1549380323-Statistical Mechanics Theory and Molecular Simulation

Functional integrals 469

The first term in the penultimate line of eqn. (12.4.21) is the classical Euclidean
action integral. The solution of eqn. (12.4.20) that satisfies the endpoint conditions is

xcl(τ) =

x

(

e−ω(τ−β ̄h)−eω(τ−β ̄h)

)

+x′(eωτ−e−ωτ)

eβ ̄hω−e−β ̄hω

, (12.4.23)

which we derive by assuming a solution of the formxcl(τ) =Aexp(ωτ) +Bexp(−ωτ)
and using the endpoint conditions to solve for the constantsAandB. When this
solution is substituted into the classical action integral, we obtain

∫β ̄h

0

dτ

[

1

2

mx ̇^2 cl(τ) +

1

2

mω^2 x^2 cl(τ)

]

=

mω

2sinh(β ̄hω)

[

(x^2 +x

′ 2

)cosh(β ̄hω)− 2 xx′

]

. (12.4.24)

Inserting eqn. (12.4.24) into eqn. (12.4.18), we obtain the density matrix for the har-
monic oscillator as

ρ(x,x′;β) =I 0 exp

[

−

mω

2 ̄hsinh(β ̄hω)

(

(x^2 +x

′ 2

)cosh(β ̄hω)− 2 xx′

)]

, (12.4.25)

whereI 0 is the path integral

I 0 =

∫y(βh ̄)=0

y(0)=0

Dy(τ) exp

[

−

1

̄h

∫β ̄h

0

dτ

(

m

2

y ̇^2 +

mω^2

2

y^2

)]

. (12.4.26)

Note that the remaining functional integralI 0 does not depend on the pointsxand
x′and therefore can only contribute an overall (temperature-dependent) constant to
the density matrix. This affects the thermodynamics but not any averages of physical
observables.^1 Nevertheless, it is instructive to see how such a functional integral is
performed.
We first note thatI 0 is a functional integral over functionsy(τ) satisfyingy(0) =
y(β ̄h) = 0. Because of these endpoint conditions, the pathsy(τ) can be expanded in
a Fourier sine series:

y(τ) =

∑∞

n=1

cnsin(ωnτ), (12.4.27)

where
ωn=

nπ

β ̄h

. (12.4.28)

Since a giveny(τ) is uniquely determined by its expansion coefficientscn, integrating
over the functionsy(τ) is equivalent to integrating over all possible values of the
expansion coefficients. Thus, we seek to change from an integral over the functions

(^1) This is only the case for the harmonic oscillator. For anharmonic potentials, a stationary-phase
approximation to the path integral, which also employs an expansion about classical paths, allows
the dependence ofI 0 onxandx′to be approximated.