Thermodynamics, Statistical Physics, and Quantum Mechanics

314 SOLUTIONS

where

and the theta function is 1 if and 0 if For the

solutions are in the form of or Instead, write it as

where the phase shift is For define a constant according to

Then the eigenfunction is

For the constraint that forces the choice of the hyberbolic

sine function. Matching the eigenfunction and slope at gives

Dividing these equations eliminates the constants A and B. The remaining
equation defines the phase shift.

b) In the limit that the argument of the arctangent vanishes, since
the hyperbolic tangent goes to unity, and

c) In the limit of zero energy, we can define

To find the part of the cross section at low energy, we start with

where the total cross section is