A Guide to Physics Problems. Part 2. Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 315

5.64 3D Delta Function (Princeton)

For a particle of wave vector Schrödinger’s equation for the radial part

of the wave function is

Only scattering is important at very low energies, so solve for

Also define and get

At is well behaved, so Thus we choose our wave

functions to be

The quantity is the phase shift. We match the wave functions at

The formula for matching the slopes is derived from (S.5.64.2):

Matching the function and slope produces the equations

which are solved to eliminate A and B and get

In the limit of low energy, we want We assume there are no bound

states so that where is a constant. We find in thislimit:

We also give the formula for the cross section in terms of the scattering
length The assumption of no bound state is that