1549380323-Statistical Mechanics Theory and Molecular Simulation

454 The Feynman path integral

ρ(x,x′;β) = lim

P→∞

(

mP

2 πβ ̄h^2

)P/ 2 ∫

dx 2 ···dxP

×exp

{

−

1

̄h

∑P

k=1

[

mP

2 β ̄h

(xk+1−xk)^2 +

β ̄h

2 P

(U(xk+1) +U(xk))

]}∣∣

∣

∣

∣

xP+1=x′

x 1 =x

. (12.2.22)

In eqn. (12.2.22), the quantum kinetic energy is present in the formof a harmonic
nearest-neighbor coupling term that acts between points along the path. The spring
constant for this interaction ismP/β^2 ̄h^2.
Eqn. (12.2.22) is the limitP→∞of adiscretized path integralrepresentation for
the density matrix. As eqn. (12.2.22) indicates, the endpoints of the paths at pointsx 1
andxP+1are fixed at the “initiation” and “detection” points,xandx′, respectively.
The intermediate integrations overx 2 ,...,xPconstitute the sum over all possible paths
fromxtox′in imaginary time−iβ ̄h. For finiteP, because the potentialUonly acts
at the discrete pointsxk, the paths are lines between successive imaginary time points,
as suggested by Fig. 12.4. Note that if the particle is confined to an intervalx∈[0,L],
then all of the coordinate integrations must be restricted to this interval as well. The
weight or amplitude assigned to each path is the value of the integrand in eqn. (12.2.22)
evaluated along the discrete path.
A path integral representation for the real-time propagator cannow be derived
from eqn. (12.2.22) by applying eqn. (12.2.4) and settingβ=it/ ̄h. This yields a path
integral expression for the coordinate-space matrix elements ofthe propagator,Uˆ(t):

U(x,x′;t) = lim

P→∞

(

mP

2 πit ̄h

)P/ 2 ∫

dx 2 ···dxP

×exp

{

i

̄h

∑P

k=1

[

mP

2 t

(xk+1−xk)^2 −

t

2 P

(U(xk+1) +U(xk))

]}∣∣

∣

∣

∣

xP+1=x′

x 1 =x

. (12.2.23)

Notice the change in relative sign between the kinetic and potential energy terms
between eqns. (12.2.22) and (12.2.23) in the path integral expressions for the den-
sity matrix and the propagator. The path sums in eqns. (12.2.22) and (12.2.23) are
represented pictorially in Fig. 12.6.
From eqn. (12.2.22), a path integral expression for the canonicalpartition function
Q(L,T) for a system confined tox∈[0,L] can be derived. Recall thatQ(L,T) =
Tr[exp(−βHˆ)]. Evaluating the trace in the coordinate basis gives

Q(L,T) =

∫L

0

dx〈x|e−β

Hˆ

|x〉=

∫L

0

dx ρ(x,x;β). (12.2.24)

In order to evaluate eqn. (12.2.24), the diagonal elements of the density matrix in the
coordinate basis are needed; these can be obtained by settingx 1 =xP+1=xin eqn.
(12.2.22). Finally, an integration over the diagonal elements must beperformed. Since