_{Q}}}{4\pi}\;\left[\exp\left(-\frac{\eta_{Q}(q+q_{0})^{2}+\eta_{C}x^{2}}{2}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\exp\left(-\frac{\eta_{Q}(q-q_{0})^{2}+\eta_{C}x^{2}}{2}\right)\right]$

where we introduced the precision $\eta:=1/\sigma^{2}$ to simplify the
formulae.

At any time $t$ during the interaction, we have

$P(x,q,\epsilon)=P(x-qk(t),q,0),$ (50)

and the probability density can be written in a form that is convenient for
calculating the statistics of the pointer by integrating with respect to $q$,

$\displaystyle P(x,q,t)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\eta_{C}\eta_{Q}}}{4\pi}\;\left[\exp\left(-\frac{\eta_{Q}(q+q_{0})^{2}+\eta_{C}(x-qk(t))^{2}}{2}\right)\right.$
(51)
$\displaystyle\qquad\qquad+\left.\exp\left(-\frac{\eta_{Q}(q-q_{0})^{2}+\eta_{C}(x-qk(t))^{2}}{2}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\eta_{C}\eta_{Q}}}{4\pi}\;\left\\{\exp\left[-\frac{\eta_{Q}+\eta_{C}k^{2}(t)}{2}\left(q+\frac{\eta_{Q}q_{0}-\eta_{C}k(t)x}{\eta_{Q}+\eta_{C}k^{2}(t)}\right)^{2}\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.-\frac{\left(x-q_{0}k(t)\right)^{2}}{2\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}\right]\right.$
$\displaystyle\qquad\qquad+\left.\exp\left[-\frac{\eta_{Q}+\eta_{C}k^{2}(t)}{2}\left(q-\frac{\eta_{Q}q_{0}+\eta_{C}k(t)x}{\eta_{Q}+\eta_{C}k^{2}(t)}\right)^{2}\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.-\frac{\left(x+q_{0}k(t)\right)^{2}}{2\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)\right/\sigma_{C}^{2})}\right]\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi\sigma_{C}\sigma_{Q}}\;\left\\{\exp\left[-\frac{\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}{2\sigma_{Q}^{2}}\left(q+\frac{q_{0}-\sigma_{Q}^{2}k(t)x/\sigma_{C}^{2}}{1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}}\right)^{2}\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.-\frac{\left(x-q_{0}k(t)\right)^{2}}{2\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}\right]\right.$
$\displaystyle\qquad+\left.\exp\left[-\frac{\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}{2\sigma_{Q}^{2}}\left(q-\frac{q_{0}+\sigma_{Q}^{2}k(t)x/\sigma_{C}^{2}}{1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}}\right)^{2}\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.-\frac{\left(x+q_{0}k(t)\right)^{2}}{2\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)\right/\sigma_{C}^{2})}\right]\right\\}.$

If we integrate Eq. (51) with respect to $q$, we get

$\displaystyle P_{C}(x,t)$ $\displaystyle=$ $\displaystyle\int dq\,P(x,q,t)$
(52) $\displaystyle=$
$\displaystyle\frac{1}{4\pi\sigma_{C}\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)^{1/2}}\;\left\\{\exp\left[-\frac{\left(x-q_{0}k(t)\right)^{2}}{2\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}\right]\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left.\exp\left[-\frac{\left(x+q_{0}k(t)\right)^{2}}{2\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}\right]\right\\}.$

It is also possible to write the joint probability $P(x,q,t)$ in a form that
is useful for determining the conditional probability $P_{Q|C}(q|x,t)$ in the
limit $\sigma_{C}<<\sigma_{Q}k(t)$. We first rewrite $P(x,q,t)$ in the form

$\displaystyle P(x,q,t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\frac{1}{\sqrt{\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}}\frac{1}{\sqrt{\sigma_{Q}^{2}/\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}}$
(53)
$\displaystyle\left\\{\exp\left[-\frac{\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}{2\sigma_{Q}^{2}}\left(q+\frac{q_{0}-\sigma_{Q}^{2}k(t)x/\sigma_{C}^{2}}{1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}}\right)^{2}\right.\right.$
$\displaystyle\left.\left.\qquad\qquad\qquad\qquad\qquad-\frac{\left(x-q_{0}k(t)\right)^{2}}{2\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}\right]\right.$
$\displaystyle\qquad\qquad+\exp\left[-\frac{\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}{2\sigma_{Q}^{2}}\left(q-\frac{q_{0}+\sigma_{Q}^{2}k(t)x/\sigma_{C}^{2}}{1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}}\right)^{2}\right.$
$\displaystyle\left.\left.\qquad\qquad\qquad\qquad\qquad-\frac{\left(x+q_{0}k(t)\right)^{2}}{2\sigma_{C}^{2}\left(1+\sigma_{Q}^{2}k^{2}(t)/\sigma_{C}^{2}\right)}\right]\right\\}\,.$

In the limit $\sigma_{C}<<\sigma_{Q}p(t)$, we get

$\displaystyle P(x,q,t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\frac{1}{\sigma_{C}\sigma_{Q}}\left\\{\exp\left[-\frac{k^{2}(t)}{2\sigma_{C}^{2}}\left(q-\frac{x}{k(t)}\right)^{2}-\frac{\left(x-q_{0}k(t)\right)^{2}}{2\sigma_{Q}^{2}k^{2}(t)}\right]\right.$
(54)
$\displaystyle\qquad\qquad+\left.\exp\left[-\frac{k^{2}(t)}{2\sigma_{C}^{2}}\left(q-\frac{x}{k(t)}\right)^{2}-\frac{\left(x+q_{0}k(t)\right)^{2}}{2\sigma_{Q}^{2}k^{2}(t)}\right]\right\\}$
$\displaystyle=$
$\displaystyle\frac{k(t)}{\sqrt{2\pi}\sigma_{C}}\exp\left[-\frac{k^{2}(t)}{2\sigma_{C}^{2}}\left(q-\frac{x}{k(t)}\right)^{2}\right]$
$\displaystyle\qquad\qquad\frac{1}{2\sqrt{2\pi}\sigma_{Q}k(t)}\exp\left[-\frac{\left(x-q_{0}k(t)\right)^{2}}{2\sigma_{Q}^{2}k^{2}(t)}-\frac{\left(x+q_{0}k(t)\right)^{2}}{2\sigma_{Q}^{2}k^{2}(t)}\right]$
$\displaystyle=$
$\displaystyle\frac{k(t)}{\sqrt{2\pi}\sigma_{C}}\exp\left[-\frac{k^{2}(t)}{2\sigma_{C}^{2}}\left(q-\frac{x}{k(t)}\right)^{2}\right]$
$\displaystyle\qquad\qquad\frac{1}{2\sqrt{2\pi}\sigma_{Q}k(t)}\exp\left[-\frac{\left(x-q_{0}k(t)\right)^{2}}{2\sigma_{Q}^{2}k^{2}(t)}-\frac{\left(x+q_{0}k(t)\right)^{2}}{2\sigma_{Q}^{2}k^{2}(t)}\right]$
$\displaystyle=:$ $\displaystyle P_{Q|C}(q|x,t)P_{C}(x|t).$

from which $P_{Q|C}(q|x,t)$ can be read off.

##  Appendix C Classical mixtures

We consider a solution $S_{C|q}(x|q,t)$ of the Hamilton-Jacobi equation that
depends on a parameter $q$ and a corresponding probability density
$P_{C|q}(x|q,t)$ which, together with $S_{C|q}(x|q,t)$, satisfies the
continuity equation. One may define then classical mixtures of states with
different values of $q$ in terms of the elements of the mixture
$\\{S_{C|q}(x|q,t),~{}P_{C|q}(x|q,t)\\}$.

Definition: A classical mixture $\rho(x,p,t)$ is defined by introducing an
additional coordinate $p$ and setting

$\rho(x,p,t)=\int\,dq\,P_{q}(q)\,\delta\left(p-\frac{\partial
S_{C|q}(x|q,t)}{\partial x}\right)\,P_{C|q}(x|q,t)$ (55)

where $P_{q}(q)\geq 0$ and $\int\,dq\,P_{q}(q)=1$ so that $\int
dx\,dp\,\rho(x,p,t)=1$. One can show that a classical mixture is strictly
equivalent to a classical phase-space distribution [9]. Notice that it f

Definition: A classical pure state is a particular type of mixture which only
contains a single configuration ensemble state labeled by a particular value
of $q$, say $q=q_{0}$. It them follows that $P_{q}(q)=\delta(q-q_{0})$ and

$\rho^{(q=q_{0})}(x,p,t)=\int\,dq\,\delta(q-q_{0})\,\delta\left(p-\frac{\partial
S_{C|q}(x|q,t)}{\partial x}\right)\,P_{C|q}(x|q,t)$ (56)

Since there are no interference effects in classical mechanics, mixtures and
pure states do not differ as strongly for classical ensembles as they do for
quantum ensembles.

Classical mixtures have a number of applications and they play an important
role in the descriptions of measurements with a classical apparatus [9], as we
have already shown in Sec. 6.

The statistics of a classical ensemble may be described in different ways
using classical mixtures. We consider a free particle and show that an
arbitrary mixture $\rho^{x_{0}}(x,p,t)$ based on particular states described
by the family of solutions

$S_{C|x_{0}}(x|x_{0},t)=m\frac{(x-x_{0})^{2}}{2t},~{}~{}~{}~{}~{}P_{C|x_{0}}(x|x_{0},t)=\frac{1}{t}F_{C|x_{0}}\left(\frac{x-x_{0}}{t}\right),$
(57)

can also be expressed as a mixture $\rho^{v}(x,p,t)$ of states described by
the family of solutions

$S_{C|v}(x|v,t)=-\frac{mv^{2}}{2}t+mvx,~{}~{}~{}~{}~{}P_{C|v}(x|v,t)=F_{C|v}(x-vt).$
(58)

In both representations, $F$ are functions that are only constrained by $P\geq
0$ and $\int dx\,P=1$. The solution of the Hamilton-Jacobi equation of Eq.
(57) is the Hamilton principal function, which describes a free particle that
follows a trajectory that passes through $x$ and $x_{0}$, while that of Eq.
(58) is the one obtained by separation of variables, which describes a free
particle of velocity $v$ [37].

In the first case the mixture can be written

$\displaystyle\rho^{x_{0}}(x,p,t)$ $\displaystyle=$
$\displaystyle\int\,dx_{0}\,P_{x_{0}}(x_{0})\,\delta\left(p-\frac{m(x-x_{0})}{t}\right)\,\frac{1}{t}F_{C|x_{0}}\left(\frac{x-x_{0}}{t}\right)$
(59) $\displaystyle=$
$\displaystyle\int\,dx_{0}\,P_{x_{0}}(x_{0})\,\frac{t}{m}\delta\left(\frac{pt}{m}-x+x_{0}\right)\,\frac{1}{t}F_{C|x_{0}}\left(\frac{x-x_{0}}{t}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{m}P_{x_{0}}\left(x-\frac{pt}{m}\right)\,F_{C|x_{0}}\left(\frac{p}{m}\right),$

while in the second case it will take the form

$\displaystyle\rho_{v}(x,p,t)$ $\displaystyle=$
$\displaystyle\int\,dv\,P_{v}(v)\,\delta\left(p-mv\right)\,F_{C|v}\left(x-vt\right)$
(60) $\displaystyle=$
$\displaystyle\int\,dv\,P_{v}(v)\,\frac{1}{m}\delta\left(\frac{p}{m}-v\right)\,F_{C|v}\left(x-vt\right)$
$\displaystyle=$
$\displaystyle\frac{1}{m}P_{v}\left(\frac{p}{m}\right)\,F_{C|v}\left(x-\frac{pt}{m}\right).$

Since we want the two mixtures to describe equivalent statistics,
$\rho^{x_{0}}(x,p,t)=\rho^{v}(x,p,t)$, we need to set the functions
$P_{v}(*):=F_{C|x_{0}}(*)$ and $F_{C|v}(*):=P_{x_{0}}(*)$, which leads to the
result

$P_{v}\left(\frac{p}{m}\right)\,F_{C|v}\left(x-\frac{pt}{m}\right)=P_{x_{0}}\left(x-\frac{pt}{m}\right)\,F_{C|x_{0}}\left(\frac{p}{m}\right).$
(61)

Let us now specialize to the case where the first mixture includes just one
state with a particular value of $x_{0}$, i.e. $x_{0}=0$, and the ensemble
moves with average velocity $v^{\prime}$ so that

$\displaystyle P_{x_{0}}\left(x_{0}\right)$ $\displaystyle=$
$\displaystyle\delta(x_{0}),$ $\displaystyle
F_{C|x_{0}}\left(\frac{x}{t}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\alpha}e^{-\frac{1}{2}\left(\frac{x-v^{\prime}t}{\alpha
t}\right)^{2}}=\frac{1}{\sqrt{2\pi}\alpha}e^{-\frac{1}{2}\left[\frac{1}{\alpha}\left(\frac{x}{t}-v^{\prime}\right)\right]^{2}}.$
(62)

Then, the second mixture will be equivalent to the first mixture if

$P_{v}\left(\frac{p}{m}\right)=\frac{1}{\sqrt{2\pi}\alpha}e^{-\frac{1}{2}\left[\frac{1}{\alpha}\left(\frac{p}{m}-v^{\prime}\right)\right]^{2}},~{}~{}~{}~{}~{}F_{C|v}\left(x-\frac{pt}{m}\right)=\delta\left(x-\frac{pt}{m}\right),$
(63)

which defines the mixture

$\displaystyle\rho^{v}(x,p,t)$ $\displaystyle=$
$\displaystyle\int\,dv\,P_{v}(v)\,\delta\left(p-\frac{\partial
S_{C|v}(x|v,t)}{\partial x}\right)\,F_{C|v}\left(x-vt\right)$ (64)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\alpha}\int\,dv\,e^{-\frac{1}{2}\left(\frac{v-v^{\prime}}{\alpha}\right)^{2}}\,\delta\left(p-\frac{\partial
S_{C|v}(x|v,t)}{\partial x}\right)\,\delta\left(x-vt\right),\qquad$

where we have made use of the fact that the delta function
$\delta\left(p-\frac{\partial S_{C|v}(x|v,t)}{\partial x}\right)$ forces
$p=mv$. Thus, the first mixture, $\rho^{x_{0}}(x,p,t)$, has now been expressed
in terms of a new mixture $\rho^{v}(x,p,t)$ that describes particles moving
with velocities $v$, where the probability of $v$ is given by a Gaussian
centered at $v^{\prime}$ with spread $\sigma=\alpha$.

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