Our goal is to investigate the differential equation which defines , and in particular to solve for explicitly, so that the expansion for can be explicitly carried out. From Huygens' principle,
A form valid for all time is
Using the chain rule and the fact that satisfies the Schrödinger equation, we have
Substituting equation 6.29 for , we have
is the Green's function equation in the Schrödinger theory. Along with the boundary conditions for this defines the retarded Green's function.
For the free-particle propagator . can depend only upon the difference of the coordinates and . This is because the wave at emerging from a unit source at which is turned on at depends only on the interval , and is precisely the amplitude of this wave.
Consider the Fourier transform
The Schrödinger equation for the Green's function becomes
and hence for ,
To complete the expression, we must have a rule for handling the singularity. We will see that the retarded boundary condition
requires us to add a positive infinitesimal imaginary part to the denominator. Lets integrate
The integral over is evaluated as a contour integral in the complex -plane as shown in figure 6.3. For , the contour may be closed along an infinite semicircle below the real axis in order to ensure exponential damping of the integrand, and the value of the integral is by Cauchy's theorem. For , the contour is closed above and the integral vanishes because the pole at lies outside the contour. The integral is thus just times the step function. Continuing our previous integration, we have
where the subscript stands for plane waves. This is a special case for plane waves. In general
where is a generalized sum and integral over the spectrum of quantum numbers , is a complete set of normalized solutions to the Schrödinger equation which satisfy a completeness statement
There is an enormous amount of information contained in . All the solutions of the Schrödinger equation, including the bound states, as required in the completeness relation, appear with equal weight. It is no wonder that is difficult to compute.
The same Green's function which propagates a solution of the Schrödinger equation forward in time propagates its complex conjugate backward in time. From (6.41) we have