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cannot be distinguished in this way then they cannot be distinguished by
an actual experiment. However, for such an ideal experiment stochastic
and quantum mechanics give the same probability density |ψ|2 for the
position of the particles at the given time.
The physical interpretation of stochastic mechanics is very di¬erent
from that of quantum mechanics. Consider, to be speci¬c, a hydrogen
atom in the ground state. Let us use Coulomb units; i.e., we set m =
e2 = h = 1, where m is the reduced mass of the electron and e is its
¯
charge. The potential is V = ’1/r, where r = |x| is the distance to the
origin, and the ground state wave function is
1
ψ = √ e’r .
π
In quantum mechanics, ψ is a complete description of the state of the sys-

111
112 CHAPTER 16

tem. According to stochastic mechanics, the electron performs a Markov
process with

x(t)
dx(t) = ’ dt + dw(t),
|x(t)|

where w is the Wiener process with di¬usion coe¬cient 1 . (The gradient
2
of ’r is ’x/r.) The invariant probability density is |ψ|2 . The electron
moves in a very complicated random trajectory, looking locally like a
trajectory of the Wiener process, with a constant tendency to go towards
the origin, no matter which direction is taken for time. The similarity to
ordinary di¬usion in this case is striking.
How can such a classical picture of atomic processes yield the same
predictions as quantum mechanics? In quantum mechanics the positions
of the electron at di¬erent times are non-commuting observables and so
(by Theorem 14.1) cannot in general be expressed as random variables.
Yet we have a theory in which the positions are random variables.
To illustrate how con¬‚ict with the predictions of quantum mechanics is
avoided, let us consider the even simpler case of a free particle. Again we
set m = h = 1. The wave function at time 0, ψ0 , determines the Markov
¯
process. To be concrete let us take a case in which the computations
are easy, by letting ψ0 be a normalization constant times exp(’|x|2 /2a),
where a > 0. Then ψt is a normalization constant times

|x|2 |x|2 (a ’ it)
exp ’ = exp ’ .
a2 + t2
2(a + it)

Therefore (by Chapter 15)
a
u=’ x,
a2 + t2
t
v= x,
a2 + t2
t’a
b= x.
a2 + t2
Thus the particle performs the Gaussian Markov process
t’a
dx(t) = xdt + dw(t),
a2 + t2
where w is the Wiener process with di¬usion coe¬cient 1 .
2
COMPARISON WITH QUANTUM MECHANICS 113

Now let X(t) be the quantum mechanical position operator at time t
(Heisenberg picture). That is,
1 1
X(t) = e 2 it∆ X0 e’ 2 it∆ ,

where X0 is the operator of multiplication by x. For each t the probability,
according to quantum mechanics, that if a measurement of X(t) is made
the particle will be found to lie in a given region B of ‚3 is just the
integral over B of |ψt |2 (where ψt is the wave function at time t in the
Schr¨dinger picture). But |ψt |2 is the probability density of x(t) in the
o
above Markov process, so this integral is equal to the probability that
x(t) lies in B.
We know that the X(t) for varying t cannot simultaneously be repre-
sented as random variables. In fact, since the particle is free,
t1 + t2 X(t1 ) + X(t2 )
X = (16.1)
2 2
for all t1 , t2 , and the corresponding relation is certainly not valid for the
random variables x(t). Thus the mathematical structures of the quan-
tum and stochastic theories are incompatible. However, there is no con-
tradiction in measurable predictions of the two theories. In fact, if one
attempted to verify the quantum mechanical relation (16.1) by measuring

t1 + t2
X , X(t1 ), X(t2 ),
2
then, by the uncertainty principle, the act of measurement would produce
a deviation from the linear relation (16.1) of the same order of magnitude
as that which is already present in the stochastic theory of the trajectories.
Although the operators on the two sides of (16.1) are the same operator,
it is devoid of operational meaning to say that the position of the particle
at time (t1 + t2 )/2 is the average of its positions at times t1 and t2 .
The stochastic theory is conceptually simpler than the quantum the-
ory. For instance, paradoxes related to the “reduction of the wave packet”
(see Chapter 14) are no longer present, since in the stochastic theory
the wave function is no longer a complete description of the state. In
the quantum theory of measurement the consciousness of the observer
(i.e., my consciousness) plays the rˆle of a deus ex machina to introduce
o
randomness, since without it the quantum theory is completely determin-
istic. The stochastic theory is inherently indeterministic.
114 CHAPTER 16

The stochastic theory raises a number of new mathematical questions
concerning Markov processes. From a physical point of view, the theory
is quite vulnerable. We have ignored a vast area of quantum mechanics”
questions concerning spin, bosons and fermions, radiation, and relativistic
covariance. Either the stochastic theory is a curious accident or it will
generalize to these other areas, in which case it may be useful.
The agreement between the predictions of quantum mechanics and
stochastic mechanics holds only for a limited class of forces. The Hamil-
tonians we considered (Chapter 15) involved at most the ¬rst power of
the velocity in the interaction part. Quantum mechanics can treat much
more general Hamiltonians, for which there is no stochastic theory. On
the other hand, the basic equations of the stochastic theory (Eq. (15.2)
with F = ma) can still be formulated for forces that are not derivable
from a potential. In this case we can no longer require that v be a gradient
and no longer have the Schr¨dinger equation. In fact, quantum mechanics
o
is incapable of describing such forces. If there were a fundamental force in
nature with a higher order dependence on velocity or not derivable from
a potential, at most one of the two theories could be physically correct.
Comparing stochastic mechanics (which is classical in its descriptions)
and quantum mechanics (which is based on the principle of complementar-
ity), one is tempted to say that they are, in the sense of Bohr, complemen-
tary aspects of the same reality. I prefer the viewpoint that Schr¨dinger
o
[30, §14] expressed in 1926:
“ . . . It has even been doubted whether what goes on in the atom
could ever be described within the scheme of space and time. From the
philosophical standpoint, I would consider a conclusive decision in this
sense as equivalent to a complete surrender. For we cannot really alter our
manner of thinking in space and time, and what we cannot comprehend
within it we cannot understand at all. There are such things”but I do
not believe that atomic structure is one of them.”

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