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We have already (Chapter 13) studied the kinematics of such a process.
We let b— be the mean backward velocity, u = (b ’ b— )/2, v = (b + b— )/2.
By (13.5) and (13.6) of Chapter 13,
‚u h
=’ grad div v ’ grad v · u,
‚t 2m (15.2)
‚v h
=a’v· v+u· u+ ∆u,
‚t 2m
where a is the mean acceleration.
Suppose that the particle is subject to an external force F . We make
the dynamical assumption F = ma, and substitute F/m for a in (15.2).
This agrees with what is done in the Ornstein-Uhlenbeck theory of Brow-
nian motion with friction (Chapter 12).
Consider the case when the external force is derived from a poten-
tial V , which may be time-dependent, so that F (x, t) = ’ grad V (x, t).
Then (15.2) becomes
‚u h
=’ grad div v ’ grad v · u,
‚t 2m (15.3)
‚v 1 h
= ’ grad V ’ v · v + u · u + ∆u.
‚t m 2m
If u0 (x) and v0 (x) are given, we have an initial value problem: to solve the
system (15.3) of coupled non-linear partial di¬erential equations subject
to u(x, 0) = u0 (x), v(x, 0) = v0 (x) for all x in ‚3 . Notice that when we
do this we are not solving the equations of motion of the particle. We are
merely ¬nding what stochastic process the particle obeys, with the given
force and the given initial osmotic and current velocities. Once u and v
are known, b, b—, and ρ are known, and so the Markov process is known.
It would be interesting to know the general solution of the initial
value problem (15.3). However, I can only solve it with the additional
assumption that v is a gradient. (We already know (Chapter 13) that u
is a gradient.) A solution of the problem without this assumption would

seem to correspond to ¬nding the Markov process of the particle when
the containing ¬‚uid, the aether, is in motion.
Let R = 2 log ρ. Then we know (Chapter 13) that
grad R = u. (15.4)
Assuming that v is also a gradient, let S be such that
grad S = v. (15.5)
Then S is determined, for each t, up to an additive constant.
It is remarkable that the change of dependent variable

ψ = eR+iS (15.6)

transforms (15.3) into a linear partial di¬erential equation; in fact, into
the Schr¨dinger equation

‚ψ h
¯ 1
∆ψ ’ i V ψ + i±(t)ψ.
=i (15.7)
‚t 2m h
(Since the integral of ρ = ψψ is independent of t, if (15.7) holds at all
then ±(t) must be real. By choosing, for each t, the arbitrary constant
in S appropriately we can arrange for ±(t) to be 0.)
To prove (15.7), we compute the derivatives and divide by ψ, ¬nding

‚R ‚S h
¯ 1
∆R + i∆S + [grad(R + iS)]2 ’ i V + i±(t).
+i =i
‚t ‚t 2m h
Taking gradients and separating real and imaginary parts, we see that
this is equivalent to the pair of equations

‚u h
=’ ∆v ’ grad v · u,
‚t 2m
‚v h
¯ 1 1 1
∆u + grad u2 ’ grad v 2 ’ grad V.
‚t 2m 2 2 m
Since u and v are gradients, this is the same as (15.3).
Conversely, if ψ satis¬es the Schr¨dinger equation (15.7) and we de¬ne
R, S, u, v by (15.6), (15.4), and (15.5), then u and v satisfy (15.3). Note
that u becomes singular when ψ = 0.
108 CHAPTER 15

Is it an accident that Markov processes of the form (15.1) with the
dynamical law F = ma are formally equivalent to the Schr¨dinger equa-
tion? As a test, let us consider the motion of a particle in an external
electromagnetic ¬eld. Let, as is customary, A be the vector potential,
• the scalar potential, E the electric ¬eld strength, H the magnetic ¬eld
strength, and c the speed of light. Then
H = curl A, (15.8)
1 ‚A
= ’ grad •.
E+ (15.9)
c ‚t
The Lorentz force on a particle of charge e is
F =e E+ V —H , (15.10)
where v is the classical velocity. We adopt (15.10) as the force on a particle
undergoing the Markov process (15.1) with v the current velocity. We do
this because the force should be invariant under time inversion t ’ ’t,
and indeed H ’ ’H, v ’ ’v (while u ’ u) under time inversion. As
before, we substitute F/m for a in (15.2). Now, however, we assume the
generalized momentum mv + eA/c to be a gradient. (This is a gauge
invariant assumption.) Letting grad R = mu/¯ as before, we de¬ne S up
to an additive function of t by
m e
grad S = v+ A,
¯ mc
and let
ψ = eR+iS .
Then ψ satis¬es the Schr¨dinger equation
‚ψ i e ie
=’ ’i¯ ’A ψ’
h •ψ + i±(t)ψ, (15.11)
‚t 2m¯h c h
where as before ±(t) is real and can be made 0 by a suitable choice of S.
To prove (15.11), we perform the di¬erentiations and divide by ψ,
‚R ‚S h
∆R + i∆S + [grad(R + iS)]2
+i =i
‚t ‚t 2m
e 1e ie
A2 ’ • + i±(t).
A · grad(R + iS) + div A ’
2m¯ c2
mc 2 mc h h

This is equivalent to the pair of equations we obtain by taking gradients
and separating real and imaginary parts. For the real part we ¬nd
m ‚u h
¯ e
=’ grad div(v + A)
h ‚t
¯ 2m mc
¯ mm e e m
’ grad 2 u · A· u
v+ A + grad
2m h
¯ h
¯ mc mc h
+ grad div A,
2 mc
which after simpli¬cation is the same as the ¬rst equation in (15.2).
For the imaginary part we ¬nd

m ‚v e ‚A
+ =
¯ ‚t mc ‚t
¯ m mm m e m e
grad div u + grad u· u ’ grad A·
v+ v+ A
2m h
¯ hh
¯¯ h
¯ mc h
¯ mc
e m e e
A2 ’ • .
A· A’
+ grad v+
mc h
¯ mc 2m¯ c
h h

Using (15.9) and simplifying, we obtain

‚v e h
¯ 1 1
grad div u + grad u2 ’ grad v 2 .
= E+
‚t m 2m 2 2
Next we use the easily veri¬ed vector identity
grad v 2 = v — curl v + v · v
and the fact that u is a gradient to rewrite this as
‚v e h
= E ’ v — curl v + u · u’v· v+ ∆u. (15.12)
‚t m 2m
But curl (v + eA/mc) = 0, since the generalized momentum mv + eA/c
is by assumption a gradient, so that, by (15.8), we can substitute eH/mc
for curl v, so that (15.12) is equivalent to the second equation in (15.2)
with F = ma.
110 CHAPTER 15


There is a long history of attempts to construct alternative theories to
quantum mechanics in which classical notions such as particle trajectories
continue to have meaning.
[46]. L. de Broglie, “Etude critique des bases de l™interpr´tation actuelle
de la m´canique ondulatoire”, Gauthiers-Villars, Paris, 1963.
[47]. D. Bohm, A suggested interpretation of the quantum theory in terms
of “hidden” variables, Physical Review 85 (1952), 166“179.
[48]. D. Bohm and J. P. Vigier, Model of the causal interpretation of
quantum theory in terms of a ¬‚uid with irregular ¬‚uctuations, Physical
Review 96 (1954), 208“216.
The theory that we have described in this section is (in a somewhat
di¬erent form) due to F´nyes:
[49]. Imre F´nyes, Eine wahrscheinlichkeitstheoretische Begr¨ndung und
e u
Interpretation der Quantenmechanik, Zeitschrift f¨r Physik 132 (1952),
[50]. W. Weizel, Ableitung der Quantentheorie aus einem klassische
kausal determinierten Model, Zeitschrift f¨r Physik 134 (1953), 264“285;
Part II, 135 (1953), 270“273; Part III, 136 (1954), 582“604.
[51]. E. Nelson, Derivation of the Schr¨dinger equation from Newtonian
mechanics, Physical Review 150 (1966).
Chapter 16

Comparison with quantum

We now have two quite di¬erent probabilistic interpretations of the
Schr¨dinger equation: the quantum mechanical interpretation of Born
and the stochastic mechanical interpretation of F´nyes. Which interpre-
tation is correct?
It is a triviality that all measurements are reducible to position mea-
surements, since the outcome of any experiment can be described in terms
of the approximate position of macroscopic objects. Let us suppose that
we observe the outcome of an experiment by measuring the exact position
at a given time of all the particles involved in the experiment, including
those constituting the apparatus. This is a more complete observation
than is possible in practice, and if the quantum and stochastic theories


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