# A model of wave function collapse in a quantum measurement of spin as the
Schr\̈cyrodinger equation solution of a system with a simple harmonic
oscillator in a bath

Li Hua Yu  Brookhaven National Laboratory

(4/6/2023)

###### Abstract

We present a set of exact system solutions to a model we developed to study
wave function collapse in the quantum spin measurement process. Specifically,
we calculated the wave function evolution for a simple harmonic oscillator of
spin $\frac{1}{2}$, with its magnetic moment in interaction with a magnetic
field, coupled to an environment that is a bath of harmonic oscillators. The
system’s time evolution is described by the direct product of two independent
Hilbert spaces: one that is defined by an effective Hamiltonian, which
represents a damped simple harmonic oscillator with its potential well divided
into two, based on the spin and the other that represents the effect of the
bath, i.e., the Brownian motion. The initial states of this set of wave
functions form an orthonormal basis, defined as the eigenstates of the system.
If the system is initially in one of these states, the final result is
predetermined, i.e., the measurement is deterministic. If the bath is
initially in the ground state,and the wave function is initially a wave packet
at the origin, it collapses into one of the two potential wells depending on
the initial spin. If the initial spin is a vector in the Bloch sphere not
parallel to the magnetic field, the final distribution among the two potential
wells is given by the Born rule applied to the initial spin state with the
well-known ground state width. Hence, the result is also predetermined. We
discuss its implications to the Bell theorem(bell_theorem, ).

We end with a summary of the implications for the understanding of the
statistical interpretation of quantum mechanics.

## Introduction

The quantum measurement problem is related to the foundations of quantum
mechanics. In particular, the collapse of the wave function during quantum
measurement is crucial to the understanding of quantum mechanics. ”The
inability to observe such a collapse directly has given rise to different
interpretations of quantum mechanics”(measurement, ). We have chosen to use a
model for the quantum measurement process to clarify the collapse of the wave
function.

To study the collapse of the wave function, we are interested in the simplest
example of a dissipative system. Such a system, i.e., a harmonic oscillator
coupled to an environment, which is a bath of harmonic oscillators, has been
the subject of extensive studies (Legget1, ; Legget2, ; caldeira, ; kanai, ;
Nakajima, ; Zwanzig, ; Senitzky, ; feynman, ; kac, ; kostin, ; louisell, ;
scully, ; yasue, ; koch, ; dekker, ; kac2, ; weisskopf, ). A special case of
this system, the Ohmic case discussed by Caldeira and Leggett (to be defined
later), is exactly solved using the path-integral with the density matrix
diagonalized in (Legget1, ; Legget2, ; caldeira, ), which indicates a wave
function collapse. Following these works, we solved the wave function
evolution directly in this special case as the solution of the
Schr\̈cyrodinger equation(yu, ), where even though we did not focus on the
description of the wave function collapse, it is obvious in the solution.

In this paper, for the Ohmic case of the system, we shall focus on the wave
function collapse and introduce spin $\frac{1}{2}$ into the simple harmonic
oscillator, with its magnetic moment interacting with a magnetic field aligned
to the axis of the oscillation in a direction $z$. The bath of harmonic
oscillators serves as a model of a detector. The potential well of the simple
harmonic oscillator is split into two potential wells according to whether the
spin is in $+z$ or $-z$ direction. Thus, this models a spin measurement
process.

We consider the problem the same as discussed by Caldeira and Leggett (CL)
(Legget1, ) and (yu, ), except that in this case the main oscillator has spin:
a harmonic-oscillator system (the dissipative system) with coordinate $q$ on
$z$ axis, mass $M$, and frequency
$(\omega_{0}^{2}+\Delta\omega^{2})^{\frac{1}{2}}$ , interacting with a bath of
$N$ harmonic oscillators of coordinates $x_{j}$, mass $m_{j}$, and frequency
$\omega_{j}$, where $\Delta\omega^{2}$ is a shift induced by the coupling
already discussed by CL. The Hamiltonian of the system and the bath is

$\displaystyle
H=\frac{p^{2}}{2M}\text{+$\frac{1}{2}M(\omega_{0}^{2}+\Delta\omega^{2})q^{2}-MB\sigma_{z}q+q\sum_{j=1}^{N}c_{j}x_{j}+\sum_{j=1}^{N}\left(\frac{p_{j}^{2}}{2m_{j}}+\frac{1}{2}m_{j}\omega_{j}^{2}x_{j}^{2}\right)$
}$
$\displaystyle\sigma_{z}\left|+\right\rangle=\left|+\right\rangle=\left(\begin{array}[]{c}1\\\
0\end{array}\right),\sigma_{z}\left|-\right\rangle=-\left|-\right\rangle=-\left(\begin{array}[]{c}0\\\
1\end{array}\right),\psi=\left(\begin{array}[]{c}\psi_{+}(q)\\\
\psi_{-}(q)\end{array}\right)$

The interaction of the spin $\frac{1}{2}\sigma$ dipole moment with the
magnetic field contributes to the term $-MB\sigma_{z}q$, where z is the
projection of the Pauli matrix $\sigma$ on $z$ and $\psi$ is the main
oscillator’s wave function. The simple harmonic oscillator’s potential well is
divided into two potential wells based on the spin.

Here, we outline the solution to this problem with examples and a possible
interpretation:

In Section 1, despite the addition of the new term $-MB\sigma_{z}q$ to the
previous Hamiltonian in (yu, ), we show that this system can be solved almost
exactly the same way. The system’s time evolution is described by the direct
product of two independent Hilbert spaces: one that is defined by an effective
Hamiltonian, representing a damped simple harmonic oscillator with its
potential well divided into two based on the spin, and the other that
represents the effect of the bath, i.e., the Brownian motion.

We present a set of exact solutions of the system that all collapse to a
$\delta$-function centered at a specified position and time depending on
whether the spin eigenstate is parallel or opposite to the magnetic field,
with a small displacement determined by the eigenstates of each bath
oscillators. The initial states of this set of wave functions form an
orthonormal basis in the Hilbert space of the system, which is defined as the
“eigenstates of the system.” Hence, if the system is initially in one of these
states, the final result is predetermined, i.e., the measurement is
deterministic.

In Section 2, if the system is initially a superposition of these states, the
final distribution is a superposition of $\delta$-functions for a sufficiently
long time after the damping time. Each has a probability amplitude determined
by the corresponding eigenstate’s probability amplitude in the initial state
at time $t=0$ , with the center position determined by the spin eigenstate of
the main oscillator and the contribution from each bath oscillator $x_{j}$.
Once the distribution is determined from the initial state by the Born rule,
the final distribution is also predetermined. As a result, we only need to
apply quantum mechanics’ statistical interpretation to the initial state; the
final distribution is already determined by the Schr\̈cyrodinger equation.

One of the examples we shall discuss is the special case where the initial
states of the bath oscillators are in their ground states at absolute
temperature zero. In this case, the initial state of the bath is not an
“eigenstate of the system,” as we defined it above, but a superposition of the
“eigenstates of the system,” even though it is in the ground state (we remark
here that the system is initially not in equilibrium even though the bath is
in the ground state). The main harmonic oscillator is assumed to be initially
a wave packet at the origin, i.e., in the middle between the two split
potential wells. If the initial spin state is one of the two eigenstates of
the main oscillator, for example, $\left|+\right\rangle$, the final state
would be at the bottom of one of the two potential wells, that is, the one
with $z>0$, with a narrow spread due to the contributions from the bath
oscillators which is the Brownian width. We find the width approximately
agrees with the well-known width of the ground state for the simple harmonic
oscillator if the damping time is much longer than its period.

In Section 3, we discuss another example where the initial spin state is a
vector in the Bloch sphere not parallel to the magnetic field, i.e., it is the
eigenstate of $\sigma_{\theta}$ where $\theta$ is the angle between the vector
and the $z$ direction. In this case, even though the initial state is a pure
state, it is not the eigenstate of the system; it is a superposition of two
eigenstates of the system. This state is not an eigenstate of the spin
measurement of $\sigma_{z}$ because the magnetic field is not in the $\theta$
direction. The probability of the two eigenstates is given by the Born rule
applied to the initial state, i.e., $\cos^{2}(\frac{\theta}{2})$ for $z>0$,
$\sin^{2}(\frac{\theta}{2})$ for $z<0$, in spite of the initial state being a
pure state. We shall demonstrate that the final distribution has two distinct
peaks at the centers of the two potential wells accordingly, as shown in
Fig.1. Again, the final result is determined by the Schr\̈cyrodinger equation
according to the initial probability distribution.

If the bath temperature is not zero, the system is in a mixed state specified
by a density matrix $\rho$ according to the Boltzmann distribution. Still, we
found the result is similarly determined by initial distribution.

In Section 4, we summarize the result and its implications in the Bell
theorem. Because the discussion is only for a specific model, the
interpretation is only suggestive. Applying the results of Sections 1, 2, and
3, we establish that the analysis of two separate spin measurements
$\sigma_{1}\centerdot\vec{a}$ and $\sigma_{2}\centerdot\vec{b}$ at remote
positions A and B on two entangled particles in the Bell theorem is equivalent
to the analysis of two measurements $-\sigma_{2}\centerdot\vec{a}$ and
$\sigma_{2}\centerdot\vec{b}$ at time $t=0$ on particle 2 at the origin.

Because $\sigma_{2}\centerdot\vec{a}$ and $\sigma_{2}\centerdot\vec{b}$ at
time $t=0$ do not form a set of commuting observables, if
$\sigma_{2}\centerdot\vec{a}$ is definite, then $\sigma_{2}\centerdot\vec{b}$
is uncertain. Hence the probability distribution is a consequence of the
uncertainty principle. So long as the uncertainty principle is valid, there is
no hidden variable theory to have a definite value for
$\sigma_{2}\centerdot\vec{a}$ and $\sigma_{2}\centerdot\vec{b}$
simultaneously. Therefore, there is no need to resort to “local hidden
variable theory” as J. Bell did to prove Bell inequality. As a result, there
is no “nonlocality” involved in the violation of Bell inequality when the
quantum mechanics prediction is confirmed. Because the result is predetermined
by the initial condition, the final distance of the two entangled particles in
the experiment in the Bell theorem does not affect the probability
distribution as long as their spin correlation is not destroyed by the
environment before detection. Thus, the experiment’s agreement with quantum
mechanics does not provide information related to non-locality. This result
suggests a viewpoint that may aid in understanding the interpretation of
Bell’s theorem. This result also implies that the Born rule at the end of the
measurement can be derived from the Schr\̈cyrodinger equation as long as the
Born rule is applied to the interpretation of the initial state.

Finally, Section 5 gives a summary of the implications for the understanding
of the statistical interpretation of quantum mechanics.

## 1\. Solution of a damped simple harmonic oscillator with spin in a bath

The dynamic equation for operators in the Heisenberg representation leads to
the following set of equations of motion:

$\displaystyle
M\ddot{q}=-M\omega_{0}^{2}q-M\Delta\omega^{2}q+MB\sigma_{z}-\sum_{j}c_{j}x_{j}$
(1) $\displaystyle m_{j}\ddot{x_{j}}=-m_{j}\omega_{j}^{2}x_{j}-c_{j}q\
(j=1,2,...,N)$

Now, applying the Laplace transform (caldeira, ) (the bars are our notations
for the Laplace transform, and $s$ is the Laplace transform of time $t$), Eqs.
(1) can be used to eliminate the bath variables $x_{j}$ to obtain the equation
for $q$,

$\displaystyle
M(s^{2}\overline{q}-sq_{0}-\dot{q}_{0})=-M\omega_{0}^{2}\overline{q}-M\Delta\omega^{2}\overline{q}+MB\sigma_{z}\frac{1}{s}-\sum_{j}c_{j}\frac{sx_{j0}+\dot{x}_{j0}}{s^{2}+\omega_{j}^{2}}+\sum_{j}\frac{c_{j}^{2}}{m_{j}(s^{2}+\omega_{j}^{2})}\overline{q}$
(2)

where $q_{0},\dot{q}_{0}$ , $x_{j0},\dot{x}_{j0}$ are the initial values of
the respective operators in the Heisenberg representation. Assuming the number
of bath oscillators is large enough so that we can replace the sum over j by
integration over $\omega_{j}$, the coefficient of the last term can then be
separated into two terms:

$\displaystyle\int_{0}^{\omega_{\text{cufoff}}}\frac{c_{j}^{2}}{m_{j}}\frac{\rho(\omega_{j})}{\omega_{j}^{2}}d\omega_{j}-s^{2}\int_{0}^{\omega_{\text{cufoff}}}\frac{c_{j}^{2}}{m_{j}}\frac{\rho(\omega_{j})}{\omega_{j}^{2}}\frac{1}{(s^{2}+\omega_{j}^{2})}d\omega_{j}$
(3)

where $\rho(\omega_{j})$ is the bath oscillator density, with an upper-
frequency limit $\omega_{\text{cufoff}}$. Following an argument similar to the
one pointed out by CL (Legget1, ), the requirement that the system becomes a
damped oscillator with frequency $\omega_{0}$ and a damping rate $\eta$ in the
classical limit, known as the "Ohmic friction" condition, leads to the
following constraint:

$\rho(\omega_{j})=\frac{2\eta M}{\pi}\frac{m_{j}\omega_{j}^{2}}{c_{j}^{2}}$
(4)

If the frequency renormalization constant $\Delta\omega^{2}$ is chosen to
satisfy

$M\Delta\omega^{2}=\int_{0}^{\omega_{\text{cufoff}}}\frac{c_{j}^{2}}{m_{j}}\frac{\rho(\omega_{j})}{\omega_{j}^{2}}d\omega_{j}=\frac{2\eta
M}{\pi}\omega_{\text{cufoff}}$ (5)

Then by observing Eqs.(3,4), it can be shown that for sufficiently large
$\omega_{\text{cufoff}}\gg\omega_{0}$, the first term of Eq.(3) represents a
frequency shift $\frac{2\eta M}{\pi}\omega_{\text{cufoff}}$, while the second
term of Eq.(3) leads to a damping term $-\eta Ms\overline{q}$, with the
damping constant $\eta$.

The frequency is shifted to $\omega_{0}$. Then Eq.(2) is simplified, and its
inverse Laplace transform yields the quantum Langevin equation, which is valid
at time $t>0_{+}$:

$\ddot{q}(t)+\eta\dot{q}(t)+\omega_{0}^{2}q(t)=f(t)$ (6)

with a constant magnetic force $B\sigma_{z}$ and the Brownian motion driving
force

$f(t)=B\sigma_{z}-\sum_{j}\frac{c_{j}}{M}\left(\text{$x_{j0}$}\cos(\omega_{j}t)+\text{$\dot{x}_{j0}$}\frac{\sin(\omega_{j}t)}{\omega_{j}}\right)$
(7)

During the derivation, in order to carry out the integral in Eq.(2), we used
the requirement of the inverse Laplace transform that $s$ must pass all the
singular points from the right side of the complex plane, and hence Re($s$) >
0\. Equations (6) and (7) are the equations of a driven damped harmonic
oscillator with a constant force, the solution of which is well known as a
linear combination of the initial values at $q_{0}-\sigma_{z}d,\dot{q}_{0}$ ,
$x_{j0},\dot{x}_{j0}$ and a displacement $\sigma_{z}d$, where
$d\equiv\frac{B}{\omega_{0}^{2}}$:

$\displaystyle
q(t)=a_{1}(t)\left(q_{0}-\sigma_{z}d\right)+a_{2}(t)\dot{q_{0}}+\sigma_{z}d+\sum_{j}\left(\text{$x_{j0}$}b_{j1}(t)+\text{$\dot{x}_{j0}$}b_{j2}(t)\right)$
(8) $\displaystyle
x_{i}(t)=\alpha_{i1}(t)\left(q_{0}-\sigma_{z}d\right)+\alpha_{i0}(t)\sigma_{z}d+\alpha_{i2}(t)\dot{q}_{0}+\sum_{j}\left(x_{j0}\beta_{ij1}(t)+\dot{x}_{j0}\beta_{ij2}(t)\right)$

with $a_{1}(t)=\frac{\mu e^{-\nu t}-\nu e^{-\mu t}}{\mu-\nu}$ ,
$a_{2}(t)=-\frac{e^{-\mu t}-e^{-\nu t}}{\mu-\nu}$,
$\mu\equiv\frac{\eta}{2}+\omega i$, $\nu\equiv\frac{\eta}{2}-\omega i$ and
here $\omega\equiv\left(\omega_{0}^{2}-\frac{\eta^{2}}{4}\right)^{1/2}$is the
frequency shifted by damping. And
$a_{1}(0)=1,a_{2}(0)=0,\dot{a}_{1}(0)=0,\dot{a}_{2}(0)=1$, (see Appendix I for
$a_{1}(t),a_{2}(t),b_{j1}(t),b_{j2}(t)$, etc). All formulas are correct
whether $\omega$ is real or imaginary. We have redefined the initial time as
$t=0_{+}$ to avoid a minor detail of the initial-value problem. The explicit
expressions for
$\alpha_{i0}(t),\alpha_{i1}(t),\alpha_{i2}(t),\beta_{ij1}(t),\beta_{ij2}(t)$
are well known in basic physics.

We emphasize that the use of the Laplace transform instead of the Fourier
transform allows us to express $q(t)$ and $x_{j}(t)$ explicitly in terms of
the initial values, as in Eqs. (8). Equations (8) serve as the starting point
of subsequent discussions. We will proceed to find the Green’s function of the
full system, and hence the solution of the wave function in the
Schr\̈cyrodinger representation. The result tells us in what sense the damped
oscillator is described by an effective Hamiltonian without the bath variables
and gives it a specific form. It also shows that under this condition, the
wave function can be factorized, and that the main factor relevant to the
damped oscillator is a solution of the Schr\̈cyrodinger equation with an
effective Hamiltonian.

Equations (8) are correct both in classical mechanics and in quantum mechanics
in the Heisenberg representation. We notice that $q(t)$ and $x_{j}(t)$ are
both linear superpositions of $q_{0}-\sigma_{z}d,\dot{q}_{0}$ ,
$x_{j0},\dot{x}_{j0}$, $\sigma_{z}d$ with c-number coefficients. The
commutation rules between $q(t),\dot{q}(t),x_{j}(t),\dot{x}_{j}(t)$ are
$[q(t),\dot{q}(t)]=\frac{i\hbar}{M}~{}$,
$[x_{j}(t),\dot{x}_{j}(t)]=\frac{i\hbar}{m_{j}}$ , and operators $q(t)$ and
$\dot{q}(t)$ commute with $x_{j}(t),\dot{x}_{j}(t)$. One can prove these
commutation rules in two ways: (1) by direct computation, using the fact that
at $t=0$, they are correct, and they all commute with operator $\sigma_{z}$
and (2) by the general principle that
$q(t),\dot{q}(t),x_{j}(t),\dot{x}_{j}(t)$ are related by a unitary
transformation to $q_{0},\dot{q}_{0},\sigma_{z},x_{j0},\dot{x}_{j0}$.

Equations (8) show that the operators $q(t)$ and $x_{j}(t)$ can each be
written as a sum of two terms:

$\displaystyle q(t)=\left(Q(t)+\sigma_{z}d\right)+\sum_{j}\xi_{j}(t)$ (9)
$\displaystyle x_{i}(t)=\zeta_{j}(t)+\sum_{j}X_{ij}(t)$

where

$\displaystyle
Q(t)=a_{1}(t)\left(q_{0}-\sigma_{z}d\right)+a_{2}(t)\dot{q}_{0}=a_{1}(t)Q_{0}+a_{2}(t)\dot{Q}_{0}$
(10)
$\displaystyle\xi_{j}(t)=\text{$x_{j0}$}b_{j1}(t)+\text{$\dot{x}_{j0}$}b_{j2}(t)$

$\left(Q(t)+\sigma_{z}d\right)$ and $\zeta(t)$ are linear in
$q_{0},\dot{q}_{0},$ and $\sigma_{z}$ and independent of
$x_{j0},\dot{x}_{j0}$, and $\xi_{j}(t)$ and $X_{ij}(t)$ are linear in
$x_{j0},\dot{x}_{j0}$, and independent of $q_{0},\dot{q}_{0},$ and
$\sigma_{z}$. Thus $Q(t),\sigma_{z}d$ and $\zeta(t)$ are operators in one
Hilbert space $S_{Q}$, while $\xi_{j}(t)$ and $X_{ij}(t)$ are in an
independent Hilbert space $S_{X}$, and the full Hilbert space is a direct
product of $S_{Q}\otimes S_{X}$.

We shall first analyze the structure of $S_{Q}$. To explicitly show that we
are discussing the $S_{Q}$ space, we define $Q_{0}\equiv q_{0}-\sigma_{z}d$,
$\dot{Q}_{0}\equiv\dot{q}_{0}=-\frac{i\hbar}{M}\frac{\partial}{\partial
q_{0}}$. Because $\sigma_{z}d$ is a constant operator and it commutes with
$q_{0}$ and $\dot{q}_{0}$, we have
$[Q_{0},\dot{Q}_{0}]=[q_{0},\dot{q}_{0}]=\frac{i\hbar}{M}$. Thus, we can write
$-\frac{i\hbar}{M}\frac{\partial}{\partial
Q_{0}}\equiv\dot{Q}_{0}=\dot{q}_{0}$ as a well-defined operator in the space
of $q_{0},\sigma_{z}$, i.e., $S_{Q}$. The eigenfunction of $Q(t)$ with an
eigenvalue denoted by $Q_{1}$, in the $Q_{0}$ representation, is easily
calculated to be (see Appendix II),

$u_{Q_{1}}(Q_{0},t)=\left(\frac{M\omega
e^{\frac{\eta}{2}t}}{2\pi\hbar\sin(\omega
t)}\right)^{\frac{1}{2}}\exp\left[-i\frac{M}{2\hbar
a_{2}}\left(a_{1}Q_{0}^{2}-2Q_{1}Q_{0}+\phi(Q_{1},t)\right)\right]$ (11)

with $\phi$ as an arbitrary phase, i.e., a real number. This eigenfunction is
related to Green’s function

$G(Q_{1},Q_{0};t,0)=<Q_{1}|U(t)|Q_{0}>,$ (12)

where we denote the evolution operator by $U(t$). To see this, we use the
relation between Schr\̈cyrodinger operator $\mathbf{Q}_{S}$ and Heisenberg
operator $Q(t)\equiv\mathbf{Q}_{H}(t)=U^{-1}(t)\mathbf{Q}_{S}U(t)$.

Let $|Q_{1}>$ be the eigenvector of $\mathbf{Q}_{S}$ with eigenvalue $Q_{1}$,
i.e.,

$\displaystyle\mathbf{Q}_{S}|Q_{1}>=Q_{1}|Q_{1}>$

we see that $U^{-1}|Q_{1}>$ is the eigenvector of $Q(t)$ of value $Q_{1}$

$\displaystyle
Q(t)U^{-1}|Q_{1}>=\mathbf{Q}_{H}U^{-1}|Q_{1}>=U^{-1}\mathbf{Q}_{S}UU^{-1}|Q_{1}>=U^{-1}\mathbf{Q}_{S}|Q_{1}>=Q_{1}U^{-1}|Q_{1}>$

So $U^{-1}|Q_{1}>$ is proportional to $u_{Q_{1}}(Q_{0},t)$. If we choose
$u_{Q_{1}}(Q_{0},t)$ as an orthonormal basis, then $U$ is unitary:
$U^{\dagger}=U^{-1}$.

Then, we have
$u_{Q_{1}}(Q_{0},t)=<Q_{0}|U^{-1}|Q_{1}>=<Q_{0}|U^{\dagger}|Q_{1}>=<Q_{1}|U(t)|Q_{0}>^{*}=G^{*}(Q_{1},Q_{0};t,0)$,
i.e., $G(Q_{1},Q_{0};t,0)=u_{Q_{1}}^{*}(Q_{0},t)$. Thus we have

$G(Q_{1},Q_{0};t,0)=\left(\frac{M\omega
e^{\frac{\eta}{2}t}}{2\pi\hbar\sin(\omega
t)}\right)^{\frac{1}{2}}\exp\left[i\frac{M}{2\hbar
a_{2}}\left(a_{1}Q_{0}^{2}-2Q_{1}Q_{0}+\phi(Q_{1},t)\right)\right]$ (13)

Next, we shall determine the arbitrary phase $\phi(Q_{1},t)$ , which is the
phase of the eigenvectors of $Q(t)$. Using Eq.(10), we find the commutation
rule for $Q$ and $\dot{Q}$ (see Appendix I):

$[Q,\dot{Q}]=\left(a_{1}(t)\dot{a_{2}}(t)-\dot{a}_{1}(t)a_{2}(t)\right)[Q_{0},\dot{Q_{0}}]=e^{-\eta
t}[Q_{0},\dot{Q_{0}}]=e^{-\eta t}\frac{i\hbar}{M}$ (14)

Thus, we define the canonical momentum $P(t)$ as

$P=Me^{\eta t}\dot{Q}=Me^{\eta
t}\left(\dot{a}_{1}(t)Q_{0}+\dot{a}_{2}(t)\dot{Q}_{0}\right)=Me^{\eta
t}\dot{a}_{1}(t)Q_{0}-i\hbar e^{\eta t}\dot{a}_{2}(t)\frac{\partial}{\partial
Q_{0}}$ (15)

and get the commutation rule $[Q(t),P(t)]=i\hbar$. The eigenfunction of $P(t)$
can be calculated in two ways: (1) we can calculate the eigenvector of $P(t)$
in the $Q_{0}$ representation using Eq.(15) and then use Green’s function
Eq.(13) to transform it into the $Q(t)$ representation and (2) the commutation
rule $[Q(t),P(t)]=i\hbar$ requires that $P(t)=-i\hbar\frac{\partial}{\partial
Q}$, so the eigenfunction of $P(t)$ with eigenvalue $P_{1}$ is
$\exp[i\frac{P_{1}}{\hbar}Q]$. By comparing these two solutions, the arbitrary
phase $\phi(Q_{1},t)$ in Green’s function is determined to be within a phase
$\phi(t)$ , which is independent of $Q_{1}$. $\phi(t)$ is an arbitrary real
function of time, except that $\phi(0)$ = 0 so that it satisfies the condition
that at $t=0$ , the Green’s function becomes $\delta(Q_{1}-Q_{0})$. Thus, we
obtain Green’s function in the $S_{Q}$ space:

$\displaystyle G(Q_{1},Q_{0};t,0)=\left(\frac{M\omega
e^{\frac{\eta}{2}t}}{2\pi i\hbar\sin(\omega
t)}\right)^{\frac{1}{2}}\exp\left[i\frac{M}{2\hbar
a_{2}}\left(a_{1}Q_{0}^{2}-2Q_{1}Q_{0}+e^{\eta
t}\dot{a}_{2}Q_{1}^{2}\right)-\frac{i}{\hbar}\phi(t)\right]$ (16)

It is then straightforward to derive the Hamiltonian $H_{Q}$ using the
following relation:

$H_{Q}=i\hbar\left(\frac{\partial}{\partial t}U_{Q}\right)U_{Q}^{-1}$ (17)

and remember that the matrix elements of $U_{Q}$ and $U_{Q}^{-1}$ are Green’s
function and its conjugate. The result is

$H_{Q}=e^{-\eta t}\frac{P^{2}}{2M}+\frac{1}{2}Me^{\eta
t}\omega_{0}^{2}Q^{2}+\dot{\phi}(t)$ (18)

Since $\phi$ is arbitrary except that $\phi((0)=0$, we can take $\phi(t)=0$.
Therefore, we have derived the well-known effective Hamiltonian for the
dissipative system. We emphasize that the expression for $H_{Q}$ is derived
here, whereas it is usually introduced by heuristic arguments.

Next, we shall analyze the effect of the bath. Similar to Eq.(10), we define
the contribution of the bath oscillator $j$ to the Brownian motion of the main
oscillator as

$\text{$\xi_{j}(t)\equiv x_{j0}$}b_{j1}(t)+\text{$\dot{x}_{j0}$}b_{j2}(t)$
(19)

where $b_{j1}(t),b_{j1}(t),\dot{b}_{j1}(t),\dot{b}_{j1}(t)$ all equal to zero
at $t=0$ (see Appendix I), so $\xi_{j}(t=0)=0$ and $\dot{\xi}_{j}(t=0)=0$, and
hence in Eq.(9), $q(0)=q_{0},\dot{q}(0)=\dot{q}_{0}$.

Similar to Eq.(11) we obtain the eigenfunctions $\theta_{\xi_{j1}}(x_{j0},t)$
for $\xi_{j}$ and $t>0$. Using Dirac’s notation we have

$\displaystyle Q|u_{Q_{1}},s\rangle=Q_{1}|u_{Q_{1}},s\rangle$
$\displaystyle\xi_{j}|\theta_{\xi_{j1}}\rangle=\xi_{j_{1}}|\theta_{\xi_{j1}}\rangle,$

where $s=\pm 1$ is the eigenvalue of $\sigma_{z}$:

$\sigma_{z}|s>=s|s>,$ (20)

and

$|u_{Q},s\rangle\equiv u_{Q}(Q_{0},t)|s>.$ (21)

Thus $|u_{Q},s\rangle\prod_{j}\otimes|\theta_{\xi_{j}}\rangle$ is an
eigenvector of $q(t)$, with eigenvalue of $q=Q+sd+\sum_{j}\xi_{j}$. Hence
$Q=q-sd-\sum_{j}\xi_{j}$. In other words, the eigenvector of $q(t)$ with
eigenvalue $q$ is

$|q,s,\\{\xi_{j}\\}\rangle=|u_{q-sd-\sum_{j}\xi_{j}},s\rangle\prod_{j}\otimes|\theta_{\xi_{j1}}\rangle$
(22)

The set of Hermitian $q,s,\\{\xi_{j}\\}$ is a set of commuting observables
distinct from the set of $q,s,\\{x_{j}$}. The set of wave functions
$|q,s,\\{\xi_{j}\\}\rangle$ forms a Hilbert space’s orthonormal basis.

We first study the evolution of the damped simple harmonic oscillator wave
function. Initially ($t=0$) it is $u_{Q_{1}}(Q_{0},t)$. Then, at time $t$,
according to the discussion following Eq.(11) ,
$G(Q_{1},Q_{0};t,0)\equiv\left\langle
Q_{1}\left|U_{Q}(t)\right|Q_{0}\right\rangle$, and
$u_{Q_{1}}(Q_{0},t)=\left\langle
Q_{0}\left|U_{Q}^{-1}(t)\right|Q_{1}\right\rangle$, it evolves into

$\displaystyle\intop dQ_{0}G(Q_{2},Q_{0};t,0)u_{Q_{1}}(Q_{0},t)=\intop
dQ_{0}\left\langle Q_{2}\left|U_{Q}(t)\right|Q_{0}\right\rangle\left\langle
Q_{0}\left|U_{Q}^{-1}(t)\right|Q_{1}\right\rangle$ $\displaystyle=\left\langle
Q_{2}\left|U_{Q}(t)U_{Q}^{-1}(t)\right|Q_{1}\right\rangle=\left\langle
Q_{2}\mid Q_{1}\right\rangle=\delta(Q_{2}-Q_{1})$

Similarly, we can confirm that if the wave function of the bath oscillator $j$
is $\theta_{\xi_{j1}}(x_{j0},t)$, at time $t>0$, it evolves into
$\delta(\xi_{j}-\xi_{j1})$ at time $t>0$.

We emphasize that if the initial state (at $t=0_{+}$) is
$|q_{1},s_{1},\\{\xi_{j1}\\}\rangle$, then the wave function will evolve at
time $t$ into
$\delta(q-q_{1})\delta_{s,s_{1}}\prod_{j}\delta(\xi_{j}-\xi_{j1})$. At time
$t=0_{+}$, because $Q(t),$$\sigma_{z}$ and $\\{\xi_{j}(t)\\}$ are linear
combinations of operators
$q_{0},\dot{q}_{0},\sigma_{z},\text{$x_{j0}$},\text{$\dot{x}_{j0}$}$, they
form a complete set of commuting observables of time $t=0_{+}$. In other
words, at time $t=0_{+}$, we consider the unitary transform from
$q_{0},\sigma_{z},\text{$x_{j0}$,}$ to $Q(t),$$\sigma_{z}$ and
$\\{\xi_{j}(t)\\}$ as a variable transform. The set of wave functions
$|q_{1},s_{1},\\{\xi_{j1}\\}\rangle$ forms an orthonormal basis in the
system’s Hilbert space, which we call the "eigenstates of the system”. As a
result, if the system is initially (that is, at $t=0_{+}$) in one of these
states, $|q_{1},s_{1},\\{\xi_{j1}\\}\rangle$, the final result at time $t$ is
predetermined, i.e., the measurement is deterministic, with a definite value
of $q_{1},s_{1},\\{\xi_{j1}\\}$ .

## 2\. The initial superposition of eigenstates determines the final
probability distribution

If the wave function is initially
$|\Psi_{0}\rangle=|\psi_{0},s\rangle\prod_{j}\otimes|\chi_{j0}\rangle$, then
to calculate the wave function at time $t$, we must expand $\Psi_{0}$ in terms
of the eigenvectors $|q,s,\\{\xi_{j}\\}\rangle$ in Eq.(22), i.e., we must
calculate

$\displaystyle\psi(Q,s,t)=<u_{Q},s,t|\psi_{0},s>=\int
u_{Q}^{*}(q_{0},t)\psi_{0}(q_{0})dq_{0}<s|s>=\int
u_{Q}^{*}(q_{0},t)\psi_{0}(q_{0})dq_{0}$ (23)
$\displaystyle\chi_{j}(\xi_{j},t)=<\theta_{\xi_{j}},t|\chi_{j0}>=\int\theta_{\xi_{j}}^{*}(x_{j0},t)\chi_{j0}(x_{j0})dx_{j0}$

Notice that even though we label these functions by the parameter $t$, they
are the wave functions at time $t=0$. We label them with $t$ only to show that
we define them as the initial state that will evolve to specified
$Q,\\{\xi_{j}\\}$ at time $t$. The probability amplitude for the system in
state $|q_{1},s_{1},\\{\xi_{j1}\\}\rangle$ at time $t=0_{+}$ is then

$\displaystyle\Psi(q_{1},s_{1},\\{\xi_{j1}\\},t)=<q_{1},s_{1},\\{\xi_{j1}\\}|\Psi_{0}>=<u_{q_{1}-s_{1}d-\sum_{j}\xi_{j1}},s_{1},t|\psi_{0}>|s_{1}>\prod_{j}<\theta_{\xi_{j1}},t|\chi_{j0}>$
$\displaystyle\equiv\psi(q_{1}-s_{1}d-\sum_{j}\xi_{j1},s_{1},t)\prod_{j}\chi_{j}(\xi_{j1},t)$
(24)

Because $|q_{1},s_{1},\\{\xi_{j1}\\}\rangle$ evolves into
$\delta(q-q_{1})\delta_{s,s_{1}}\prod_{j}\delta(\xi_{j}-\xi_{j1})$ at time
$t$, the wave function at time $t$ is

$\displaystyle\int dq_{1}\prod_{j}\int
d\xi_{j1}\sum_{s_{1}}\delta(q-q_{1})\delta_{s,s_{1}}\prod_{j}\delta(\xi_{j}-\xi_{j1})\Psi(q_{1},s_{1},\\{\xi_{j1}\\},t)=\Psi(q,s,\\{\xi_{j}\\},t)$
(25)

This equation seems to be redundant because it simply replaces
$q_{1},s_{1},\\{\xi_{j1}\\}$ by $q,s,\\{\xi_{j}\\}$, so in the following, we
shall only use Eq.(24) to get the wave function. However, the purpose of this
redundant step is to point out that the final wave function is obtained by
projecting the initial state onto the system eigenstates, which themselves are
taken as the initial states. We emphasize here again that this probability
amplitude is completely determined by the initial wave function
$|\Psi_{0}\rangle$. More examples will be provided to demonstrate the
implications of this point.

Notice that $\psi(Q,t)$ of Eq.(23) is the wave function in the Schrödinger
representation with the effective Hamiltonian Eq.(18). Thus, we have connected
the effective Hamiltonian approach to the dissipative system problem with the
other approaches that take both the system and the bath into account. We also
notice that, despite the fact that our $\Psi(q,s,\\{\xi_{j}\\},t)$ is in a
different representation than $\Psi(q,s,\\{x_{j}\\},t)$, the usual probability
interpretation remains valid:
$\int\int...\int|$$\Psi(q,s,\\{\xi_{j}\\},t)|^{2}\prod_{j}d\xi_{j}$ is the
probability density of finding the particle at $q$. Since this solution is
very simple, it provides a simple way to analyze other complicated problems,
e.g., studying the influence of Brownian motion on interference, on which we
shall not elaborate.

Under certain conditions, for example, at low temperatures and when the system
$q$ is in highly excited states, the range of $q$ is large enough that we can
approximately write
$\Psi(q,s,\\{\xi_{j}\\},t)=\psi(q-sd,s,t)\prod_{j}\chi_{j}(\xi_{j},t)$ for all
the values of $j$ that do not have a vanishingly small probability,
$q\gg|\sum_{j}\xi_{j}|$. That is, the wave function is factorized, the
dissipative system $q$ can be described by the wave function $\psi(q-sd,s,t)$
only, and the Brownian motion can be ignored. As a result, it is worthwhile to
investigate the width of the argument of the wave function due to Brownian
motion, i.e., the mean value of $\left(\sum_{j}\xi_{j}\right)^{2}$ at time
$t$. It can be calculated using the expression $\psi(Q,s,t)$ and
$\chi_{j}(\xi_{j},t)$ in Eq.(23), as shown in the example below.

As one example of the superposition of the “eigenstates of the system,” we
assume the absolute temperature is zero; the initial state of the bath
oscillators is in the ground state; the initial state of the damped simple
harmonic oscillator is a wave packet at the origin ($q=0$) with width same as
the width of ground state of the simple harmonic oscillator
$\sigma_{q}^{2}=\frac{\hbar}{2M\omega_{0}}$ and the spin state is $|s>=|+1>$.
The initial state of the damped harmonic oscillator is

${\displaystyle\psi_{0}(Q_{0})=\left(\frac{1}{2\pi\sigma_{q}^{2}}\right)^{\frac{1}{4}}\exp\left(-\frac{1}{4\sigma_{q}^{2}}\left(Q_{0}+d\right)^{2}\right)}$
(26)

where $Q_{0}=q_{0}-d$ and $d\equiv\frac{B}{\omega_{0}^{2}}$. Following the
step of Eq.(24) by Gaussian integration, we find the wave function at time $t$
as a Gaussian,

$\displaystyle\psi(Q,t)=\phi_{0}(Q,t)\exp\left(\frac{M\omega_{0}}{\hbar}\frac{-2Qd-
a_{1}d^{2}}{2\left(a_{1}+i\omega_{0}a_{2}\right)}\right)$
$\displaystyle\phi_{0}(Q,t)=\left(\frac{M\omega_{0}}{\pi\hbar}\right)^{\frac{1}{4}}\left(\frac{1}{a_{1}(t)+i\omega_{0}a_{2}(t)}\right)^{\frac{1}{2}}\exp\left(-\frac{M\omega_{0}}{2\hbar}\frac{\omega_{0}a_{2}(t)-i\dot{a}_{2}(t)}{\omega_{0}a_{2}(t)-ia_{1}(t)}e^{\eta
t}Q^{2}\right)$

where $\phi_{0}(Q,t)$ is the solution when the magnetic field is turned off
($B=0$). The functions $a_{1}(t),a_{1}(t),\dot{a}_{1}(t),\dot{a}_{2}(t),$are
given in Appendix I. The initial function
$\psi_{0}(Q_{0})=\phi_{0}(Q_{0}+d,t=0)$ is the function of the ground state of
the oscillator with the centroid displaced to $-d$ but $Q_{0}$ is the position
relative to $d$, so the centroid is at $q_{0}=0$.

The probability distribution of $Q$ ignoring the contribution from the bath,
is given by

$\displaystyle|\psi(Q,t)|^{2}=\left(\frac{1}{2\pi\sigma_{Q}^{2}}\right)^{\frac{1}{2}}\exp\left(-\frac{\left(Q+a_{1}d\right)^{2}}{2\sigma_{Q}^{2}}\right)$
(27)
$\displaystyle\sigma_{Q}^{2}\equiv\frac{\hbar}{2M\omega_{0}}\left(\omega_{0}^{2}a_{2}^{2}+a_{1}^{2}\right)=\sigma_{q}^{2}\left(1+\frac{\eta^{2}}{4\omega^{2}}+\frac{1}{2}\frac{\eta}{\omega}\sin(2\omega
t)-\frac{\eta^{2}}{4\omega^{2}}\cos(2\omega t)\right)e^{-\eta t}$

This is a wave packet that begins at $Q=-d$ (i.e., $q=0$), oscillates around
$Q=0$ (i.e. $q=d)$ , and has its amplitude dampened to zero because
$a_{1}(t)=e^{-\frac{\eta}{2}t}\left(\cos(\omega
t)+\frac{\eta}{2\omega}\sin(\omega t)\right)$. With damping time $2/\eta$, the
width $\sigma_{Q}(t)$ also damps to zero,so the wave function collapses to the
point at $d$. If the initial spin state is $s=-1$, it collapses to $-d$.

To include the Brownian motion from the bath, we have the bath oscillator
initial wave function in ground state

${\displaystyle\chi_{j0}(x_{j0})=\left(\frac{m_{j}\omega_{j}}{\pi\hbar}\right)^{\frac{1}{4}}\exp\left(-\frac{m_{j}\omega_{j}}{2\hbar}x_{j0}^{2}\right)}$
(28)

The eigenfunction of
$\xi_{j}|\theta_{\xi_{j1}}\rangle=\xi_{j_{1}}|\theta_{\xi_{j1}}\rangle$ gives
$\theta_{\xi_{j1}}(x_{j0},t)=\left(\frac{m_{j}}{2\pi\hbar
b_{j2}(t)}\right)^{\frac{1}{2}}\exp\left[-i\frac{m_{j}}{2\hbar
b_{j2}(t)}\left(b_{j1}(t)x_{j0}^{2}-2\xi_{j1}x_{j0}\right)\right]$. The
derivation of $\theta_{\xi_{j1}}(x_{j0},t)$ is very similar to the derivation
of the eigenfunction $u_{Q}$ of $Q(t)$ in Appendix II, (see Appendix I for the
expression for $b_{j1}(t),b_{j2}(t)$ ). The projection of the ground state
onto the eigenvector in Eq.(24) is found by another Gaussian integral as

$\displaystyle\chi_{j}(\xi_{j},t)=<\theta_{\xi_{j}},t|\chi_{j0}>=\left(\frac{m_{j}\omega_{j}}{\pi\hbar}\right)^{\frac{1}{4}}\left(\frac{1}{-ib_{j1}+\omega_{j}b_{j2}(t)}\right)^{\frac{1}{2}}\exp\left(-\frac{m_{j}}{2\hbar
b_{j2}}\frac{\xi_{j1}^{2}}{\left(-ib_{j1}+\omega_{j}b_{j2}(t)\right)}\right)$
(29)

As a result, the contribution to the Brownian motion width derived from
$\chi_{j}(\xi_{j},t)$ is
$<\xi_{j}^{2}>=\sigma_{\xi_{j}}^{2}=\frac{1}{2}\left(b_{j1}^{2}+\omega_{j}^{2}b_{j2}^{2}\right)\frac{\hbar}{m_{j}\omega_{j}}$.

According to the discussion that follows Eq.(24,25), the width of the wave
function

$\Psi(q_{1},s_{1},\\{\xi_{j1}\\},t)=\psi(q-sd-\sum_{j}\xi_{j},s,t)|s>\prod_{j}\chi_{j}(\xi_{j},t)$
(30)

can be calculated from the widths of $\psi(Q,t)$ in Eq.(27) and
$\chi_{j}(\xi_{j},t)$ in Eq.(29) as
$\sigma_{Q}^{2}+<\left(\sum_{j}\xi_{j}\right)^{2}>$ (see Appendix III for the
derivation). Long after the damping time $2/\eta$ , the width
$\sigma_{Q}^{2}\rightarrow 0$ , and thus the Brownian width is given by

$<\left(\sum_{j}\xi_{j}(t)\right)^{2}>=\int\int...\int\left(\sum_{j}\xi_{j}(t)\right)^{2}\prod_{j}|\chi_{j}(\xi_{j},t)|^{2}d\xi_{j}=\sum_{j}<\xi_{j}^{2}>=\sum_{j}\sigma_{\xi_{j}}^{2}$
(31)

From the bath’s spectral density Eq.(4), and the expressions for
$b_{j1},b_{j2},\omega,\omega_{0},\eta$ in Appendix I, and Eq.(8), we find

$\displaystyle\sigma_{\xi}^{2}\equiv\sum_{j}\sigma_{\xi_{j}}^{2}=\sum_{j}\frac{\hbar}{2m_{j}\omega_{j}}\left(b_{j1}^{2}+\omega_{j}^{2}b_{j2}^{2}\right)=\int
d\omega_{j}\rho(\omega_{j})\frac{\hbar}{2m_{j}\omega_{j}}\left(b_{j1}^{2}+\omega_{j}^{2}b_{j2}^{2}\right)$
(32) $\displaystyle=\frac{\eta\hbar}{\pi M}\int
d\omega_{j}\omega_{j}\frac{M^{2}}{c_{j}^{2}}\left(b_{j1}^{2}+\omega_{j}^{2}b_{j2}^{2}\right)$

As $t\rightarrow\infty$, this approaches

$\displaystyle\sigma_{\xi}^{2}=\sum_{j}\sigma_{\xi_{j}}^{2}=\frac{\eta\hbar}{2\pi
M}\int_{0}^{\infty}d\omega_{j}^{2}\frac{1}{\text{($\omega_{0}^{2}$-$\omega_{j}^{2})^{2}$+$\eta^{2}\omega_{j}^{2}$}}=\frac{\hbar}{2\pi\omega
M}\left(\frac{\pi}{2}+\arctan(\frac{\omega^{2}-\frac{\eta^{2}}{4}}{\eta\omega})\right)$

If $\eta\ll$$\omega$, i.e., if the damping time is much longer than the main
oscillator period.

$\displaystyle<\left(\sum_{j}\xi_{j}(t)\right)^{2}>\approx\frac{\hbar}{2\pi\omega
M}\left(\frac{\pi}{2}+\arctan(\frac{\omega}{\eta})\right)\approx\frac{\hbar}{2\omega
M}$

This Brownian width approximately equals the ground state of the simple
harmonic oscillator.

As a result, depending on whether $s=1$ or $-1$, the wave function collapses
to one of the two split potential wells at $\pm d$ with a spread
$\sigma_{\xi}=\sqrt{\frac{\hbar}{2\omega M}}$. As long as
$d\gg\sqrt{\frac{\hbar}{2\omega M}}$ , the spin measurement has a definite
answer. From the discussion following Eq.(24,25), this result is completely
predetermined by the initial spin state.

Figure 1: (a) Wave function probability density without Brownian motion (b)
with Brownian motion. The initial spin state is specified by a vector
$\vec{a}$ on the Block sphere with $\theta=\pi/4$ for this case. The
probability for the spin up and down is both non-zero. The color scale is the
probability density. The two wave packets of the spin up and down are well
separated near the beginning. When Brownian motion is ignored, the wave
function profile width approaches zero at $t\geq 5$. The wave function
collapses to either at $q=d=3$, or $q=-d=-3$ for $t>5$, with the probability
of being up and down as $\cos^{2}(\theta/2)=0.8535$ , and
$\sin^{2}(\theta/2)=0.1464$.

## 3\. An example of the case when the Bloch vector of the pure state spinor
is not aligned with the $z$ axis

When the spin is in a pure state, the initial state can always be specified by
a vector $\vec{a}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ on the
Bloch sphere(bloch, ) as

$\displaystyle|\vec{a}>=\cos(\theta/2)\left|+\right\rangle+e^{i\phi}\sin(\theta/2)\left|-\right\rangle$

Then the initial state $\Psi_{0}=|\psi_{0}>|\vec{a}>\prod_{j}|\chi_{j0}>$.

$\vec{a}$ is not parallel to the $z$ axis unless $\theta=0$, or $\pi$. Its
projection onto the initial system eigenstate $|q_{1},s_{1},\\{\xi_{j1}\\}>$
in Eq.(22) , given by Eq.(24,29) is

$\displaystyle\Psi(q_{1},s_{1},\\{\xi_{j1}\\},t)=<q_{1},s_{1},\\{\xi_{j1}\\}|\Psi_{0}>=<u_{q_{1}-s_{1}d-\sum_{j}\xi_{j1}},s_{1},t|\psi_{0}><s_{1}|\vec{a}>\prod_{j}<\theta_{\xi_{j1}},t|\chi_{j0}>$
$\displaystyle=\psi(q_{1}-s_{1}d-\sum_{j}\xi_{j1},s_{1},t)<s_{1}|\vec{a}>\prod_{j}\chi_{j}(\xi_{j1},t)$

Figure 2: (a) the width $\sigma_{Q\xi}(t)$, $,\sigma_{\xi}(t),$ and
$\sigma_{Q}(t)$; (b)Probability in the wave packet with spin up (magenta),
down(cyan), and total (blue);

Figure 3: (a) the wave function density profile of the wave packets at $t=0$
and very close to the beginning at $t=0.05$. The two wave packets are already
starting to separate. (b) Wave function density profile at $t=0$ and $t=2$
compares with and without Brownian motion included. We also indicate the
positions of the two potential wells corresponding to spin up or down, with an
arbitrary scale. The two wave packets are close but do not reach the bottom of
the potential wells yet at $t=2$.

According to the discussion following Eq.(24,25), at time $t$, $\Psi_{0}$
evolves into the wave function

$\displaystyle\Psi_{+}(q,t)\equiv\Psi(q,1,\\{\xi_{j}\\},t)=\psi(q-d-\sum_{j}\xi_{j},1,t)\cos(\theta/2)\prod_{j}\chi_{j}(\xi_{j},t)$
(33)
$\displaystyle\Psi_{-}(q,t)\equiv\Psi(q,-1,\\{\xi_{j}\\},t)=\psi(q+d-\sum_{j}\xi_{j},-1,t)e^{i\phi}\sin(\theta/2)\prod_{j}\chi_{j}(\xi_{j},t)$

The wave packet split into two, with $s=1$ collapsing to $q=d$ with a
probability of $\cos^{2}(\theta/2)$, and $s=-1$ collapsing to $q=-d$ with a
probability of $\sin^{2}(\theta/2)$, both with a width
$\sigma_{\xi}=\sqrt{\frac{\hbar}{2\omega M}}$, given by Eq.(32)

At temperature $T$ the contribution from the Brownian width of the bath to the
width of $q$ is calculated by the bath density matrix as

$<\left(\sum_{j}\xi_{j}(t)\right)^{2}>=\sum_{j}\frac{\hbar}{2m_{j}\omega_{j}}\left(b_{j1}^{2}+\omega_{j}^{2}b_{j2}^{2}\right)\coth(\frac{\hbar\omega_{j}}{2kT})$
(34)

This approaches the width in Eq.(32) as $T$ approaches zero. The width
increases with temperature.

When we apply Eq.(41) and Eq.(27) for $|\psi(Q,t)|^{2}$ given in Appendix III,
we have

$\displaystyle|\Psi_{+}(q,t)|^{2}=\cos^{2}(\theta/2)\frac{1}{\sqrt{2\pi}}\frac{1}{\sigma_{Q\xi}}\exp\left(-\frac{\left(q-d+a_{1}d\right)^{2}}{2\sigma_{Q\xi}^{2}}\right)$
$\displaystyle|\Psi_{-}(q,t)|^{2}=\sin^{2}(\theta/2)\frac{1}{\sqrt{2\pi}}\frac{1}{\sigma_{Q\xi}}\exp\left(-\frac{\left(q+d-a_{1}d\right)^{2}}{2\sigma_{Q\xi}^{2}}\right)$

where the width of the wave function density is
$\sigma_{Q\xi}^{2}=\sigma_{Q}^{2}+\sigma_{\xi}^{2}$.

For this abstract model, for a case with $\hbar=1,M=1$, and the main
oscillator period $T_{0}=1,\omega_{0}=2\pi/T_{0}$, $\eta=2$, we choose $d=3$,
at temperature $T=0$. The angle between the spin direction and the $z$ axis is
chosen to be $\theta=\pi/4$. We plot the
$|\Psi_{+}(q,t)|^{2},|\Psi_{-}(q,t)|^{2}$ versus $t$ in Fig.1a when the
Brownian motion is ignored, i.e. with width $\sigma_{Q}$, and in Fig.1b with
Brownian motion included, i.e., with width $\sigma_{Q\xi}$. The separation of
the two wave packets is visible from the very beginning near $t=0$.

The width in Eq.(27) $\sigma_{Q}^{2}(t)\rightarrow 0$ as $t\rightarrow\infty$.
As a result, the wave packet is already approaching $\delta$ function at $t=5$
in Fig.1a.

The width of $|\psi(Q,t)|^{2}$, i.e., $\sigma_{Q}(t)$, Brownian width
$\sigma_{\xi}(t)$ and total width $\sigma_{Q\xi}(t)$, as function of $t$ are
also shown in Fig.2a.

The probability of falling into an upper and lower potential well is shown in
Fig.2b, i.e. they are constants starting from $t=0_{+}$: for $\theta=\pi/4$,
they are $\cos^{2}(\theta/2)=0.8535$ (magenta); $\sin^{2}(\theta/2)=0.1464$
(cyan). That is, they are predetermined. Their sum is $1$(blue).

The profile of wave packets with spin up and down begins to separate at
$t=0.05$, as shown in Fig.3a. The wave packets at $t=2$ are shown in Fig.3b,
where we compare the profiles with and without Brownian motion when they are
well separated.

The conclusion is: the probability distribution $\cos^{2}(\theta/2)$,
$\sin^{2}(\theta/2)$ is predetermined by the initial state $|\vec{a}>$.

## 4\. Implications for applying the result to the Bell theorem

We can now apply the results of Sections 1,2,3 to the Bell theorem. The main
results of these sections are that the final result of the measurement is
determined by the wave function of the initial state, as stressed at the end
of Section 1, and clearly shown in Fig.2b. Since this is only for a specific
model, the interpretation and the implication here are only suggestive.

### 4.1. We first repeat the statement of the Bell theorem to clarify the
notations used here.

In the Bell theorem experiment (bell, ; bell_theorem, ), we assume that the
two entangled particles leave the origin at point O and move in opposite
directions, so that particle 1 reaches position A and measures
$\sigma_{1}\centerdot\vec{a}$ and particle 2 reaches position B and measures
$\sigma_{2}\centerdot\vec{b}$ ($\vec{a}$ ,$\vec{b}$ are some unit vectors).

The paper assumed a hidden variable $\lambda$ such that the result of
measuring $\sigma_{1}\centerdot\vec{a}$ is determined by $\vec{a}$ and
$\lambda$ as $A(\vec{a},\lambda)$, and the result of measuring
$\sigma_{2}\centerdot\vec{b}$ is determined by $\vec{b}$ and $\lambda$ as
$B(\vec{b},\lambda)$ in the same instance, and

$A(\vec{a},\lambda)=\pm 1;B(\vec{b},\lambda)=\pm 1.$ (35)

According to Bell, “The vital assumption is that the result B for particle 2
does not depend on the setting $\vec{a}$ of the magnet for particle 1, nor A
on $\vec{b}$”(bell, ). Bell assumed that if $\rho(\lambda)$ is the probability
distribution of $\lambda$, then the expectation value of the product of the
two components $\sigma_{1}\centerdot\vec{a}$ and $\sigma_{2}\centerdot\vec{b}$
is

$P(\vec{a},\vec{b})=\intop
d\lambda\rho(\lambda)A(\vec{a},\lambda)B(\vec{b},\lambda)$ (36)

Bell supposes that the experimenter has a choice of settings for the second
detector: it can be set either to $\vec{b}$ or to $\vec{c}$ . Bell proves that
if Eq.(35) and Eq.(36) are correct, the difference in correlation between
these two choices of detector settings must satisfy the inequality

$|P(\vec{a},\vec{b})-P(\vec{a},\vec{c})|\leq 1+P(\vec{b},\vec{c})$ (37)

Then, Bell shows it is easy to find situations where quantum mechanics
violates the inequality, i.e., the “Bell inequality”, thus Eq.(35) and Eq.(36)
derived from the “local hidden variable theory,” is incompatible with quantum
mechanics.

### 4.2. The analysis of two separate measurements at remote positions A and B
on two particles is equivalent to the analysis of two measurements on one
particle at the origin

When we apply the result of Sections 1, 2, and 3, because the result is
predetermined by the initial condition, it is clear that the measurement at
any later time would not affect the initial probability amplitude. And, any
conclusion from the result does not provide any information about whether the
measurement at $A$ influences $B$. Consequently, the experiment does not carry
any information about “action at a distance,” or “spooky action,” or
“nonlocal” property.

Therefore, we only need to compare the probability amplitude at time $t=0$,
i.e., we compare the measurement of $\sigma_{1}\centerdot\vec{a}$ and
$\sigma_{2}\centerdot\vec{b}$ at time $t=0$. When we compare these two
measurements with the experiment in the Bell theorem, we can compare them at
$t=0$, which means we compare them at the origin $O$.

Furthermore, because the two entangled particles are completely correlated
such that $\sigma_{1}\centerdot\vec{a}=-\sigma_{2}\centerdot\vec{a}$, we only
need to compare $-\sigma_{2}\centerdot\vec{a}$ with
$\sigma_{2}\centerdot\vec{b}$ at time $t=0$ and investigate the condition for
both to have definite values. In the following, when applying the results of
Section 3, we can consider $\vec{z}$ as $\vec{b}$, and take $\vec{a}$ the same
as the $\vec{a}$ in Section 3.

### 4.3 The statement “There are no hidden variables for non-commuting
observables to have definite values at the same time” is the consequence of
the uncertainty principle. We discuss its implications for the Bell theorem
and “Schr\̈cyrodinger’s cat problem”.

Because we start with $\sigma_{1}\centerdot\vec{a}=\pm 1$, and measure
$\sigma_{2}\centerdot\vec{b}$, the complete correlation between particles 1
and 2 requires particle 2’s initial state to be either
$\sigma_{2}\centerdot\vec{a}=1$, or $\sigma_{2}\centerdot\vec{a}=-1$. So as
long as $\sin\theta\neq 0$, particle 2’s initial state is not an eigenstate of
$\sigma_{2}\centerdot\vec{b}$. Comparing with the Bell theorem, this is
equivalent to measuring $\sigma_{2}\centerdot\vec{a}$ first and then
immediately measuring $\sigma_{2}\centerdot\vec{b}$.

The initial state for the measurement of $\sigma_{2}\centerdot\vec{b}$ is then
an eigenstate of $\sigma_{2}\centerdot\vec{a}$, not an eigenstate of
$\sigma_{2}\centerdot\vec{b}$, even though it is a pure state, i.e., it is the
result of the measurement of $\sigma_{2}\centerdot\vec{a}$. As a result, as
long as $\sin\theta\neq 0$, the measurement of $\sigma_{2}\centerdot\vec{b}$
must have two possible outcomes. This is not determined by any external
observer, but is because the initial state is not a definite state for the
measurement of $\sigma_{2}\centerdot\vec{b}$, even though it is a definite
state for $\sigma_{2}\centerdot\vec{a}$.

It is also clear that if $\sin\theta\neq 0$, there are no hidden variables
that can give both $\sigma_{2}\centerdot\vec{a}$ and
$\sigma_{2}\centerdot\vec{b}$ definite values. This is the Uncertainty
Principle. Further, the probability of $\sigma_{2}\centerdot\vec{b}=1$ is
$\cos^{2}(\theta/2)$, and the probability of $\sigma_{2}\centerdot\vec{b}=-1$
is $\sin^{2}(\theta/2)$ according to the Born rule.

In the present model case, it is very natural to understand why if
$\sin\theta\neq 0$, there are two outcomes from $\sigma_{2}\centerdot\vec{b}$.
Since the initial state is a pure state with $\vec{a}$ on the Bloch sphere, it
is a superposition of two components. The one with
$\sigma_{2}\centerdot\vec{b}=1$ will be deflected by the magnetic field to
direction $\vec{b}$, the other with $\sigma_{2}\centerdot\vec{b}=-1$ will be
deflected to direction $-\vec{b}$. This probability distribution between the
two outcomes, on the other hand, exists from the start as a physical reality
embodied later by magnetic deflection.

As a result, the measurement of $\sigma_{2}\centerdot\vec{a}$ is insufficient
to prepare a definite state for the second measurement; we must still specify
that the measurement setup in the environment after the start is still a
measurement of $\sigma_{2}\centerdot\vec{a}$, or in other words, a measurement
of $\sigma_{2}\centerdot\vec{b}$ but with $\vec{b}=\pm\vec{a}$.

The implication of this is that even though the initial wave function is the
eigenvector of a complete set of commuting observables and has definite
values, this alone cannot guarantee a definite status; because, in addition to
this, we must first make sure the measurement device is arranged to measure
this same set of observables. From this point of view, the wave function is
not a complete description of the system; a description of the measurement
environment must be included in order to have a complete description. One of
the foundations of quantum mechanics, the Uncertainty Principle, is based on
the fact that no physical system can be built to let two non-commuting
observables have definite values; consequently, no one can prepare such an
initial state in any experiment.

If we consider the case in Section 3 with $\vec{a}=\vec{x}=(1,0,0)$, then
$\theta=\frac{\pi}{2}$,$\phi=0$,

$\displaystyle|\vec{a}>=\cos(\theta/2)\left|+\right\rangle+e^{i\phi}\sin(\theta/2)\left|-\right\rangle=\frac{1}{\sqrt{2}}\left(\left|+\right\rangle+\left|-\right\rangle\right)$

we may consider this as a model to compare with an extremely simplified
“Schr\̈cyrodinger’s cat problem”. It is not an eigenstate of $\sigma_{z}$
which is the measurement we are taking. According to our discussion above, the
initial state is an eigenstate of the measurement of
$\sigma\centerdot\vec{a}$. It is not a definite state for the measurement of
$\sigma_{z}$ from the beginning, because it has a probability of 50% for spin
up or down. So the initial measurement of spin $\sigma\centerdot\vec{a}$
produced an ensemble of initial states for the following measurement of
$\sigma_{z}$. We emphasize that this probability distribution is not generated
at the end by someone observing it when the measurement of $\sigma_{z}$ is
completed, but is already determined right after the measurement of
$\sigma\centerdot\vec{a}$. The role of a final measurement of $\sigma_{z}$ is
to sort out the distribution, sample $\left|+\right\rangle$ or
$\left|-\right\rangle$, and put them into the centers of the two split
potential wells.

Therefore, there is no need to resort to any “external observer”.

No matter whether there is any one to observe the result, the probability
distribution is certain from the very beginning. $|\vec{a}>$ is called a “pure
state” only in the sense that it will give a definite result if one measures
$\sigma\centerdot\vec{a}$. The main point is no one can prepare a definite
state for $\sigma_{z}$ by measuring $\sigma\centerdot\vec{a}$.

### 4.4. At this point, one may ask why the Bell theorem sometimes seems to
generate a sense of non-locality in quantum mechanics.

As we discussed above, the violation of Bell inequality by quantum mechanics
is because the Uncertainty Principle contradicts the hidden-variable theory.
It was unnecessary to replace the “hidden variable theory” with the “local
hidden variable theory”. There is no need to resort to the word “local” in the
derivation of Bell inequality.

According to reference (bell_theorem, ), “In the words of physicist John
Stewart Bell, for whom this family of results is named, ’If [a hidden-variable
theory] is local, it will not agree with quantum mechanics, and if it agrees
with quantum mechanics, it will not be local’ ”.

This conclusion seems to generate a sense of some non-locality in quantum
mechanics when the experiment indeed agrees with quantum mechanics. However,
with a careful examination of the derivation in 4.1, as long as we assume
there is a hidden variable theory, i.e., Eq.(35) and Eq.(36), we can follow
Bell’s derivation and reach the same conclusion as the Bell theorem without
resorting to locality or non-locality.

The the basic assumption by Bell for his derivation is “the result B for
particle 2 does not depend on the setting $\vec{a}$ of the magnet for particle
1, nor A on $\vec{b}$”(bell, ), as we refered to in 4.1. We emphasize here
that communication between A and B (or not) is irrelevant.

Even if A does communicate with B such that B receives information in advance
that the setting chosen by A is $\vec{a}$, as long as B does not choose the
setting $\vec{b}$ to be parallel to $\vec{a}$, Bell’s assumption will still
lead to Eq.(35) and Eq.(36), which then leads to Bell inequality without
resorting to locality or non-locality, i.e., without resorting to whether
there is any communication between A and B.

So we may rephrase the above statement (1) as a new statement (2): “If a
hidden-variable theory can give definite values to non-commuting observables,
the result will not agree with quantum mechanics, and if it agrees with
quantum mechanics, then no hidden-variable theory can give definite values to
non-commuting observables.”

As long as the Uncertainty Principle is correct, it is natural that quantum
mechanics will violate Bell inequality, and also it rules out a “hidden
variable theory”. Both the Bell theory and our discussion has nothing to do
with whether there is non-locality, “action-at-a distance”, or not.

The theory we used is non-relativistic, which is justified because there is
nothing close to light speed involved.

The comparison of statement (1) and statement (2) and the discussion above
demonstrates the experiment’s agreement with quantum mechanics does not
provide information related to non-locality. The inclusion of the word "local"
in Bell’s statement here shifts the emphasis from "hidden variables theory" to
"local hidden variable theory". This contradicts quantum mechanics, giving the
impression that the confirmation of quantum mechanics prediction leads to some
non-locality. Hence, the discussion above suggests this statement may be
misleading if one does not carefully examine the basic ground underlying the
derivation of the Bell theorem.

We conclude that as long as the Uncertainty Principle is established,
violation of Bell inequality in the experiment entirely rules out the
existence of any hidden variable theory, irrelevant of any locality or non-
locality.

## 5\. Summary: The implications for the understanding of the statistical
interpretation of quantum mechanics

The preceding discussion in Section 4 appears to be simply repeating the
fundamentals of quantum mechanics; thus, as quantum mechanics predicts, it
leads to the violation of Bell Inequality. Indeed, we have demonstrated that
in quantum mechanics, when the Born Rule is applied to the initial state, the
result is a self-consistent theory that includes using the Schr\̈cyrodinger
equation to describe the quantum measurement process itself, without resorting
to non-locality, hidden variable, or any external observer; without resorting
to whether the measurement at A immediately influences the measurement at B or
the distance between A and B. Thus, it removes many mysteries surrounding the
statistical interpretation of quantum mechanics.

We emphasize that because the discussion is only for a specific model, the
implications of the discussion in Section 4 are only suggestive and summarized
as:

  1. 1.

The derivation of the wave function collapse in the measurement process in
Sections 1, 2, and 3 shows the probability distribution is determined from the
very beginning (i.e., $t=0$), not at the end.

  2. 2.

As a result, when Bell inequality is violated in an experimental test of the
Bell theorem, there is no paradox arising from the concepts of “non-locality”
or “action at a distance”.

  3. 3.

A linear superposition of the eigenstates of a complete set of commuting
observables is an ensemble of systems with a probability distribution
determined by the Born rule. A wave function will give a definite measurement
result only if it is the eigenstate of the observables being measured. This
crucial point removes many mysteries surrounding the statistical
interpretation of quantum mechanics.

  4. 4.

This suggests that as long as the Born rule is applied to the interpretation
of the initial state, the Born rule at the end of the measurement can be
derived from the Schr\̈cyrodinger equation.

It is important to note, in the model case we have discussed here, 1) the
correlation between the two entangled particles is assumed to be completely
preserved, and 2) the fact that the discussion about the violation of Bell
inequality without resorting to the distance between A and B does not imply
that the distance between A and B is irrelevant to the experiment on the
entangled particles. On the contrary, the longer distance between A and B
means the experiment design needs to preserve the correlation between the two
particles over a longer distance.

We thank Prof. Y. Hao, Dr. T. Shaftan, Dr. V.Smaluk and Mrs. K. Stolle for
discussion and suggestions on the manuscript.

###### Acknowledgements.

We thank Prof. Y. Hao, Dr. T. Shaftan, Dr. V.Smaluk and Mrs. K. Stolle for
discussion and suggestions on the manuscript.

## Appendix I

The following expressions are used in the derivation in Sections 1, 2, and 3.
In Eq.(8). with $a_{1}(t)=\frac{\mu e^{-\nu t}-\nu e^{-\mu t}}{\mu-\nu}$ ,
$a_{2}(t)=-\frac{e^{-\mu t}-e^{-\nu
t}}{\mu-\nu}$,$\mu\equiv\frac{\eta}{2}+\omega
i$,$\nu\equiv\frac{\eta}{2}-\omega i$ and here
$\omega\equiv\left(\omega_{0}^{2}-\frac{\eta^{2}}{4}\right)^{1/2}$, we have

$\displaystyle a_{1}(t)=e^{-\frac{\eta}{2}t}\left(\cos(\omega
t)+\frac{\eta}{2\omega}\sin(\omega t)\right)$ $\displaystyle
a_{2}(t)=\frac{1}{\omega}e^{-\frac{\eta}{2}t}\sin(\omega t)$
$\displaystyle\dot{a}_{1}(t)=-\omega_{0}^{2}a_{2}(t)$
$\displaystyle\dot{a}_{2}(t)=e^{-\frac{\eta}{2}t}\cos(\omega
t)-\frac{\eta}{2\omega}e^{-\frac{\eta}{2}t}\sin(\omega t)$ $\displaystyle
a_{1}(t)\dot{a}_{2}(t)-\dot{a}_{1}(t)a_{2}(t)=e^{-\eta t}$

where we can confirm that
$a_{1}(0)=1,a_{2}(0)=0,\dot{a}_{1}(0)=0,\dot{a}_{2}(0)=1$.

Then, use Eq.(10) and the expressions of
$a_{1}(t),a_{12}(t),\dot{a}_{1}(t),\dot{a}_{2}(t)$, we find the commutiator

$[Q,\dot{Q}]=[a_{1}(t)Q_{0}+a_{2}(t)\dot{Q}_{0},\dot{a}_{1}(t)Q_{0}+\dot{a_{2}}(t)\dot{Q}_{0}]=e^{-\eta
t}\frac{i\hbar}{M}$ (38)

Also in the solution of the damped oscillator equation Eq.(6), i.e., Eq.(8),
the coefficients of the $j$’th bath oscillator contribution to $q$

$\displaystyle
b_{j1}(t)=-\frac{c_{j}}{M}\text{InverseLaplace}(\frac{s}{\left(s^{2}+\omega_{j}^{2}\right)(s+\mu)(s+\nu)})$
$\displaystyle=-\frac{c_{j}}{M}\left[\frac{\omega_{j}(\mu+\nu)\sin(t\omega_{j})}{\left(\mu^{2}+\omega_{j}^{2}\right)\left(\nu^{2}+\omega_{j}^{2}\right)}+\frac{\left(\mu\nu-\omega_{j}^{2}\right)\cos(t\omega_{j})}{\left(\mu^{2}+\omega_{j}^{2}\right)\left(\nu^{2}+\omega_{j}^{2}\right)}+\frac{\mu
e^{-\mu t}}{(\mu-\nu)\left(\mu^{2}+\omega_{j}^{2}\right)}-\frac{\nu e^{-\nu
t}}{(\mu-\nu)\left(\nu^{2}+\omega_{j}^{2}\right)}\right]$ $\displaystyle
b_{j2}(t)=-\frac{c_{j}}{M}\text{
InverseLaplace($\frac{1}{\left(s^{2}+\omega_{j}^{2}\right)(s+\mu)(s+\nu)}$)}$
$\displaystyle=-\frac{c_{j}}{M}\left[\frac{\left(\mu\nu-\omega_{j}^{2}\right)\sin(t\omega)}{\omega_{j}\left(\mu^{2}+\omega_{j}^{2}\right)\left(\nu^{2}+\omega_{j}^{2}\right)}-\frac{(\mu+\nu)\cos(t\omega_{j})}{\left(\mu^{2}+\omega_{j}^{2}\right)\left(\nu^{2}+\omega_{j}^{2}\right)}-\frac{e^{\mu(-t)}}{(\mu-\nu)\left(\mu^{2}+\omega_{j}^{2}\right)}+\frac{e^{\nu(-t)}}{(\mu-\nu)\left(\nu^{2}+\omega_{j}^{2}\right)}\right]$
$\displaystyle
b_{j1}(t{\color[rgb]{0,0,0}\rightarrow}\infty)=-\frac{c_{j}}{M}\left[\frac{\omega_{j}(\mu+\nu)\sin\left(t\omega_{j}\right)}{\left(\omega_{j}^{2}+\mu^{2}\right)\left(\omega_{j}^{2}+\nu^{2}\right)}+\frac{\left(\mu\nu-\omega_{j}^{2}\right)\cos\left(t\omega_{j}\right)}{\left(\omega_{j}^{2}+\mu^{2}\right)\left(\omega_{j}^{2}+\nu^{2}\right)}\right]$
$\displaystyle\omega_{j}b_{j2}(t{\color[rgb]{0,0,0}\rightarrow}\infty)=-\frac{c_{j}}{M}\left[\frac{\left(\mu\nu-\omega_{j}^{2}\right)\sin\left(t\omega_{j}\right)}{\left(\omega_{j}^{2}+\mu^{2}\right)\left(\omega_{j}^{2}+\nu^{2}\right)}-\frac{\omega_{j}(\mu+\nu)\cos\left(t\omega_{j}\right)}{\left(\omega_{j}^{2}+\mu^{2}\right)\left(\omega_{j}^{2}+\nu^{2}\right)}\right]$
$\displaystyle
b_{j1}(t{\color[rgb]{0,0,0}\rightarrow}\infty)^{2}+\omega_{j}^{2}b_{j2}(t{\color[rgb]{0,0,0}\rightarrow}\infty)^{2}=\frac{c_{j}^{2}}{M^{2}}\frac{1}{\text{($\omega_{0}^{2}$-$\omega_{j}^{2})^{2}$+$\eta^{2}\omega_{j}^{2}$}}$

where we can confirm that
$b_{j1}(0)=b_{j1}(0)=\dot{b}_{j1}(0)=\dot{b}_{j1}(0)=0$.

## Appendix II

Denote the eigenvector of $Q(t)=a_{1}(t)Q_{0}+a_{2}(t)\dot{Q}_{0}$ with
eigenvalue $Q_{1}$ as $u_{Q_{1}}(Q_{0},t)$, where $Q_{0}\equiv
q_{0}-\sigma_{z}d,\dot{Q}_{0}\equiv-\frac{i\hbar}{M}\frac{\partial}{\partial
q_{0}}\equiv-\frac{i\hbar}{M}\frac{\partial}{\partial Q_{0}}$, $Q(t)$ and
$\dot{Q}(t)$ are understood as a function of $q_{0}-\sigma_{z}d,\dot{q}_{0}$
from Eq.(10). Since $\sigma_{z}$ commutes with $Q_{0},\dot{Q}_{0}$, there is
no ambiguity for the functions of $Q_{0},\dot{Q}_{0}$ regarding the order of
$\sigma_{z}$ in any product in the function. The eigenequation

$\displaystyle\left(a_{1}(t)Q_{0}-\frac{i\hbar
a_{2}}{M}\frac{\partial}{\partial Q_{0}}\right)u_{Q_{1}}=Q_{1}u_{Q_{1}}$

leads to

$u_{Q_{1}}=C(Q_{1},t)\exp\left[-i\frac{M}{2\hbar
a_{2}}\left(a_{1}Q_{0}^{2}-2Q_{1}Q_{0}\right)\right]$ (39)

$C(Q_{1},t)$ is determined by orthonormal condition on $u_{Q}$, i.e., $\int
dQ_{0}u_{Q_{2}}^{*}(Q_{0})u_{Q_{1}}(Q_{0})=\delta(Q_{1}-Q_{2})$ as
$C(Q_{1},t)=\left(\frac{M}{2\pi\hbar
a_{2}}\right)^{\frac{1}{2}}e^{-i\frac{M}{2\pi\hbar a_{2}}\phi(Q_{1},t)}$in
Eq.(11).

## Appendix III Wave function probability density with Brownian motion

We abbreviate the wave function density as
$f(\sum_{j}\xi_{j1})=|\psi(q_{1}-s_{1}d-\sum_{j}\xi_{j1},s_{1},t)|^{2}$, then
using the Taylor expansion of $f$ it is straight forward to show that

$\displaystyle\int\int..\int\int
f(\sum_{j}\xi_{j})\prod_{j}|\chi_{j}(\xi_{j},t)|^{2}d\xi_{j}$
$\displaystyle=\frac{1}{\prod_{j}\sqrt{2\pi}\sigma_{\xi_{j}}}\int\int..\int\int
f(\sum_{j}\xi_{j})\exp\left(-\sum_{j}\frac{\xi_{j}^{2}}{2\sigma_{\xi_{j}}^{2}}\right)\prod_{j}d\xi_{j}$
(40) $\displaystyle=\frac{1}{\sqrt{2\pi}\sigma_{\xi}}\int
f(\xi)\exp\left(-\frac{\xi^{2}}{2\sigma_{\xi}^{2}}\right)d\xi$
$\displaystyle\sigma_{\xi}^{2}=\sum_{j}\sigma_{\xi_{j}}^{2}$

Thus

$\displaystyle|\Psi(q,s,t)|^{2}\equiv\int\int..\int\int|\Psi(q,s,\\{\xi_{j}\\},t)|^{2}\prod_{j}d\xi_{j}$
$\displaystyle=\int\int..\int\int|\psi(q-sd-\sum_{j}\xi_{j},s,t)|^{2}\prod_{j}|\chi_{j}(\xi_{j},t)|^{2}d\xi_{j}$
(41)
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma_{\xi}}\int|\psi(q-sd-\xi,s,t)|^{2}\exp\left(-\frac{\xi^{2}}{2\sigma_{\xi}^{2}}\right)d\xi$

For probability density of Eq.(33),with $|\psi(Q,t)|^{2}=|\psi(q-d,t)|^{2}$ in
Eq.(27) we have

$\displaystyle|\Psi_{+}(q,t)|^{2}=\cos^{2}(\theta/2)\frac{1}{\sqrt{2\pi}\sigma_{\xi}}\int|\psi(q-d-\xi,1,t)|^{2}\exp\left(-\frac{\xi^{2}}{2\sigma_{\xi}^{2}}\right)d\xi$
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma_{\xi}}\int\left(\frac{1}{2\pi\sigma_{Q}^{2}}\right)^{\frac{1}{2}}\exp\left(-\frac{\left(q-sd-\xi-
a_{1}z\right)^{2}}{2\sigma_{Q}^{2}}\right)\exp\left(-\frac{\xi^{2}}{2\sigma_{\xi}^{2}}\right)d\xi$
$\displaystyle=\cos^{2}(\theta/2)\frac{1}{\sqrt{2\pi}}\frac{1}{\sigma_{Q\xi}}\exp\left(-\frac{\left(q-d+a_{1}d\right)^{2}}{2\sigma_{Q\xi}^{2}}\right)$

where the width of the distribution is
$\sigma_{Q\xi}^{2}=\sigma_{Q}^{2}+\sigma_{\xi}^{2}$. The same way the
probability of the lower potential well with spin down is

$|\Psi_{-}(q,t)|^{2}=\sin^{2}(\theta/2)\frac{1}{\sqrt{2\pi}}\frac{1}{\sigma_{Q\xi}}\exp\left(-\frac{\left(q+d-a_{1}d\right)^{2}}{2\sigma_{Q\xi}^{2}}\right)$
(42)

## References

  * (1) “Bell’s theorem - Wikipedia”, https://en.wikipedia.org/wiki/Bell%27s_theorem#cite_note-1
  * (2) “Measurement problem”, https://en.wikipedia.org/wiki/Measurement_problem
  * (3) A.O. Caldeira and A.J. Leggett, Ann. Phys. 149, 374 (1983); Physica 121A, 587 (1983).
  * (4) A.O. Caldeira and A.J. Leggett, PHYSICAL REVIEW A VOLUME 31, NUMBER 2 FEBRUARY 198
  * (5) A.O. Caldeira, Helv. Phys. Acta 61, 611 (1988).
  * (6) E. Kanai, Prog. Theor. Phys. 3, 440 (1948).
  * (7) S. Nakajima, Prog. Theor. Phys. 20, 948 (1958).
  * (8) R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).
  * (9) I.R. Senitzky, Phys. Rev. 119, 670 (1960).
  * (10) R.P. Feynman and F.L. Vernon, Ann. Phys. 24, 118 (1963).
  * (11) G.W. Ford, M. Kac, and P. Mazur, J. Math. Phys. 6,504 (1965).
  * (12) M. D. Kostin, J. Chem. Phys. 57, 3589 (1972).
  * (13) W.H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
  * (14) M. Sargent III, M.O. Scully, and W.E. Lamb, Laser Physics (Addison-Wesley, Reading, MA, 1974).
  * (15) K. Yasue, Ann. Phys. 114, 479 {1978).
  * (16) R.H. Koch, D.J. Van Harlingen, and J. Clarke, Phys. Rev. Lett. 45, 2132 (1980).
  * (17) H. Dekker, Phys. Rep. 80, 1 (1981).
  * (18) R. Benguria and M. Kac, Phys. Rev. Lett. 46, 1 (1981).
  * (19) V.F. Weisskopf and F.P. Wigner, Z. Phys. 63, 54 (1930); 65, 18 (1930).
  * (20) Li Hua Yu,Chang-Pu Sun, PHYSICAL REVIEW A VOLUME 49, NUMBER 1 JANUARY 1994
  * (21) “Bloch sphere”, https://en.wikipedia.org/wiki/Bloch_sphere
  * (22) J. S. BELL, “ON THE EINSTEIN PODOLSKY ROSEN PARADOX”, Physics Vol. 1, No. 3, pp. 195-290, 1964 Physics Publishing Co. https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf