Bound states for a Spherically Symmetric Schrodinger equation

1. The problem statement, all variables and given/known data
A particle of mass m moves in three dimesions in a potential energy field
V(r) = -V0 r< R
0 if r> R

where r is the distance from the origin. Its eigenfunctions psi(r) are governed by

[tex] \frac{\hbar^2}{2m} \nabla^2 \psi + V(r) \psi = E \psi [/tex]
ALL in spherical coords.

Consider a spherically symmertic eigenfunction with no angular dependance of the form

[tex] \psi(r) = \frac{u(r)}{r} [/tex]

Solve for u(r) in teh regions r< R and r > R and yb imposiing boundary conditions, find the eigenfunction of a bound state with energy [itex] E = \hbar^2 \alpha^2 / 2m [/itex]

Show taht there is one bound state of this kind if the depth of the weel obeys
[tex] \frac{\hbar^2 \pi^2}{8mR^2} < V_{0} < \frac{9\hbar^2 \pi^2}{8 mR^2} [/tex]

2. Relevant equations
Ok i found te solution of the wavefunction to be
[tex] C \sin (k_{0}r) /r [/tex] if r < R
[tex] A e^{\alpha r}/ r [/tex] if r > R
The solutions are such because the solutions are found a bound state that is E <= V0. Also the solutions are spherically symmetric.

where [tex] k_{0} = \sqrt{\frac{2m}{\hbar^2} (V_{0} + E)} [/tex]

3. The attempt at a solution
Furthermore i found that
[tex] k_{0} \cot k_{0} R = -\alpha [/tex]
[tex] k_{0}^2 + \alpha^2 = \frac{2m}{\hbar^2} V_{0} [/tex]

How would i prove the condition for V0?? Would i do this graphically assuming different values for R???

Thanks for your help!!!
 

OlderDan

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