14 Path Integral Formulation of Quantum Mechanics
A quantum system associated with classical mechanics admits a formulation in terms of functional
integrations over all possible paths. This path-integral formulation was pioneered by Dirac in 1933,
and popularized by Feynman in the 1940’s. It is quite useful in modern quantum mechanics, but
it has become an indispensable tool especially in quantum field theory.
The key intuitive idea behind the path integral construction of the probability amplitudes of
quantum mechanics is as follows. Classically, the system isdeterministic, and we know the precise
path followed by a particle, given initialqand finalq′conditions. Quantum mechanically, each
geometric path is a possible “trajectory” for the quantum particle, and its contribution to the full
quantum probability amplitude will be given by some weight factor which remains to be determined.
The full quantum probability amplitude will then be given bya sum over all possible paths, and a
weight factor for each path.
probability amplitude =∑pathsweight factor for path (14.1)A sketch of the (unique) path followed by a classical particle and possible quantum paths is depicted
in the Figure below.
q
q’
Figure 15: The black line represents the classical trajectory; thered lines represent various
path which contribute to the path integral for the quantum probability amplitude.
In the present chapter, we shall give a detailed derivation of the sum over paths and the
associated weight factor. The standard reference is the book by Feynman and Hibbs, but I will
follow rather the more general derivation in phase space given originally by Dirac.
14.1 The time-evolution operator
Time evolution in the Schr ̈odinger and Heisenberg picturesare given by a differential equation
respectively for states and observables,
i ̄h
d
dt|φ(t)〉=H(t)|φ(t)〉 Schroedinger picturei ̄hd
dt
A(t) = [A(t),H(t)] Heisenberg picture (14.2)