1549380323-Statistical Mechanics Theory and Molecular Simulation

558 Quantum time-dependent statistical mechanics

Let us begin with a standard nonsymmetrized time correlation function defined by

CAB(t) =

〈

Aˆ(0)Bˆ(t)

〉

=

1

Q(N,V,T)

Tr

[

e−β

Hˆˆ

Aei

Hˆt/ ̄hˆ

Be−i

Hˆt/h ̄]

, (14.6.1)

whereAˆandBˆare quantum mechanical operators in the interaction picture with
unperturbed HamiltonianHˆ.^4 If we evaluate the trace in the basis of the eigenvectors
ofHˆ, then a simple formula for the quantum time correlation function results:

CAB(t) =

1

Q(N,V,T)

∑

n

〈En|e−β

Hˆˆ

Aei

Hˆt/ ̄hˆ

Be−i

Hˆt/h ̄

|En〉

=

1

Q(N,V,T)

∑

n,m

〈En|e−β

Hˆˆ

A|Em〉〈Em|ei

Hˆt/ ̄hˆ

Be−i

Hˆt/h ̄

|En〉

=

1

Q(N,V,T)

∑

n,m

e−βEnei(Em−En)t/ ̄h〈En|Aˆ|Em〉〈Em|Bˆ|En〉. (14.6.2)

Thus, if we are able to calculate all of the eigenvalues and eigenvectors ofHˆ, as
well as the full set of matrix elements ofAˆandBˆ, then the calculation of the time
correlation function just requires carrying out the two sums in eqn. (14.6.2). Generally,
however, this can only be done for systems having just a few degrees of freedom. In the
condensed phase, for example, it is simply not possible to solve the eigenvalue problem
forHˆdirectly.
In the Feynman path-integral formalism of Chapter 12, the eigenvalue problem is
circumvented by computing thermal traces in the basis of coordinate eigenstates. We
will now apply this approach to the quantum time correlation function. For simplicity,
we will consider a single particle in one dimension, and we will letAˆandBˆbe functions
of the position operator ˆx,Aˆ=Aˆ(ˆx),Bˆ=Bˆ(ˆx). Taking the coordinate-space trace,
we obtain

CAB(t) =

1

Q(N,V,T)

∫

dx〈x|e−β

Hˆˆ

A(ˆx)ei

Hˆt/ ̄hˆ

B(ˆx)e−i

Hˆt/ ̄h

|x〉 (14.6.3)

=

1

Q(N,V,T)

∫

dxdx′dx′′〈x|e−β

Hˆ

|x′〉a(x′)〈x′|ei

Hˆt/ ̄h

|x′′〉b(x′′)〈x′′|e−i

Hˆt/ ̄h

|x〉.

If each of the matrix elements〈x|e−βHˆ|x′〉,〈x′|eiHˆt/ ̄h|x′′〉, and〈x′′|e−iHˆt/ ̄h|x′〉were
expressed as path integrals, we would interpret eqn. (14.6.3) as follows: Starting atx,
propagate along a real-time path to the pointx′′using the propagator exp(−iHˆt/ ̄h)
and evaluate the eigenvalueb(x′′) ofBˆat that point; fromx′′, propagate backward
in real time using the propagator exp(iHˆt/ ̄h) to the pointx′and evaluate the eigen-
valuea(x′) ofAˆ; finally, propagate in imaginary time using the propagator fromx′
to the original starting pointx. This is represented schematically in Fig. 14.6(a).
Unfortunately, standard Monte Carlo or molecular dynamics schemes cannot be used

(^4) For the remainder of this chapter, we will drop the “0” subscript on the Hamiltonian, since it is
assumed thatHˆrepresents the unperturbed Hamiltonian.