A model for the stochastic origins of Schrodinger's equation

Abstract

A model for the motion of a charged particle in the vacuum is presented which, although purely classical in concept, yields Schrodinger's equation as a solution. It suggests that the origins of the peculiar and nonclassical features of quantum mechanics are actually inherent in a statistical description of the radiative reactive force.

Full text

arXiv:quant-ph/0112157v1 25 Dec 2001
A model for the stochastic origins of Schr¨odinger’s equation
∗
Mark Davidson
†
(Received 2 March 1979; accepted for publication 23 April 1979)
Abstract
A model for the motion of a charged particle in the vacuum is presented which,
although purely classical in concept, yields Schr¨odinger’s equation as a solution. It
suggests that the origins of the peculiar and nonclassical features of quantum mechanics
are actually inherent in a statistical description of the radiative reactive force.
1 Introduction
Stochastic models of quantum mechanics attempt to reconcile the postulates of quantum
theory with modern probability theory, and to provide a space-time picture of quantum
phenomena. The traditional inspiration for this effort is rooted in the extensive debates of
the 1920’s and 30’s over the interpretation of quantum mechanics. A standoff developed,
which persists to this day, between the Bohr complementarity school and the statistical
school usually associated with Einstein [1]–[3].
Today the Bohr interpretation is much more widely accepted. It asserts, in a nutshell,
that given a physical state, then there is a state vector of some Hilbert space which de-
scribes this state completely, but only statistical properties about the physical system can
be deduced from this presumed complete description. A number of forceful (and unresolved)
completeness arguments against the Bohr view have been made [4]–[7], and a number o f
the founders of modern quantum theory did not accept this view, including Einstein [2],
Schr¨odinger [5], and De Broglie [8].
There are several reasons why the Bohr view is dominant. Rigorous no-go t heorems
make stochastic or hidden variable models difficult to construct [9]–[12]. Despite these,
there are statistical theories which do reproduce all of the statistical assertions o f quantum
mechanics, such as the differential space theory of Wiener and Siegel [13]. Any such theory
must have some nonlocal features to avoid conflict with Bell’s theorem [12], and this can
present conceptual problems. The Bohr view provides a j ustification for ignoring the puzzling
questions of the origins of quantum mechanics, and for concentrating on applications of the
theory. The accomplishments of the last half century have validated this point of view.
∗
J. Math. Phys. 20(9) September 1979, 1865–1869.
c
1979 American Institute of Physics.
†
Current Address: Spectel Research Corporation, 807 Rorke Way, Palo Alto, CA 94303
Email: mdavid@spectelresearch.com, Web: www.spectelresearch.com
1

Most stochastic or hidden variable models have some nonclassical or difficult to under-
stand features to them. For example, Bohm’s early hidden variable theory [14] required the
existence of a nonclassical quantum mechanical potential to be consistent with Schr¨odinger’s
equation. The Fenyes-Nelson stochastic model [1 5]–[17] a lso has a nonclassical quality about
it. The dynamical assumption of Nelson [16], for example, is not derived from first principles,
and implies the existence of nonclassical forces acting on the particle. In most statistical
models of quantum mechanics t here is a gap in the derivatio n of quantum mechanical laws
from classical laws, usually in the form of postulating a quantum mechanical potential o r
its equivalent. These g aps make the models unconvincing. An exception is the derivation
of Schr¨odinger’s equation from stochastic electrodynamics (SED) [18], where all quantum
behavior is derived from a classical Langevin equation. The mathematics of this derivation
are quite complicated, however, and there are several points of nonrigor owing to the singular
nature of the random force in this model. Moreover, the SED model yields a Moyal type
of phase space picture [19], whereas the Markov model of Fenyes and Nelson seems better
adapted to describing quantum mechanics.
This paper presents a simple model, within the Fenyes-Nelson scheme, which provides
an explanation of the origin of the quantum mechanical potential, and of the steady state
Schr¨odinger’s equation. This model describes the diffusion of charged particles, and it in-
cludes the radiative reactive force. Neutral particles are not considered, but all known finite
mass neutral particles are believed to be bound states of charged particles, so the results de-
rived are not limited by this. The vacuum in which these charged particles move is assumed
to have a finite temperature, but this temperature may be taken to zero. Inherent in the
derivation is the concept of a vacuum alive with fluctuations and randomness. This concept
of a nonempty vacuum has been slowly creeping back into physics with the work of Wheeler
[20]. Boyer [21], the models of Bohm and Vigier [22], and De Broglie [23], and more subtly
in the whole quantum field effort with its infinite vacuum fluctuations.
The model presented is not a complete treatment of the problem. It r elies on two rea-
sonable postulates: The charged part icles are described by a continuous Markov process in
configuration space, and they are assumed to satisfy Gibbs’ classical distribution, where the
radiative reactive force is included. In the limit of zero temperature, these postulates imply
the Schr¨odinger equation and t he existence of a quantum mechanical potential, provided the
diffusion constant of the theory has a certain value.
2 The model
Consider the Schr¨odinger equation for a single particle in a potential V :
"
−
¯h
2
2m
∆ + V
#
ψ = i¯h
∂ψ
∂t
, ψ = e
R+iS
. (1)
It is equivalent to the f ollowing two equations:
∂ρ
∂t
= −∇ ·
¯h
m
∇Sρ, ρ = ψ
∗
ψ (2)
2

and
¯h
2
2m
(∇S)
2
+ V −
¯h
2
2m
∆ρ
1/2
ρ
1/2
= −¯h
∂S
∂t
, (3)
where R and S are chosen to be real. Equation (2) simply reflects the conservatio n of prob-
ability, a nd Eq. (3) is the Hamilton-Jacobi equation, but with an extra quantum mechanical
potential:
V
QM
= −
¯h
2
2m
∆ρ
1/2
ρ
1/2
. (4)
Were it not for this potential term, Schr¨odinger’s equation could be interpreted as the dif-
fusion of Newtonian particles whose initial conditions were not completely specified. This
potential term is required in most classical models of quantum mechanics. For example,
Madelung’s hydrodynamic model requires it [24], Bohm’s early hidden variable model re-
quires it [14], and De Broglie’s theory of the double solution requires it [8]. This term is also
implicit in the dynamical assumption of Nelson [16], where Eqs. (2) and (3) are interpreted
as diffusion equations for a continuous Markov process. It is the possible origin of this extra
term which shall be examined in t his paper.
The quantum mechanical potential implies an unusual fo r ce, which acts on the particle,
but which depends on the statistical properties of an ensemble of particle trajectories. This
kind of behavior is difficult to understand in classical statistical mechanics. Indeed, it is
this extra potential term which leads to quantum interference effects, and the difficulty of
describing quantum interference in terms of classical statistical theories has been forcefully
stated by Feynman [25]. Despite this, it appears that the model presented does give a
possible explanation of this extra potential in a classical statistical theory. The reason is
that the r adiative reactive force plays a large role in the theory about to be presented.
Preacceleration associated with this radiative reactive force was not considered by Feynman
in his arguments.
Consider a charged particle in motion in the physical vacuum. Let this particle be
described by classical mechanics, and let its motion be nonrelativistic. Then it satisfies the
equation
m
0
a(t) =
Z
∞
0
ds e
−s
F (x(t + τ s), t), (5)
where
τ =
2
3
q
2
m
0
c
3
, (6)
and where q is the charge of the particle, m
0
its mass, and τ ha s units of time. For an
electron, τ ≈ 10
−22
s, if q and m
0
are taken to be t he observed charge and mass of the
electron. For most practical calculations, such a brief preacceleration can be ignored. It has
played little role in Newtonian physics. As is shown by Rohrlich [2 6], Eq. (5) is the unique
nonrelativistic limit of a perfectly well-defined relativistic theory.
There are two ways that the preacceleration effect can become amplified in the model to
be presented. First of all, if q and m
0
are not the observed charge and mass, but rather are
bare quantities, then τ can b e much larger. If the diffusion constant of the Markov theory,
which will be used to describe the pa rt icle is large, then preacceleration also becomes more
important.
3

Suppose that the vacuum is alive with random field fluctuations, and suppose that it has
a small temperature T . A more precise definition of this concept will not be att empted. It
will only be assumed that the classical Gibbs distribution is satisfied. If the radiative force
were ignored, then the particle would reach a state of equilibrium at temperature T , and its
spatial density would be given by the classical Gibbs distribution,
ρ(x) = e
−V (x)/kT
, (7)
up to a normalization constant, where k is Bo ltzman’s constant. This equation may be
written
kT ∇ln(ρ) = −∇V = F
ext
. (8)
Equation (8) would not be satisfied by a charged particle which experiences a significant ra-
diative force. The statistical distribution in this case is simply not known. Two assumptions
shall be made to generalize Eq. ( 8) to include radiative forces in the simplest possible way.
The first a ssumption is that t he charged particle, in thermal equilibrium with the vacuum,
is described by a continuous Markov process on configuration space. Using Nelson’s notation
[16] x is assumed to satisfy the stochastic differential equation
dx(t) = b(x(t))dt + dW (t), (9)
where W is a three-dimensional Wiener process with
E(dW
i
(t)dW
j
(t)) = 2ν δ
i,j
dt (10)
and where ν is called the diffusion constant. This type of process was studied by Nelson
[16, 17], and he showed that Schr¨odinger’s equation could be derived, with a dynamical
assumption, provided ν = ¯h/2m. In fact, this result can be generalized [27], and any value
of ν greater than zero can be used to develop a model of Schr¨odinger’s equation. The
solutions to (9) are Markov processes on configuration space, and in general, velocities are
not well defined. This Markov description must be viewed as an approximation to the actual
motion of the particle, valid so long a s dt is not too small in Eq. (9). If Eq. (9) were taken to
be true for arbitrarily small dt, then the particle would be relativistic, and the nonrelativistic
approximation would be inaccurate.
Imagine that the charged particle, in interaction with the finite temperature vacuum and
subject to an external potential V , has reached a stationary state of thermal equilibrium
described by a probability density ρ(x). Consider the following conditional expectation:
F
E
(x) = −E

Z
∞
0
ds e
−s
∇V (x(t + τs))




x(t) = x

. (11)
From Eq. (5), it is seen that this expresses the expected value of t he total force on the particle,
including preacceleration, given that at time t the par t icle’s trajectory passed t hro ugh the
point x. This equation represents the best estimate that can be made of the instantaneous
force acting on the particle at position x and time t.
By analogy with the classical Gibbs distribution [Eq. (8)], the following equation for the
charged particle is postulated:
kT ∇ ln(ρ) = F
E
(x). (12)
4

This constitutes the second postulate. All it says is that t he classical Gibbs distribution is
satisfied fo r the total force given by Eq. (11), and including radiative effects. Implicit in Eq.
(12) is the assumption that F
E
has vanishing curl. This will prove to be consistent.
F
E
, as expressed in Eq. (11), will depend on b in Eq. (9), and therefore Eq. (12) will be
a differential equation for ρ. To derive this equation, the Markov transition function is used:
P
t−u
(y, x) = lim
d
3
y →0
1
d
3
y
P (x(t) ∈ d
3
y|x(u) = x), (13)
which satisfies the forward and backward equations of Kolmogorov [28]:
∂
∂t
P
t−u
(y, x) + ∇
y
· b(y)P
t−u
(y, x) − ν∆
y
P
t−u
(y, x) = 0, t > u, (14)
∂
∂u
P
t−u
(y, x) + b(x) · ∇
x
P
t−u
(y, x) + ν∆
x
P
t−u
(y, x) = 0, t > u. (15)
F
E
may be written as
F
E
(x) = −
Z
∞
0
ds e
−s
Z
d
3
yP
τs
(y, x)∇V (y). (16)
P must satisfy two limiting conditions: The first is a statement of continuity, and the second
is a statement of ergodicity:
P
0
(y, x) = δ
3
(y − x), (17)
P
∞
(y, x) = ρ(y). (18)
Equation (18) requires some qualifications. If t he density ρ vanishes as some point, then
Eq. (18) is not quite valid, as has been shown by Albeverio and Hoegh-Krohn [29]. In this
case, space is divided up into disjoint regions bounded by surfaces ρ(x) = 0, and the Markov
transition function vanishes unless x and y are in the same region. Equation (18) is true if
x and y are in the same region in this case, and this is sufficient for the results below.
¿From Eq. (18) and Eq. (14 ) it follows that, taking the limit t → ∞ in (14),
b = ν∇ ln(ρ). (19)
Now, using the backward equation [Eq. (15)] together with the expression for F
E
[Eq. (16)]
one obtains
(b · ∇ + ν∆)F
E
(x) = −
Z
∞
0
ds e
−s
Z
d
3
y
∂
∂τs
P
τs
(y, x)∇V (y), (20)
where it has been assumed that the order of differentiation and integration can be freely
interchanged. Integrating (20) by parts, and using (17) then yields
[1 − τ (b · ∇ + ν∆) ] F
E
(x) = −∇V (x). (21)
At this point, the Gibbs distribution [Eq. (12)] is used to substitute for F
E
in Eq. (21). One
finds
[1 − τ (b · ∇ + ν∆)] ∇ ln [ρ (x)] = −
1
kT
∇V (x). (22)
5

Defining R by
R =
1
2
ln(ρ), (23)
and using (19) and (22), one finds:
∇
h
R − τν

(∇R)
2
+ ∆R
i
= −
1
2kT
∇V. (24)
Integrating this expression, and rewriting it, one obtains
[−2τνkT ∆ + V + 2kT R] e
R
= λe
R
, λ = const. (25)
This can also be written as
ρ(x) = exp
"
−
1
kT
V (x) − 2τ νkT
∆ρ
1/2
ρ
1/2
− λ
!#
. (26)
This last expression clearly displays the existence of an extra, and unusual, potential given
by
− 2τνkT ∆ρ
1/2
/ρ
1/2
. (27)
This extra potential term is due to the radiative reactive force, and it has exactly the same
form (including the right sign) as t he quantum mechanical potential [Eq. (4)]. Equation (25)
bears a remarkable similarity in form to the Schr¨odinger steady state equation.
The strength of the radiative preacceleration effects depend on the magnitude of the
gradient of (27) relative to the gradient of V . This depends on the factor
γ = 2τνkT =
4
3
q
2
νkT
m
0
c
3
. (28)
This factor γ determines the magnitude of radiative effects. It is interesting that one cannot
distinguish between different values of q
2
, ν, and m
0
, but only different values of γ. For small
T , γ can be large if the ratio q
2
ν/m
0
is large. Since ν is a free parameter in this model, a
large radiative correction is possible for la r ge ν, regardless of the size of the other factors.
Suppose that
γ =
¯h
2
2m
=
4
3
q
2
νkT
m
0
c
3
, (29)
where m is the physical mass, and the possibility that it is different from the bare mass m
0
has been allowed. Then Eq. (25) becomes
−
¯h
2
2m
∆ + V + 2kT R
!
e
R
= λe
R
(30)
and (2 6) becomes
ρ(x) = exp
"
−
1
kT
V (x) −
¯h
2
2m
∆ρ
1/2
ρ
1/2
− λ
!#
. (31)
Equation (30) has the same form as Schr¨odinger’s equation, except for the extra term in
the potential, 2kT R. This extra term can be interpreted as representing the diffusion force.
6

It prevents the occurrence of zeroes in ψ = e
R
. Equation (31) is the a nalo g of the Gibbs
distribution for neutral particles [Eq. (7)], with the quantum mechanical potential included,
but due to radiative forces. If T is very small, then (30) becomes
h
−¯h
2
/2m

∆ + V
i
ψ ≈ λψ (32)
which is just Schr¨odinger’s stationary state equation.
Equation (30) is a classical model for steady state quantum mechanics with one fr ee
parameter, the temperature of the vacuum T . It is nonlinear, and in general difficult to
solve. In the limit T → 0, Eq. (32) becomes exact. It is an experimental question what T
is, assuming that the model is taken seriously.
Although the possibility that m and m
0
are different has been allowed, it is interesting
to note that if m = m
0
or if m and m
0
are proportional with a fixed factor, t hen both sides
of Eq. (29) have the same mass dependence. This means that νq
2
may be chosen to be mass
independent. If q is taken to be the electronic charge, then ν could be mass independent.
This is consistent with the generalization of the Fenyes-Nelson model [27], where any value
of ν can be used to construct a model of quantum mechanics. If ν is mass independent, then
the underlying thermal agitation could be gravitational in nature. This could be consistent
with Wheeler’s concepts of superspace [20].
If Eq. (32) is a good approximation, that is if T is small, then energy levels are quantized,
provided the usual Ha miltonian operator is taken as the energy operator. Quantization of
the energy levels of harmonic oscillators leads, through fairly well known arguments [30],
to a derivation of the Planck radiation law. The present theory, if correct, could influence
the equilibrium of radiation at finite temperature. This could provide a way out of the
Rayleigh-Jeans spectrum.
The question that remains is what could determine T , and how could a more complete
model be constructed. If T is nonzero, then it is reasonable to expect to see black-body
electromagnetic radiation at this temperature in the vacuum. The spectrum of radiation
in the vacuum is not exactly black-body, but in the microwave region, a Planck spectrum
has been observed at a temperature of 2 .7 6
◦
K [31]. The problem with this is that the
radiation may not yet have reached thermal equilibrium. It is possible that T equals this
radiation temperature, and this deserves some consideration, but this does not appear to be
a necessity, and this possibility will not be considered here.
The results of this section should be compared with the SED Langevin approach [18]. In
that model, Schr¨odinger’s equation is derived fo r the diffusion of an electron in interaction
with zero point background radiation. A number of approximations are made to derive
general results, and the radiative reactive force plays a crucial role. It is hoped that the
present model complements and perhaps sheds some light on the SED calculation. Although
less complete, the present model is much simpler than the SED model, and it is felt that
this simplicity helps to isolate the essential ingredients in the relationship between quantum
mechanics and stochastic theories with radiative reactive forces.
7

3 Conclusion
Charged particles in interaction with a low temperature vacuum can be expected to satisfy
a Schr¨odinger type equation. This result offers an explanation of the quantum mechanical
potential as essentially due to radiative reactive forces in a stochastic theory. It also suggests
that an extra t erm may be present and possibly observable in Schr¨odinger’s equation if the
vacuum temperature is not zero.
The main limitation of the model presented is that it makes no attempt to account in a
detailed way for the Markov motion of the particles fro m, say, a Langevin approa ch in terms
of random forces. However, by using only simple postulates, independent of the details of
the vacuum’s structure, it is felt that the derivation of Schr¨odinger’s equation is less model
dependent and more straightforward than, say, the SED calculation [18], although both
calculations are similar in spirit. Moreover, the SED approach may not contain all of the
relevant vacuum fluctuations. It does not include gravitational fluctuations or fluctuations
in the vector fields which mediate the weak interactions, both of which could be important
for the electron. The model presented here does not really care what fields are involved,
so long as the generalized Gibbs distribution is satisfied and the motion is described by a
Markov process. In this sense it may be more general than the SED approach.
The future of this model will hinge on the ability to generalize it to the time dependent
case, and to make it relativistic. These are major problems at the present. The importance
of the preacceleration in the model helps to explain the nonlocal character of hidden variable
models of quantum mechanics. In the classical theory, the acceleration at a particular time
depends on the force for all future times. Treating this type of dynamical system statistically,
one is forced to conclude that the most likely value for the force which will be experienced
by a particle at a given position and time will depend on the properties of the ensemble,
that is, it will depend on ρ. Any measurement made on the system will change ρ, and
this will change the expected f orce on the particle instantaneously. This peculiar property is
understood in terms of the preacceleration of charged particles, and should not be considered
unphysical, unless preacceleration is also considered unphysical.
It is believed that the results presented can be generalized to many-particle systems.
The possibilities that T is the temperature of the cosmic background radiation, or that the
thermal agita tion o f the vacuum is g r avitational in nature, with ν independent of mass, are
intriguing and should provide fertile areas for exploration.
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9

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10