Lecture 35.

We introduced the ideas of the propagator, the Feynman path integral
and Feynman diagrams, in the context of quantum mechanics.

First we introduced the propagator K(x, t_0 + t ; x_0, t_0) as the
answer to the following physical question in quantum mechanics: if we
have a particle at position x_0 at time t_0, what is the probability
amplitude to find it at position x at some later time t_0 + t? We
showed how the propagator can be calculated by evolving the initial
state | x_0 > forward a time t, then taking its projection onto the
position eigenstate | x > .

We noted that the propagator contains all the information about the
time evolution of a quantum system. We showed that we can obtain any
time-evolved wave function \psi (x, t_0 + t) from an initial \psi (x, t_0), 
by integrating the initial wavefunction over x', times the propagator
from x' to x, all over the spatial variable x'.  (We showed this
simply, in Dirac notation, by inserting the complete set of position
eigenstates x'.)  That is, the propagator is the Green's function for
Schrodinger's equation: it gives us an object to integrate an initial
wavefunction against to convert it into a final wavefunction. Physically,
the propagator implements Huygens' principle: the amplitude to find
the particle at x at time t_0 + t is the sum over amplitudes for the
particle to be at all points x' at t_0 which propagate to x in time t.

We considered splitting the particle's propagation from x_0 to x
into two steps, by inserting a complete set of position eigenstates 
| x_1 > at time t_0 + t/2. We found that the propagator, or amplitude
to go from x_0 to x, was the sum of the amplitudes for all
two-step paths from x_0 to x, including paths that passed through
*all* intermediate points x_1. We broke the propagation into
more and more intermediate steps, ending with a limit in which the
number of steps is infinite, and the paths are continuous curves. Even
in that case, the propagator from x_0 to x is the sum of the
amplitudes for all trajectories (covering all intermediate points)
that go from x_0 to x in time t.

We noted that Feynman showed that the propagator in this limit could
be written as a functional integral over all trajectories x(t), with
each trajectory weighted by the phase factor e^{iS/\hbar}, where S
is the classical action of the path. We noted why, in the classical
limit where \hbar goes to zero, this results in the classical
trajectory, for which the action is extremal --- thus making many
contributions to the path integral which are all in phase, and
which reinforce each other instead of cancelling due to a rapidly changing
phase.

Finally, we discussed the perturbative calculation of the path
integral.  Since we can usually solve for the propagator in the case
of a free theory (no interactions), we write a perturbative expansion
of the full propagator by Taylor expanding the contribution 
e^{-i\int d^3 x  V/\hbar} to the path integral due to interactions. This
expansion told us that the full propagator from x_0 to x is the
sum of the following terms: at zeroth order, the free propagator from
x_0 to x; at first order, an integral over a single intermediate
point x_1, of the free propagator from x_0 to x_1, times the
potential V(x_1), times the free propagator from x_1 to x; at
second order, an integral over two intermediate points, of products of
propagators between all points, and potentials evaluated at all
intermediate points; and similarly for higher order terms j with j
intermediate points. We mentioned how we could associate with each
term a ``Feynman diagram,'' which associates line segments with
propagators, and points with potential insertions.

We will continue with this pertubative expansion, next time.

--- 
KB