Phase Shift for Inverse Quadratic

A particle of mass m moves in one dimension in the right half-space (x > 0) with the potential

(i)

where the dimensionless constant determines the strength of the potential. Use WKB to calculate the phase shift as a function of energy.

SOLUTION

The turning point is C = λ/k. The phase integral is

The last integral is found in standard tables. To evaluate the phase shift, we need to evaluate this expression in the limit x → ∞, which gives kx – πλ/2. So the final expression for the phase shift is

(2)

The phase shift is independent of energy.