1549380323-Statistical Mechanics Theory and Molecular Simulation

456 The Feynman path integral

x

Imaginary time

0 ℏ/2 ℏ

x

β β

Fig. 12.7 Representative paths in the discrete path sum for the canonical partition function.

be expressed compactly as the limit of an expression that resemblesa classical config-
urational partition function

Q(L,T) = lim

P→∞

(

mP

2 πβ ̄h^2

)P/ 2 ∫

D(L)

dx 1 ···dxPe−βφ(x^1 ,...,xP), (12.2.27)

an analogy we will revisit when we discuss numerical methods for evaluating path
integrals in Section 12.6. Here,

φ(x 1 ,...,xP) =

∑P

k=1

[

1

2

mωP^2 (xk−xk+1)^2 +

1

P

U(xk)

]

, (12.2.28)

whereωP=

√

P/β ̄handxP+1=x 1.
An analytical calculation of the density matrix, partition function, or propagator
via path integration proceeds first by carrying out theP-dimensional integration and
then taking the limit of the result asP → ∞. As a simple example, consider the
density matrix for a free particle (U(x) = 0). Assumex∈(−∞,∞). The density
matrix in this case is given by

ρ(x,x′;β) = lim

P→∞

(

mP

2 πβ ̄h^2

)P/ 2

×

∫

dx 2 ···dxPexp

{

−

∑P

k=1

[

mP

2 β ̄h^2

(xk+1−xk)^2

]}∣∣

∣

∣

∣

xP+1=x′

x 1 =x

. (12.2.29)

In fact, we previously solved this problem in Section 4.5. Eqn. (4.5.33)is the partition
function for a classical polymer with harmonic nearest-neighbor particle couplings and
fixed endpoints. Applying the result of eqn. (4.5.33) to eqn. (12.2.29), recognizing the