What Determines the Range of Bound States in a Spherical Finite Well?

[Void123]

Homework Statement

I was reviewing some homework problems and looking at the solutions. There is one problem with a tiny step I just cannot rationalize and I am hoping someone can point me in the right direction.

I have a spherical finite well:

V = {- V_{0}: 0 < r < a},

= {0: r \geq a}

- k_{2} = k_{1} cot (k_{1} a) (1)

Refining the notation,

\alpha = a \sqrt{(2m(E + V_{0})}/hbar = k_{1} a

R = a \sqrt{(2m(V_{0})}/hbar and k_{2} = \sqrt{(2m(V_{0})}/hbar

So (1) may be rewritten as \sqrt{R^{2} - \alpha^{2}} = - \alpha cot (\alpha)

Homework Equations

From part 1.

The Attempt at a Solution

I don't understand how at R = \pi/2 there are no bound states.

Also, I am given this restriction: -V_{0} < E < 0

How is this justified and how is the precise range of bound states determined?

 

[vela]

When R=\pi/2, the only solutions are at \alpha=\pm\pi/2. In these cases, you get k₂=0.