Attractive Delta Function in 3D

A particle moves in three dimensions. The only potential is an attractive delta function at r = a of the form

(i)

where D is a parameter which determines the strength of the potential.

a) What are the matching conditions at r = a for the wave function and its derivative?

b) For what values of D do bound states exist for s-wave (ℓ = 0).

SOLUTION

a) The amplitude of the wave function is continuous at the point r = a of the delta function. For the derivative we first note that the eigenfunctions are written in terms of a radial function R(r) and angular functions:

(1)

Since the delta function is only for the radial variable r, only the function R(r) has a discontinuous slope. From the radial part of the kinetic energy operator we integrate from r = a^- to r = a⁺:

(2)

(3)

This formula is used to match the slopes at r = a.

b) In order to find bound states, we assume that the particle has an energy given by E = -h²α²/2m, where α needs to be determined by an eigenvalue equation. The eigenfunctions are combinations of exp (±α)/r. In order to be zero at r = 0 and to vanish at infinity, we must choose the form

(4)

We match the values of R(r) at r = a. We match the derivative, using the results of part (a):

(5)

(6)

We eliminate the constants A and B and obtain the eigenvalue equation for α, which we proceed to simplify:

(7)

(8)

(9)

This is the eigenvalue equation which determines α as a function of parameters such as a, D, m etc. In order to find the range of allowed values of D for bound states, we examine αa → 0. The right-hand side of (9) goes to 1, which is its largest value. So, the constraint for the existence of bound states is

(10)