dp iˆ iˆ

= [H, p] =

ˆ [V , p ]

ˆ

dt

i ‚ i ‚ dV

’ =’

= V V . (3.15)

i ‚x i ‚x dx

This is the expectation of the force over the wave function, not the force

44 From quantum to classical mechanics: when and how

at the expectation of x! When the force is constant, there is no di¬erence

between the two values and the motion is classical, as far as the expectations

of x and p are concerned. This is even true if the force depends linearly on

x:

F (x) = F0 + F (x ’ x ),

where F0 = F ( x ) and F is a constant, because

F = F0 + F x ’ x = F0 .

Expanding the force (or potential) in a Taylor series, we see that the

leading correction term on the force is proportional to the second derivative

of the force times the variance of the wave packet:

d3 V

dV dV 1

(x ’ x )2 + · · · .

= + (3.16)

dx3

dx dx 2!

x x

The motion is classical if the force (more precisely, the gradient of the force)

does not vary much over the quantum width of the particle. This is true

even for electrons in macroscopic ¬elds, as they occur in accelerators and in

dilute or hot plasmas; this is the reason that hot plasmas can be treated with

classical equations of motion, as long as the electromagnetic interactions are

properly incorporated. For electrons near point charges the force varies

enormously over the quantum width and the classical approximation fails

completely.

It is worth mentioning that a harmonic oscillator moves in a potential

that has no more than two derivatives, and “ as given by the equations

derived above “ moves according to classical dynamics. Since we know that

a quantum oscillator behaves di¬erently from a classical oscillator (e.g., it

has a zero-point energy), this is surprising at ¬rst sight! But even though

the classical equations do apply for the expectation values of position and

momentum, a quantum particle is not equal to a classical particle. For

example, p2 = p 2 . A particle at rest, with p = 0, can still have a

kinetic and potential energy. Thus, for classical behavior it is not enough

that the expectation of x and p follow classical equations of motion.

3.3 Path integral quantum mechanics

3.3.1 Feynman™s postulate of quantum dynamics

While the Schr¨dinger description of wave functions and their evolution in

o

time is adequate and su¬cient, the Schr¨dinger picture does not connect

o

3.3 Path integral quantum mechanics 45

smoothly to the classical limit. In cases that the particles we are inter-

ested in are nearly classical (this will often apply to atoms, but not to

electrons) the path integral formulation of quantum mechanics originating

from Feynman2 can be more elucidating. This formulation renders a solu-

tion to the propagation of wave functions equivalent to that following from

Schr¨dinger™s equation, but has the advantage that the classical limit is

o

more naturally obtained as a limiting case. The method allows us to ob-

tain quantum corrections to classical behavior. In particular, corrections

to the classical partition function can be obtained by numerical methods

derived from path integral considerations. These path integral Monte Carlo

and molecular dynamics methods, PIMC and PIMD, will be treated in more

detail in Section 3.3.9.

Since the Schr¨dinger equation is linear in the wave function, the time

o

propagation of the wave function can be expressed in terms of aGreen™s

function G(r f , tf ; r 0 , t0 ), which says how much the amplitude of the wave

function at an initial position r 0 at time t0 contributes to the amplitude of

the wave function at a ¬nal position r f at a later time tf . All contributions

add up to an (interfering) total wave function at time tf :

Ψ(r f , tf ) = dr 0 G(r f , tf ; r 0 , t0 )Ψ(r 0 , t0 ). (3.17)

The Green™s function is the kernel of the integration.

In order to ¬nd an expression for G, Feynman considers all possible paths

{r(t)} that run from position r 0 at time t0 to position r f at time tf . For

each path it is possible to compute the mechanical action S as an integral

of the Lagrangian L(r, r, t) = K ’ V over that path (see Chapter 15):

™

tf

L(r, r, t) dt.

™

S= (3.18)

t0

Now de¬ne G as the sum over all possible paths of the function exp(iS/ ),

which represents a phase of the wave function contribution:

def

eiS/ .

G(r f , tf ; r 0 , t0 ) = (3.19)

all paths

This, of course, is a mathematically dissatisfying de¬nition, as we do not

know how to evaluate “all possible paths” (Fig. 3.1a). Therefore we ¬rst

approximate a path as a contiguous sequence of linear paths over small

time steps „ , so a path is de¬ned by straight line segments between the

2 Although Feynman™s ideas date from 1948 (Feynman, 1948) and several articles on the subject

are available, the most suitable original text to study the subject is the book by Feynman and

Hibbs (1965).

46 From quantum to classical mechanics: when and how

tn = tf

r f (tf ) Si r

r

r¨

¨

c r$

$$

$$

r

r

ti

r ti’1

r(t)

r

¡

r

¡

r

¡

(a) (b)

r 0 (t0 ) t0

Figure 3.1 (a) Several paths connecting r 0 at time t0 with r f at time tf . One path

(thick line) minimizes the action S and represents the path followed by classical

mechanics. (b) One path split up into many linear sections (r i’1 , r i ) with actions

Si .

initial point r 0 at time t0 , the intermediate points r 1 , . . . , r n’1 at times

„, 2„, . . . , (n ’ 1)„ , and the ¬nal point r n = r f at time tn = t0 + n„ , where

n = (tf ’ t0 )/„ (Fig. 3.1b). Then we construct the sum over all possible

paths of this kind by integrating r 1 , . . . , r n’1 over all space. Finally, we take