Attractive Delta Function Potential II

A particle of mass m is confined to the right half-space, in one dimension, by an infinite potential at the origin. There is also an attractive delta function potential V(x) = -V₀ aδ(x – a), where a > 0 (see Figure 1.1).

a) Find the expression for the energy of the bound state.

b) What is the minimum value of V₀ required for a bound state?

Figure 1.1

SOLUTION

a) In order to construct the wave function for the bound state, we first review its properties. It must vanish at the point x = 0. At the point x = a, it is continuous:

(1)

Away from the points x = (0, a) it has an energy E = -h²α²/2m and wave functions that are combinations of e^-αx and e^αx. These constraints dictate that the eigenfunction has the form

(2)

At the point x = a, we match the two eigenfunctions and their derivatives, using (1). This yields two equations, which are solved to find an equation for α:

(3)

(4)

We use the first equation to eliminate A in the second equation. Then each term has a factor of Be^-αa which is canceled:

(5)

Multiplying both sides of (5) by sinh αa gives

(6)

(7)

This last equation determines α, which determines the bound state energy. There is only one solution for sufficiently large values of V₀.

b) The minimum value of V₀ for creating a bound state is called V_c. It is found by assuming that the binding energy E → 0, which means α → 0. We examine (7) for small values of α and find that

(8)

(9)