Repulsive Square Well

Consider in three dimensions a repulsive (V₀ > 0) square well at the origin of width a. The potential is

(1)

Figure 1.1

A particle of energy E = h²k²/2m < V₀ is incident upon the square well (see Figure 1.1).

a) Derive the phase shift for s-waves.

b) How does the phase shift behave as V₀ → ∞?

c) Derive the total cross section in the limit of zero energy.

SOLUTION

a) If the radial part of the wave function is R(r) then define χ(r) = rR(r). Since R is well behaved at r → 0, χ = 0 in this limit. The function χ(r) obeys the following equation for s-waves:

(1)

where

(2)

and the theta function Θ(a – r) is 1 if a > r and 0 if a < r. For r > a the solutions are in the form of sin kr or cos kr. Instead, write it as sin(kr + δ) where the phase shift is δ(k). For r < a define a constant α according to α² = k²₀ – k² > 0. Then the eigenfunction is

(3)

For r < a the constraint that χ(0) = 0 forces the choice of the hyberbolic sine function. Matching the eigenfunction and slope at r = a gives

(4)

(5)

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

(6)

(7)

b) In the limit that V₀ → ∞, the argument of the arctangent vanishes, since the hyperbolic tangent goes to unity, and δ = -ka.

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

(8)

(9)

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

(10)

where the total cross section is σ.