Thermodynamics, Statistical Physics, and Quantum Mechanics

312 SOLUTIONS

The last integral is found in standard tables. To evaluate the phase shift, we

need to evaluate this expression in the limit which gives

So the final expression for the phase shift is

The phase shift is independent of energy.

Scattering Theory

5.61 Step-Down Potential (Michigan State, MIT)

Denote by the momentum of the particle to the right of the origin, and

is momentum on the left. Since energy is conserved, we have

Now we set up the most general form for the wave function, assuming the

incoming wave has unit amplitude:

Matching the wave function and its derivative at the origingives two equa-

tions for the unknownsRandTwhich aresolved to findR:

5.62 Step-Up Potential (Wisconsin-Madison)

Write the energy as where is the wave vector on the
left of zero. Since define a wave vector on the right as