Bound states for a Spherically Symmetric Schrodinger equation

[stunner5000pt]

Homework Statement

A particle of mass m moves in three dimensions 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 dependence of the form

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

Solve for u(r) in the 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
$$\frac{\hbar^2 \pi^2}{8mR^2} < V_{0} < \frac{9\hbar^2 \pi^2}{8 mR^2}$$

Homework 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 $$k_{0} = \sqrt{\frac{2m}{\hbar^2} (V_{0} + E)}$$

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?

Thanks for your help!

 

[OlderDan]

You will need a graphical solution. The problem has been discussed here.

https://www.physicsforums.com/showthread.php?t=140083&highlight=well+transcendental

And just for emphasis, the link in that disucssion is to here

http://musr.physics.ubc.ca/~jess/p200/sq_well/sq_well.html

 

[stunner5000pt]

You will need a graphical solution. The problem has been discussed here.

https://www.physicsforums.com/showthread.php?t=140083&highlight=well+transcendental

And just for emphasis, the link in that disucssion is to here

http://musr.physics.ubc.ca/~jess/p200/sq_well/sq_well.html

Thanks a lot