1549380323-Statistical Mechanics Theory and Molecular Simulation

450 The Feynman path integral

With this heuristic introduction to the sum over paths in mind, we now proceed
to derive the Feynman path integral more rigorously and, in the process, learn how to
determine the path amplitudes.

12.2 Derivation of path integrals for the canonical densitymatrix

and the time evolution operator

In this and subsequent sections, the path integral concept will begiven a more precise
mathematical formulation and its computational advantages elucidated. For simplicity,
the discussion will initially focus on a single particle moving in one spatial dimension
with a Hamiltonian

Hˆ= ˆp

2

2 m

+U(ˆx)≡Kˆ+U.ˆ (12.2.1)

As noted in Section 12.1, the path integral describes a process in which a particle
moves unobserved between an initiation pointxand a detection pointx′. That is,
the particle is initially prepared in an eigenstate|x〉of the position operator, which is
subsequently allowed to evolve under the action of the propagatorexp(−iHˆt/ ̄h). After
a timet, we ask what the amplitude will be for detection of a particle at a pointx′.
This amplitudeAis given by

A=〈x′|e−i

Hˆt/ ̄h

|x〉≡U(x,x′;t). (12.2.2)

Therefore, what we seek are the coordinate-space matrix elements of the quantum
propagator. More generally, if a system has an initial state vector|Ψ(0)〉, then from
eqn. (9.2.33), at timet, the state vector is|Ψ(t)〉= exp(−iHˆt/ ̄h)|Ψ(0)〉. Projecting
this into the coordinate basis gives

〈x′|Ψ(t)〉= Ψ(x′,t) =〈x′|e−i

Hˆt/ ̄h

|Ψ(0)〉

=

∫

dx〈x′|e−i

Hˆt/h ̄

|x〉〈x|Ψ(0)〉

=

∫

dx〈x′|e−i

Hˆt/h ̄

|x〉Ψ(x,0), (12.2.3)

which also requires the coordinate-space matrix elements of the propagator. The Feyn-
man path integral provides a technique whereby these matrix elements can be com-
puted via a sum over all possible paths leading fromxtox′in timet.
Before presenting the detailed derivation of the path integral, it is worth noting an
important connection between the propagator and the canonicaldensity matrix. If we
denote the latter by ˆρ(β) = exp(−βHˆ), then it is clear that

ρˆ(β) =Uˆ(−iβ ̄h), Uˆ(t) = ˆρ(it/ ̄h). (12.2.4)

Eqn. (12.2.4) implies that the canonical density matrix can be obtained by evaluating
the propagator at an imaginary timet=−iβ ̄h. For this reason, the density matrix is
often referred to as animaginary time propagator. Similarly, the real-time propagator