12 The Feynman path integral
12.1 Quantum mechanics as a sum over paths
The strangeness of the quantum world is evident in systems as simpleas ideal gases
of the previous chapter, where nothing more than the spin statistics leads to remark-
ably complex behavior. We still have not yet included interactions in our treatment of
quantum systems, and as noted in Chapter 10, the eigenvalue problem for the Hamil-
tonian when interactions are included can only be solved for very small systems. For
large systems, we need a statistical approach, and this brings us to the formulation
of quantum mechanics proposed by Feynman (Feynman, 1948; Feynman and Hibbs,
1965). Not surprisingly, quantum strangeness is no less apparentin Feynman’s for-
mulation of quantum mechanics than it is in the pictures we have studied thus far.
Although mathematically equivalent to the Heisenberg and Schr ̈odinger pictures of
quantum mechanics, Feynman’s view represents a qualitative departure from these
formulations.
In order to introduce Feynman’s picture, consider a particle prepared in a state
initially localized at a pointxthat evolves unobserved to a pointx′. Invoking the
quantum wave-like nature of the particle, the wave packet representing the initial
state evolves under the action of the propagatorU(t) = exp(−iHˆt/ ̄h), and the wave
packet spreads in time, causing the state to become increasingly delocalized spatially
until it is finally observed at the pointx′through a measurement of position, where it
again localizes due to the collapse of the wave function. In contrast, Feynman’s view
vaguely resembles a classical particle picture, in which the particle evolves unobserved
fromxtox′. There is a key difference, however, from the classical view. Classically, if
we do not observe the particle, we will not know what path it will take,but we do know
that it will follow a definite path. In quantum mechanics, by contrast, it is not our
ignorance that prevents us from specifying a particle’s path (whichwe could, but it is
the very quantum nature of the particle itself that makes specifying a path impossible.
Instead of following a unique path betweenxandx′, the particle follows a myriad of
paths, specifically all possible paths, simultaneously. These paths representinterfering
alternatives, meaning that the total amplitude for the particle to be observed atx′
at timetis the sum of the amplitudes associated with all possible paths betweenx
andx′. Thus, according to the Feynman picture, we must calculate the amplitude for
each of the infinitely many paths the particle follows and then sum them to obtain the
complete amplitude for the process. Recall that the probabilityP(x′,t) for the particle
to be observed atx′is the square modulus of the amplitude; since the latter is a sum
of amplitudes for individual paths, the cross terms constitute theinterference between