3.1 Introduction 41

Table 3.2 Critical point characteristic for various isotopes of helium,

hydrogen and water

Vc (cm3 mol’1 )

Tc (K) pc (bar)

4

He 5.20 2.26 57.76

3

He 3.34 1.15 72.0

H2 33.18 12.98 66.95

HD 35.9 14.6 62.8

D2 38.3 16.3 60.3

H2 O 647.14 220.64 56.03

D2 O 643.89 216.71 56.28

constants as dissociation or association constants or partition coe¬cients

depend on isotopic composition for classical substances. If such properties

appear to be dependent on isotopic composition, this is a sure sign of the

presence of quantum e¬ects on atomic behavior.

Look at a few examples. Table 3.2 lists critical constants for di¬erent

isotopes of helium, hydrogen and water. Table 3.3 lists some equilibrium

properties of normal and heavy water. It is not surprising that the proper-

ties of helium and hydrogen at (very) low temperatures are strongly isotope

dependent. The di¬erence between H2 O and D2 O is not negligible: D2 O

has a higher temperature of maximum density, a higher enthalpy of vapor-

ization and higher molar heat capacity; it appears more “structured” than

H2 O. The most likely explanation is that it forms stronger hydrogen bonds

as a result of the quantum-mechanical zero-point energy of the intermolec-

ular vibrational and librational modes of hydrogen-bonded molecules. Ac-

curate simulations must either incorporate this quantum behavior, or make

appropriate corrections for it.

It is instructive to see how the laws of classical mechanics, i.e., Newton™s

equations of motion, follow from quantum mechanics. We consider three

di¬erent ways to accomplish this goal. In Section 3.2 we derive equations of

motion for the expectation values of position and velocity, following Ehren-

fest™s arguments of 1927. A formulation of quantum mechanics which is

equivalent to the Schr¨dinger equation but is more suitable to approach the

o

classical limit, is Feynman™s path integral formulation. We give a short in-

troduction in Section 3.3. Then, in Section 3.4, we consider a formulation

of quantum mechanics, originally proposed by Madelung and by de Broglie

in 1926/27, and in 1952 revived by Bohm, which represents the evolution

42 From quantum to classical mechanics: when and how

Table 3.3 Various properties of normal and heavy water

H2 O D2 O

melting point (—¦ C) 0 3.82

boiling point (—¦ C) 100 101.4

temperature of maximum density (—¦ C) 3.98 11.19

vaporization enthalpy at 3.8 —¦ C (kJ/mol) 44.8 46.5

molar volume at 25 —¦ C (cm3 /mol) 18.07 18.13

molar heat capacity at 25 —¦ C (J K’1 mol’1 ) 74.5 83.7

ionization constant ’ log[Kw /(mol2 dm’6 )] at 25 —¦ C 13.995 14.951

of the wave function by a ¬‚uid of particles which follow trajectories guided

by a special quantum force. The application of quantum corrections to

equilibrium properties computed with classical methods, and the actual in-

corporation of quantum e¬ects into simulations, is the subject of following

chapters.

3.2 From quantum to classical dynamics

In this section we ask the question: Can we derive classical equations of

motion for a particle in a given external potential V from the Schr¨dinger

o

equation? For simplicity we consider the one-dimensional case of a particle

of mass m with position x and momentum p = mx. The classical equations

™

of Newton are

dx p

= , (3.3)

dt m

dp dV (x)

=’ . (3.4)

dt dx

Position and momentum of a quantum particle must be interpreted as the

expectation of x and p. The classical force would then be the value of the

gradient of V taken at the expectation of x. So we ask whether

dx p

? =? , (3.5)

dt m

dp dV

? =? ’ . (3.6)

dt dx x

We follow the argument of Ehrenfest (1927). See Chapter 14 for details of

the operator formalism and equations of motion.

Recall that the expectation A of an observable A over a quantum system

3.2 From quantum to classical dynamics 43

with wave function Ψ(x, t) is given by

Ψ— AΨ dx,

ˆ

A= (3.7)

ˆ

where A is the operator of A. From the time-dependent Schr¨dinger equa-

o

tion the equation of motion (14.64) for the expectation of A follows:

dA i ˆˆ

= [H, A] . (3.8)

dt

ˆˆ ˆ ˆ

Here [H, A] is the commutator of H and A:

ˆˆ ˆ ˆ ˆˆ

[H, A] = H A ’ AH. (3.9)

We note that the Hamiltonian is the sum of kinetic and potential energy:

ˆ2

ˆ = K + V = p + V (x),

ˆ ˆ

H (3.10)

2m

ˆ ˆ ˆ

and that p commutes with K but not with V , while x commutes with V but

ˆ ˆ

ˆ

not with K. We shall also need the commutator

[ˆ, x] = px ’ xp =

pˆ ˆˆ ˆ ˆ . (3.11)

i

This follows from inserting the operator for p:

‚(xψ) ‚ψ

’x = ψ. (3.12)

i ‚x i ‚x i

Now look at the ¬rst classical equation of motion (3.5). We ¬nd using

(3.8) that

dx i p

[ˆ2 , x] =

= pˆ , (3.13)

dt 2m m

because

[ˆ2 , x] = ppx ’ xpp = ppx ’ pxp + pxp ’ xpp

pˆ ˆˆˆ ˆ ˆˆ ˆˆˆ ˆˆ ˆ ˆˆ ˆ ˆ ˆˆ

2

= p[ˆ, x] + [ˆ, x]ˆ =

ˆp ˆ p ˆp p.

ˆ (3.14)

i

Hence the ¬rst classical equation of motion is always valid.