QuantumPhysics.dvi

15 Applications and Examples of Path Integrals

In this section, we shall illustrate the use of path integrals and give examples of their calculation
for simple examples. They include the harmonic oscillator,the Aharonov-Bohm effect, the Dirac
magnetic monopole, imaginary time path integrals and the path integral formulation of statistical
mechanics. We shall end by introducing the Ising model and solving it for the one dimensional
case.

15.1 Path integral calculation for the harmonic oscillator

The classical Lagrangian is by now familiar,

L=

1

2

mq ̇^2 −

1

2

mω^2 q^2 (15.1)

The dominant path is the classical trajectory between (qa,ta) and (qb,tb), which obeys the classical
equation ̈q 0 +ω^2 q 0 = 0, and is given by

q 0 (τ) = qacosω(t−ta) +Bsinω(t−ta)

qb = qacosωT+BsinωT (15.2)

whereT=tb−ta, and the solution forBis

B=

qb−qacosωT

sinωT

(15.3)

The associated classical action may be evaluated by exploiting the extremality of the action,

S=

m

2

∫tb

ta

dt

(

d

dt

(q 0 q ̇ 0 )−q 0 q ̈ 0 −ω^2 q^20

)

=

m

2

(q 0 q ̇ 0 )(t)

∣∣

∣∣

t=tb

t=ta

(15.4)

which yields explicitly,

eiS/ ̄h= exp

{

imω

2 ̄hsinωT

[

(q^2 a+q^2 b) cosωT− 2 qaqb

]}

(15.5)

To compute the functional determinant, we solve for the eigenfunctions of

−m

d^2

dt^2

φn(t)−mω^2 φn(t) =λnφn(t) (15.6)

subject toφn(ta) =φn(tb) = 0. The normalized solutions are

φn(t) =

√

2

T

sin

(

nπ(t−ta)

T

)

n= 1, 2 , 3 ,··· (15.7)

and the eigenvalues are

λn=m

(

π^2 n^2

T^2

−ω^2

)

=

mπ^2 n^2

T^2

(

1 −

ω^2 T^2

π^2 n^2

)

(15.8)