dinate transformation U:

ρ = U† ρU. (14.74)

On a basis on which H is diagonal (i.e., on a basis of eigenfunctions of

ˆ

H : Hnm = En δnm ) the solution of ρ(t) is

i

(Em ’ En )t ,

ρnm (t) = ρnm (0) exp (14.75)

implying that ρnn is constant.

394 Vectors, operators and vector spaces

14.8.1 The ensemble-averaged density matrix

The density matrix can be averaged over a statistical ensemble of systems

without loss of information about ensemble-averaged observables. This is

in contrast to the use of c(t) which contains a phase factor and generally

averages out to zero over an ensemble.

In thermodynamic equilibrium (in the canonical ensemble) the probability

of a system to be in the n-th eigenstate with energy En is proportional to

its Boltzmann factor:

1

Pn = e’βEn , (14.76)

Q

where β = 1/kB T and

e’βEn

Q= (14.77)

n

is the partition function (summed over all quantum states). On a basis set

ˆ

of eigenfunctions of H, in which H is diagonal,

1 ’βH

ρeq = e , (14.78)

Q

Q = tr e’βH , (14.79)

implying that o¬-diagonal elements vanish, which is equivalent to the as-

sumption that the phases of ρnm are randomly distributed over the ensemble

(random phase approximation).

But (14.78) and (14.79) are also valid after any unitary coordinate trans-

formation, and thus these equations are generally valid on any orthonormal

basis set.

14.8.2 The density matrix in coordinate representation

The ensemble-averaged density matrix gives information on the probabil-

ity of quantum states φn , but it does not give direct information on the

probability of a con¬guration of the particles in space. In the coordinate

representation we de¬ne the equilibrium density matrix as a function of

(multiparticle) spatial coordinates r:

φ— (r)e’βEn φn (r ).

ρ(r, r ; β) = (14.80)

n

n

This is a square continuous “matrix” of ∞ — ∞ dimensions. The trace of ρ

is

tr ρ = ρ(r, r; β) dr, (14.81)

14.8 The density matrix 395

which is equal to the partition function Q.

A product of such matrices is in fact an integral, which is itself equal to

a density matrix:

ρ(r, r 1 ; β1 )ρ(r 1 , r ; β2 ) dr 1 = ρ(r, r ; β1 + β2 ), (14.82)

as we can check by working out the l.h.s.:

φ— (r)e’β1 En φn (r 1 ) φ— (r 1 )e’β2 Em φm (r ) dr 1

n m

n m

φ— (r)e’β1 En ’β2 Em φm (r ) φn (r 1 )φ— (r 1 ) dr 1

= n m

n,m

φ— (r)e’(β1 +β2 )En φn (r ) = ρ(r, r ; β1 + β2 ).

= n

n

A special form of this equality is

ρ(r, r ; β) = ρ(r, r 1 ; β/2)ρ(r 1 , r ; β/2) dr 1 , (14.83)

which can be written more generally as

ρ(r, r ; β) (14.84)

= ρ(r, r 1 ; β/n)ρ(r 1 , r 2 ; β/n) . . . ρ(r n’1 , r ; β/n) dr 1 , . . . , dr n’1 .

Applying this to the case r = r , we see that

Q = tr ρ (14.85)

= ρ(r, r 1 ; β/n)ρ(r 1 , r 2 ; β/n) . . . ρ(r n’1 , r; β/n) dr dr 1 , . . . , dr n’1 .

Thus the partition function can be obtained by an integral over density

matrices with the “high temperature” β/n; such density matrices can be

approximated because of the small value in the exponent. This equality is

used in path integral Monte Carlo methods to incorporate quantum distri-

butions of “heavy” particles into simulations.

15

Lagrangian and Hamiltonian mechanics

15.1 Introduction

Classical mechanics is not only an approximation of quantum mechanics,

valid for heavy particles, but historically it also forms the basis on which

quantum-mechanical notions are formed. We also need to be able to describe

mechanics in generalized coordinates if we wish to treat constraints or in-

troduce other ways to reduce the number of degrees of freedom. The basis

for this is Lagrangian mechanics, from which the Hamiltonian description is

derived. The latter is not only used in the Schr¨dinger equation, but forms

o

also the framework in which (classical) statistical mechanics is developed. A

background in Lagrangian and Hamiltonian mechanics is therefore required

for many subjects treated in this book.

After the derivation of Lagrangian and Hamiltonian dynamics, we shall

consider how constraints can be built in. The common type of constraint

is a holonomic constraint that depends only on coordinates, such as a bond

length constraint, or constraints between particles that make them behave as

one rigid body. An example of a non-holonomic constraint is the total kinetic

energy (to be kept at a constant value or at a prescribed time-dependent

value).

We shall only give a concise review; for details the reader is referred to text

books on classical mechanics, in particular to Landau and Lifschitz (1982)

and to Goldstein et al. (2002).

There are several ways to introduce the principles of mechanics, leading to

Newton™s laws that express the equations of motion of a mechanical system.

A powerful and elegant way is to start with Hamilton™s principle of least

action as a postulate. This is the way chosen by Landau and Lifshitz (1982).

397

398 Lagrangian and Hamiltonian mechanics

15.2 Lagrangian mechanics

Consider a system described by n degrees of freedom or coordinates q =

q1 , . . . , qn (not necessarily the 3N cartesian coordinates of N particles) that

evolve in time t. A function L(q, q, t) exists with the property that the

™

action

t2

def

L(q, q, t) dt

S= ™ (15.1)

t1