[p2 , x] = ppx ’ xpp = ppx ’ pxp + pxp ’ xpp = 2p[p, x] = ’2i p,

because [p, x] = ’i .

ˆ

Next we compute the matrix element for [H0 , r] (for one component):

ˆ ˆ ˆ

m|[H0 , x]|n = m|H0 x|n ’ m|xH0 |n .

The last term is simply equal to En m|x|n . The ¬rst term rewrites by using

ˆ

the Hermitian property of H0 :

—ˆ ˆ— —

ψm H0 (xψn ) dr = (H0 ψm )xψn dr = Em m|x|n .

Collecting terms, (2.92) is obtained.

2.5 Fermions, bosons and the parity rule

There is one further basic principle of quantum mechanics that has far-

reaching consequences for the fate of many-particle systems. It is the rule

that particles have a de¬nite parity. What does this mean?

Particles of the same type are in principle indistinguishable. This means

that the exchange of two particles of the same type in a many-particle sys-

tem should not change any observable, and therefore should not change the

probability density Ψ— Ψ. The wave function itself need not be invariant for

particle exchange, because any change of phase exp(iφ) does not change the

probability distribution. But if we exchange two particles twice, we return

exactly to the original state, so the phase change can only be 0—¦ (no change)

or 180—¦ (change of sign). This means that the parity of the wave function

(the change of sign on exchange of two particles) is either positive (even) or

negative (odd).

Exercises 37

The parity rule (due to Wolfgang Pauli) says that the parity is a ba-

sic, invariant, property of a particle. Thus there are two kinds of particle:

fermions with odd parity and bosons with even parity. Fermions are par-

ticles with half-integral spin quantum number; bosons have integral spins.

Some examples:

• fermions (half-integral spin, odd parity): electron, proton, neutron, muon,

positron, 3 He nucleus, 3 He atom, D atom;

• bosons (integral spin, even parity): deuteron, H-atom, 4 He nucleus, 4 He

atom, H2 molecule.

The consequences for electrons are drastic! If we have two one-electron

orbitals (including spin state) •a and •b , and we put two non-interacting

electrons into these orbitals (one in each), the odd parity prescribes that the

total two-particle wave function must have the form

ψ(1, 2) ∝ •a (1)•b (2) ’ •a (2)•b (1). (2.95)

So, if •a = •b , the wave function cannot exist! Hence, two (non-interacting)

electrons (or fermions in general) cannot occupy the same spin-orbital. This

is Pauli™s exclusion principle. Note that this exclusion has nothing to do

with the energetic (e.g., Coulomb) interaction between the two particles.

Exercises

2.1 Derive (2.30) and (2.31).

2.2 Show that (2.35) is the Fourier transform of (2.34). See Chapter 12.

Show that the width of g — g does not change with time.

2.3

Show that c2 (dt )2 ’ (dx )2 = c2 (dt)2 ’ (dx)2 when dt and dx trans-

2.4

form according to the Lorentz transformation of (2.37).

3

From quantum to classical mechanics: when and

how

3.1 Introduction

In this chapter we shall ask (and possibly answer) the question how quan-

tum mechanics can produce classical mechanics as a limiting case. In what

circumstances and for what kind of particles and systems is the classical

approximation valid? When is a quantum treatment mandatory? What er-

rors do we make by assuming classical behavior? Are there indications from

experiment when quantum e¬ects are important? Can we derive quantum

corrections to classical behavior? How can we proceed if quantum mechanics

is needed for a speci¬c part of a system, but not for the remainder? In the

following chapters the quantum-dynamical and the mixed quantum/classical

methods will be worked out in detail.

The essence of quantum mechanics is that particles are represented by a

wave function and have a certain width or uncertainty in space, related to an

uncertainty in momentum. By a handwaving argument we can already judge

whether the quantum character of a particle will play a dominant role or not.

Consider a (nearly) classical particle with mass m in an equilibrium system

at temperature T , where it will have a Maxwellian velocity distribution (in

each direction) with p2 = mkB T . This uncertainty in momentum implies

that the particle™s width σx , i.e., the standard deviation of its wave function

distribution, will exceed the value prescribed by Heisenberg™s uncertainty

principle (see Chapter 2):

σx ≥ √ . (3.1)

2 mkB T

There will be quantum e¬ects if the forces acting on the particle vary appre-

ciably over the width1 of the particle. In condensed phases, with interparticle

√

1 The width we use here is proportional to the de Broglie wavelength Λ = h/ 2πmkB T that

¬gures in statistical mechanics. Our width is ¬ve times smaller than Λ.

39

40 From quantum to classical mechanics: when and how

Table 3.1 The minimal quantum width in ˚ of the electron and some

A

atoms at temperatures between 10 and 1000 K, derived from Heisenberg™s

uncertainty relation. All values above 0.1 ˚ are given in bold type

A

m(u) 10 K 30 K 100 K 300 K 1000 K

e 0.000545 47 27 15 8.6 4.7

H 1 1.1 0.64 0.35 0.20 0.11

D 2 0.078

0.78 0.45 0.25 0.14

C 12 0.058 0.032

0.32 0.18 0.10

O 16 0.087 0.050 0.028

0.28 0.16

I 127 0.098 0.056 0.031 0.018 0.010

distances of a few ˚, this is the case when the width of the particle exceeds,

A

say, 0.1 ˚. In Table 3.1 the particle widths are given for the electron and

A

for several atoms for temperatures between 10 and 1000 K.

It is clear that electrons are fully quantum-mechanical in all cases (except

hot, dilute plasmas with interparticle separations of hundreds of ˚). Hydro-

A

gen and deuterium atoms are suspect at 300 K but heavier atoms will be

largely classical, at least at normal temperatures. It is likely that quantum

e¬ects of the heavier atoms can be treated by quantum corrections to a

classical model, and one may only hope for this to be true for hydrogen as

well. There will be cases where the intermolecular potentials are so steep

that even heavy atoms at room temperature show essential quantum e¬ects:

this is the case for most of the bond vibrations in molecules. The criterion

for classical behavior here is that vibrational frequencies should not exceed

kB T /h, which at T = 300 K amounts to about 6 THz, or a wave number of

about 200 cm’1 .

We may also consider experimental data to judge the importance of quan-

tum e¬ects, at least for systems in thermal equilibrium. In classical mechan-

ics, the excess free energy (excess with respect to the ideal gas value) of a

conservative system depends only on the potential energy V (r) and not on

the mass of the particles (see Chapter 17):

A = Aid ’ kB T ln V ’N e’βV (r) dr. (3.2)

Since the ideal gas pressure at a given molar density does not depend on

atomic mass either, the phase diagram, melting and boiling points, criti-

cal constants, second virial coe¬cient, compressibility, and several molar

properties such as density, heat capacity, etc. do not depend on isotopic