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.”