# Bound states for a Spherically Symmetric Schrodinger equation

1. Dec 11, 2006

### stunner5000pt

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

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

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

$$\psi(r) = \frac{u(r)}{r}$$

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 $E = \hbar^2 \alpha^2 / 2m$

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

2. Relevant equations
Ok i found te solution of the wavefunction to be
$$C \sin (k_{0}r) /r$$ if r < R
$$A e^{\alpha r}/ r$$ 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 $$k_{0} = \sqrt{\frac{2m}{\hbar^2} (V_{0} + E)}$$

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

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