and w— by

dx(t) = b— x(t), t dt + dw— (t).

85

86 CHAPTER 13

‚ +1.

Let f be a smooth function on Then

f x(t + ∆t), t + ∆t ’ f x(t), t =

‚f

x(t), t ∆t + [x(t + ∆t) ’ x(t)] · f x(t), t

‚t

‚2f

1

[xi (t + ∆t) ’ xi (t)][xj (t + ∆t) ’ xj (t)] i j x(t), t

+

2 i,j ‚x ‚x

+ o(t),

so that

‚

+b·

Df x(t), t = ( + ν∆)f x(t), t . (13.1)

‚t

Let ν— be the di¬usion coe¬cient of w— . (A priori, ν— might depends on

x and t, but we shall see shortly that ν— = ν.) Similarly, we ¬nd

‚

+ b— · ’ ν— ∆ f x(t), t .

D— f x(t), t = (13.2)

‚t

If f and g have compact support in time, then Theorem 11.12 shows

that

∞ ∞

EDf x(t), t · g x(t), t dt = ’ Ef x(t), t D— g x(t), t dt;

’∞ ’∞

that is,

∞

‚

+ b · + ν∆ f (x, t) · g(x, t)ρ(x, t) dxdt =

‚ ‚t

’∞

∞

‚

’ + b— · ’ ν— ∆ g(x, t) · ρ(x, t) dxdt.

f (x, t)

‚ ‚t

’∞

For A a partial di¬erential operator, let A† be its (Lagrange) adjoint with

respect to Lebesgue measure on ‚ +1 and let A— be its adjoint with respect

to ρ times Lebesgue measure. Then (Af )gρ is equal to both f A† (gρ)

and f (A— g)ρ, so that

A— = ρ’1 A† ρ.

Now

†

‚ ‚

+b· =’ ’b· ’ div b + ν∆,

+ ν∆

‚t ‚t

KINEMATICS OF MARKOVIAN MOTION 87

so that

†

‚ ‚

ρ’1 ρg = ρ’1 ’ ’b· ’ div b + ν∆ (ρg) =

+ b· + ν∆

‚t ‚t

‚g ‚ρ

’ ρ’1 g ’ b · g ’ ρ’1 b · (grad ρ)g ’ (div b)g

’

‚t ‚t

’1

+ ρ ν (∆ρ)g + 2 grad ρ · grad g + ρ∆g .

Recall the Fokker-Planck equation

‚ρ

= ’ div(bρ) + ν∆ρ. (13.3)

‚t

Using this we ¬nd

‚ρ div(bρ) ∆ρ grad ρ ∆ρ

’ρ’1 ’ν = div b + b · ’ν

= ,

‚t ρ ρ ρ ρ

so we get

‚ ‚ grad ρ

’ ’ b— · + ν— ∆ = ’ ’b· ·

+ 2ν + ν∆.

‚t ‚t ρ

Therefore, ν— = ν and b— = b ’ 2ν(grad ρ)/ρ. If we make the de¬nition

b ’ b—

u= ,

2

we have

grad ρ

u=ν .

ρ

We call u the osmotic velocity cf. Chapter 4, Eq. (6) .

There is also a Fokker-Planck equation for time reversed:

‚ρ

= ’ div(b— ρ) ’ ν∆ρ. (13.4)

‚t

If we de¬ne

b + b—

v= ,

2

we have the equation of continuity

‚ρ

= ’ div(vρ),

‚t

88 CHAPTER 13

obtained by averaging (13.3) and (13.4). We call v the current velocity.

Now

grad ρ

u=ν = ν grad log ρ.

ρ

Therefore,

‚ρ

‚u ‚

=ν grad log ρ = ν grad ‚t =

‚t ‚t ρ

’ div(vρ) grad ρ

= ’ν grad div v + v ·

ν grad =

ρ ρ

’ ν grad div v ’ grad v · u.

That is,

‚u

= ’ν grad div v ’ grad v · u. (13.5)

‚t

Finally, from (13.1) and (13.2),

‚

b— x(t), t + b ·

Db— x(t), t = b— x(t), t + ν∆b— x(t), t ,

‚t

‚

D— b x(t), t = b x(t), t + b— · b x(t), t ’ ν∆b x(t), t ,

‚t

so that the mean acceleration as de¬ned in Chapter 11, Eq. (11.15) is

given by a x(t), t where

b ’ b—

‚ b + b— 1 1

+ b· b— + b— · b ’ ν∆

a= .

‚t 2 2 2 2

That is,

‚v

=a+u· u’v· v + ν∆u. (13.6)

‚t

Chapter 14

Remarks on quantum

mechanics

In discussing physical theories of Brownian motion we have seen that

physics has interesting ideas and problems to contribute to probability

theory. Probabilities also play a fundamental rˆle in quantum mechanics,

o

but the notion of probability enters in a new way that is foreign both

to classical mechanics and to mathematical probability theory. A mathe-

matician interested in probability theory should become familiar with the

peculiar concept of probability in quantum mechanics.

We shall discuss quantum mechanics from the point of view of the rˆle o

of probabilistic concepts in it, limiting ourselves to the non-relativistic

quantum mechanics of systems of ¬nitely many degrees of freedom. This

theory was discovered in 1925“1926. Its principal features were estab-

lished quickly, and it has changed very little in the last forty years.

Quantum mechanics originated in an attempt to solve two puzzles:

the discrete atomic spectra and the dual wave-particle nature of matter

and radiation. Spectroscopic data were interpreted as being evidence for

the fact that atoms are mechanical systems that can exist in stationary

states only for a certain discrete set of energies.

There have been many discussions of the two-slit thought experiment

illustrating the dual nature of matter; e.g., [28, §12] and [29, Ch. 1].

Here we merely recall the bare facts: A particle issues from — in the

¬gure, passes through the doubly-slitted screen in the middle, and hits

the screen on the right, where its position is recorded. Particle arrivals

are sharply localized indivisible events, but despite this the probability of

arrival shows a complicated di¬raction pattern typical of wave motion. If

89

90 CHAPTER 14

one of the holes is closed, there is no interference pattern. If an observa-

tion is made (using strong light of short wave length) to see which of the

two slits the particle went through, there is again no interference pattern.

Figure 3

The founders of quantum mechanics can be divided into two groups:

the reactionaries (Planck, Einstein, de Broglie, Schr¨dinger) and the radi-

o

cals (Bohr, Heisenberg, Born, Jordan, Dirac). Correspondingly, quantum

mechanics was discovered in two apparently di¬erent forms: wave me-

chanics and matrix mechanics. (Heisenberg™s original term was “quan-

tum mechanics,” and “matrix mechanics” is used when one wishes to

distinguish it from Schr¨dinger™s wave mechanics.)

o

In 1900 Planck introduced the quantum of action h and in 1905 Ein-

stein postulated particles of light with energy E = hν (ν the frequency).

We give no details as we shall not discuss radiation. In 1924, while a

graduate student at the Sorbonne, Louis de Broglie put the two formulas

E = mc2 and E = hν together and invented matter waves. The wave

nature of matter received experimental con¬rmation in the Davisson-

Germer electron di¬raction experiment of 1927, and theoretical support

by the work of Schr¨dinger in 1926. De Broglie™s thesis committee in-

o

cluded Perrin, Langevin, and Elie Cartan. Perhaps Einstein heard of de

REMARKS ON QUANTUM MECHANICS 91

Broglie™s work from Langevin. In any case, Einstein told Born, “Read it;

even though it looks crazy it™s solid,” and he published comments on de

Broglie™s work which Schr¨dinger read.

o

Suppose, with Schr¨dinger, that we have a particle (say an electron)

o

of mass m in a potential V . Here V is a real function on ‚3 representing

the potential energy. Schr¨dinger attempted to describe the motion of

o

the electron by means of a quantity ψ subject to a wave equation. He

was led to the hypothesis that a stationary state vibrates according to

the equation

h2

¯

∆ψ + (E ’ V )ψ = 0, (14.1)

2m

where h is Planck™s constant h divided by 2π, and E (with the dimensions

¯

of energy) plays the rˆle of an eigenvalue.

o

This equation is similar to the wave equation for a vibrating elastic

¬‚uid contained in a given enclosure, except that V is not a constant.

Schr¨dinger was struck by another di¬erence [30, p. 12]:

o

“A simpli¬cation in the problem of the ˜mechanical™ waves (as com-

pared with the ¬‚uid problem) consists in the absence of boundary con-

ditions. I thought the latter simpli¬cation fatal when I ¬rst attacked

these questions. Being insu¬ciently versed in mathematics, I could not

imagine how proper vibration frequencies could appear without boundary

conditions.”

Despite these misgivings, Schr¨dinger found the eigenvalues and eigen-

o

functions of (14.1) for the case of the hydrogen atom, V = ’e/r where

e is the charge of the electron (and ’e is the charge of the nucleus) and

r2 = x2 + y 2 + z 2 . The eigenvalues corresponded precisely to the known

discrete energy levels of the hydrogen atom.

This initial triumph, in which discrete energy levels appeared for the

¬rst time in a natural way, was quickly followed by many others. Be-

fore the year of 1926 was out, Schr¨dinger reprinted six papers on wave

o

mechanics in book form [30]. A young lady friend remarked to him (see

the preface to [30]): “When you began this work you had no idea that

anything so clever would come out of it, had you?” With this remark

Schr¨dinger “wholeheartedly agreed (with due quali¬cation of the ¬‚atter-

o

ing adjective).”

Shortly before Schr¨dinger made his discovery, the matrix mechanics

o

of Heisenberg appeared. In this theory one constructs six in¬nite matrices