Computational Physics

12.4 Path-integral Monte Carlo 401



R

Figure 12.4. The path integral for a one-dimensional system. The vertical axes

areR-axes at different times. A path is a set of points given on these axes. The

heavy drawn path is the stationary path of the action, which is the solution to the

classical equations of motion. The thin lines represent neighbouring paths. For

these paths, the action is not stationary, but they are taken into account in the

quantum mechanical path integral.

seen that the prefactor in the exponent occurring before the sum was
τ/
instead
of
τ. The classical limit corresponds to
=0, which implies that the path with
minimal action dominates all the other paths. This is in fact Hamilton’s principle:
the classical path corresponds to the minimal action. If we ‘switch Planck’s con-
stant on’, we see a contribution from the nonminimal paths emerging. If we had
not identifiedR 0 withRMand if we had not integrated over this coordinate, we
would have a system with fixed end points, which brings the analogy with classical
mechanics even closer.Figure 12.4gives a pictorial representation of the idea of
the path integral.
In this section and in the previous one, we have assumed that the errors in the
individual short-time approximations do not add up to significant errors for large
times. The justification of this assumption is a theorem, which is usually denoted
as the Lie–Trotter–Suzuki formula, which says that for a HamiltonianHwhich can
be written as the sum ofKoperators:

H=

∑K

k= 1

Hk (12.72)

it holds that
e−αH→(e−αH^1 /Me−αH^2 /M...e−αHK/M)M (12.73)