Consider the quantum-mechanical canonical partition function of an N -

particle system

Q= exp(’βEn ), (3.42)

n

where the sum is to be taken over all quantum states (not energy levels!)

and β equals 1/kB T . Via the free energy relation (A is the Helmholtz free

energy)

A = ’kB T ln Q (3.43)

and its derivatives with respect to temperature and volume, all relevant

thermodynamic properties can be obtained. Unfortunately, with very few

exceptions under idealized conditions, we cannot enumerate all quantum

states and energy levels of a complex system, as this would mean the deter-

mination of all eigenvalues of the full Hamiltonian of the system.

Since the eigenfunctions φn of the system form an orthonormal set, we

can also write (3.42) as

φn (r)φ— (r) exp(’βEn ).

Q= dr (3.44)

n

n

Now compare this with the expression for the path integral kernel of (3.41).

Apart from the fact that initial and ¬nal point are the same (r), and the

form is integrated over dr, we see that instead of time we now have ’i β.

So the canonical partition function is closely related to a path integral over

negative imaginary time. The exact relation is

drG(r, ’i β; r, 0),

Q= (3.45)

with (inserting „ = ’i β/n into (3.38))

G(r, ’i β; r, 0) = lim C(n) dr 1 · · · dr n’1

n’∞

n

nm 1

exp ’β (r k ’ r k’1 )2 + V (r k ) , (3.46)

2 2β2 n

k=1

3.3 Path integral quantum mechanics 53

q

gg

y

q q

g

¡ q

©

! #

£

¡ g £

q

s ©

q

t

¡ £

£ t

q¡ tq q

” £

£

¨

B

q¨ q 0

I

q££ rr r

s

d

dq r jq

q r n’1

G

f

w

f q d

g

fq

A q

yg d

i d

r 1 r r = r 0 = r n

‚

g

g)

q

r2

Figure 3.2 A closed path in real space and imaginary time, for the calculation of

quantum partition functions.

where

’3nN/2

h2 β

C(n) = , (3.47)

2πnm

and

r 0 = r n = r.

Note that r stands for a 3N -dimensional cartesian vector of all particles in

the system. Also note that all paths in the path integral are closed: they end

in the same point where they start (Fig. 3.2). In the multiparticle case, the

imaginary time step is made for all particles simultaneously; each particle

therefore traces out a three-dimensional path.

A path integral over imaginary time does not add up phases of di¬erent

paths, but adds up real exponential functions over di¬erent paths. Only

paths with reasonably-sized exponentials contribute; paths with highly neg-

ative exponents give negligible contributions. Although it is di¬cult to

imagine what imaginary-time paths mean, the equations derived for real-

time paths can still be used and lead to real integrals.

3.3.6 Classical and nearly classical approximations

Can we easily see what the classical limit is for imaginary-time paths? As-

sume that each path (which is closed anyway) does not extend very far from

its initial and ¬nal point r. Assume also that the potential does not vary

54 From quantum to classical mechanics: when and how

appreciably over the extent of each path, so that it can be taken equal to

V (r) for the whole path. Then we can write, instead of (3.46):

G(r, ’i β; r, 0) = exp(’βV (r)) lim C(n) dr 1 · · · dr n’1

n’∞

n

nm

exp ’ 2 (r k ’ r k’1 )2 . (3.48)

2β

k=1

The expression under the limit sign yields (2πmkB T /h2 )3N/2 , independent

of the number of nodes n. The evaluation of the multiple integral is not

entirely trivial, and the proof is given below. Thus, after integrating over r

we ¬nd the classical partition function

3N/2

2πmkB T

e’βV (r) dr.

Q= (3.49)

h2

Since the expression is independent of n, there is no need to take the limit for

n ’ ∞. Therefore the imaginary-time path integral without any intervening

nodes also represents the classical limit.

Note that the integral is not divided by N !, since the indistinguishabil-

ity of the particles has not been introduced in the path integral formalism.

Therefore we cannot expect that path integrals for multiparticle systems

will treat exchange e¬ects correctly. For the application to nuclei in con-