How Many Bound States Does a Radially Symmetric Delta Potential with l=0 Admit?

[mathman44]

Dirac "bubble potential"

Homework Statement

Consider a radially symmetric delta potential V(r) = −Vo * δ(r − a) with l=0. How many bound states does this system admit?

The Attempt at a Solution

With l=0, the radial equation reduces to the one dimensional TISE. So, solving the 1D TISE with a delta potential V(r) = −Vo * δ(r − a):

I have $$R_{in} = A\exp{kr}$$ for r < a
$$R_{out} = A\exp{k(2a-r)}$$ for r > a

which I obtained my matching the condition R_in = R_out at r=a. Also, the "discontinuity equation" gives me that

$$k = \frac{mV_o}{\hbar^2}$$

meaning that there is only one energy and only one bound state. I don't believe this to be correct... especially since the question hints that the number of bound states should depend on "a".

Any help please?

 

[mathman44]

Bump.