or, in terms of G:

dr 1 G— (r 2 , t2 ; r 1 , t1 )G(r 2 , t2 ; r 1 , t1 )Ψ— (r 1 , t1 )Ψ(r 1 , t1 )

dr 2 dr 1

dr 1 Ψ— (r 1 , t1 )Ψ(r 1 , t1 )

= (3.34)

must be valid for any Ψ. This is only true if

dr 2 G— (r 2 , t2 ; r 1 , t1 )G(r 2 , t2 ; r 1 , t1 ) = δ(r 1 ’ r 1 ) (3.35)

for any pair of times t1 , t2 for which t2 > t1 . This is the normalization

condition for G.

Since the normalization condition must be satis¬ed for any time step, it

must also be satis¬ed for an in¬nitesimal time step „ , for which the path is

linear from r 1 to r 2 :

m(r 2 ’ r 1 )2

i

G(r 2 , t + „ ; r 1 , t) = C(„ ) exp + V (r, t)„ . (3.36)

2„

If we apply the normalization condition (3.35) to this G, we ¬nd that

’3/2

ih„

C(„ ) = , (3.37)

m

just as we already found while proving the equivalence with the Schr¨dinger

o

equation. The 3 in the exponent relates to the dimensionality of the wave

function, here taken as three dimensions (one particle in 3D space). For N

particles in 3D space the exponent becomes ’3N/2.

50 From quantum to classical mechanics: when and how

Proof Consider N particles in 3D space. Now

im

G— G dr 2 = C — C {(r 2 ’ r 1 )2 ’ (r 2 ’ r 1 )2 }

dr 2 exp

2„

im 2 im

= C — C exp (r 1 ’ r 12 ) (r ’ r 1 ) · r 2 .

dr 2 exp

„1

2„

Using one of the de¬nitions of the δ-function:

+∞

exp(±ikx) dx = 2πδ(k),

’∞

or, in 3N dimensions:

exp(±ik · r) dr = (2π)3N δ(k),

where the delta function of a vector is the product of delta functions of its

components, the integral reduces to

3N

m(r 1 ’ r 1 ) „

δ(r 1 ’ r 1 ).

3N 3N

(2π) δ = (2π)

„ m

Here we have made use of the transformation δ(ax) = (1/a)δ(x). The

presence of the delta functions means that the exponential factor before the

integral reduces to 1, and we obtain

3N

h„

— —

δ(r 1 ’ r 1 ).

G G dr 2 = C C

m

Thus the normalization condition (3.35) is satis¬ed if

’3N

h„

—

C C= .

m

This is a su¬cient condition to keep the integrated probability of the wave

function invariant in time. But there are many solutions for C, di¬ering

by an arbitrary phase factor, as long as the absolute value of C equals the

square root of the right-hand side (real) value. However, we do not wish

a solution for G that introduces a changing phase into the wave function,

and therefore the solution for C found in the derivation of the Schr¨dinger

o

— Ψ, but also Ψ itself invariant in the limit

equation, which leaves not only Ψ

of small „ , is the appropriate solution. This is (3.37).

Considering that we must make n = (tf ’ t0 )/„ steps to evolve the system

3.3 Path integral quantum mechanics 51

from t0 to tf = tn , we can rewrite (3.32) for a N -particle system as

’3nN/2

ih„

dr 1 · · ·

G(r f , tf ; r 0 , t0 ) = lim dr n’1

„ ’0 m

n

m(r k ’ r k’1 )2

i

’ V (r k , tk )„

exp . (3.38)

2„

k=1

Here, r is a 3N -dimensional vector. Note that in the limit „ ’ 0, the number

of steps n tends to in¬nity, keeping n„ constant. The potential V may still

be an explicit function of time, for example due to a time-dependent source.

In most cases solutions can only be found by numerical methods. In simple

cases with time-independent potentials (free particle, harmonic oscillator)

the integrals can be evaluated analytically.

In the important case that the quantum system is bound in space and not

subjected to a time-dependent external force, the wave function at time t0

can be expanded in an orthonormal set of eigenfunctions φn of the Hamil-

tonian:

Ψ(r, t0 ) = an φn (r). (3.39)

n

As the eigenfunction φn develops in time proportional to exp(’iEn t/ ), we

know the time dependence of the wave function:

i

an φn (r) exp ’ En (t ’ t0 ) .

Ψ(r, t) = (3.40)

n

¿From this it follows that the kernel must have the following form:

— ’ i En (t ’ t0 ) .

G(r, t; r 0 , t0 ) = n φn (r)φn (r 0 ) exp (3.41)

This is easily seen by applying G to the initial wave function m am φm (r 0 )

and integrating over r 0 . So in this case the path integral kernel can be

expressed in the eigenfunctions of the Hamiltonian.

3.3.5 Evolution in imaginary time

A very interesting and useful connection can be made between path inte-

grals and the canonical partition function of statistical mechanics. This

connection suggests a numerical method for computing the thermodynamic

properties of systems of quantum particles where the symmetry proper-

ties of wave functions and, therefore, e¬ects of exchange can be neglected.

This is usually the case when systems of atoms or molecules are consid-

ered at normal temperatures: the repulsion between atoms is such that the

52 From quantum to classical mechanics: when and how

quantum-mechanical exchange between particles (nuclei) is irrelevant. The

quantum e¬ects due to symmetry properties (distinguishing fermions and

bosons) are completely drowned in the quantum e¬ects due to the curva-

ture of the interatomic potentials within the de Broglie wavelength of the