1549380323-Statistical Mechanics Theory and Molecular Simulation

452 The Feynman path integral

SubstitutingΩ into eqn. (12.2.7) givesˆ

ρ(x,x′;β) = lim

P→∞

〈x′|ΩˆP|x〉= lim

P→∞

〈x′|ΩˆΩˆΩˆ···Ωˆ|x〉. (12.2.9)

In order to simplify the evaluation of eqn. (12.2.9), we introduce an identity operator
in the form of

Iˆ=

∫

dx|x〉〈x| (12.2.10)

(see also eqn. (9.2.38)) between each factor ofΩ. Since there areˆ Pfactors ofΩ,ˆ P− 1
insertions of the identity operator are needed. This will introduceP−1 integrations
over coordinate labels giving the following expression for the densitymatrix:

ρ(x,x′;β) = lim

P→∞

∫

dx 2 ···dxP

×〈x′|Ωˆ|xP〉〈xP|Ωˆ|xP− 1 〉〈xP− 1 |···|x 2 〉〈x 2 |Ωˆ|x〉. (12.2.11)

Inserting the identity operatorP−1 times is analogous to insertingP−1 gratings
with many holes in Fig. 12.4. The integration over eachxiis analogous to summing
over all possible ways a particle can pass through the infinitely many holes in each
grating.
The advantage of employing the Trotter theorem is that the matrixelements in
eqn. (12.2.11) can be evaluated in closed form. Consider the general matrix element

〈xk+1|Ωˆ|xk〉=〈xk+1|e−β

U/ˆ 2 P

e−β

K/Pˆ

e−β

U/ˆ 2 P

|xk〉. (12.2.12)

Note thatUˆ=U(ˆx) is a function of the coordinate operator. Thus,|xk〉and|xk+1〉,
being coordinate eigenvectors, are eigenvectors of exp(−βU(ˆx)/ 2 P) with eigenvalues
exp(−βU(xk)/ 2 P) and exp(−βU(xk+1)/ 2 P), respectively. Hence, eqn. (12.2.12) sim-
plifies to

〈xk+1|Ωˆ|xk〉= e−βU(xk+1)/^2 P〈xk+1|e−β

K/Pˆ

|xk〉e−βU(xk)/^2 P. (12.2.13)

SinceKˆis a function of the momentum operator, the matrix element of exp(−βK/Pˆ )
is less trivial to evaluate. However, if we insert another identity operator, this time
expressed in terms of momentum eigenvectors as

Iˆ=

∫

dp|p〉〈p|, (12.2.14)

into eqn. (12.2.13), we obtain

〈xk+1|e−β

K/Pˆ

|xk〉=

∫

dp〈xk+1|e−β

K/Pˆ

|p〉〈p|xk〉. (12.2.15)

Now the operator exp(−βK/Pˆ ) acts on one of its eigenvectors|p〉to yield