Energy levels of exponential potential

[castlemaster]

Homework Statement

Find the eigenfunctions (with angular momentum 0) and the estimation of the 3 first energy levels (given g and a) of a particle in a exponential potential such as

V = -ge^-r/a

Homework Equations

Time independent Schrodinger equation (SE)

The Attempt at a Solution

Did a first change of variables for the radial part of the SE R= u/r
Did a second change $\sigma$ = Ke^-r/2a to reach the Bessel equation

Then the solutions are Bessel functions and cannot diverge at r = 0. Therefore I end up with

$\Phi(r) = A J_{\nu}(Ke^{-r/2a})$

First question is: how I calculate the normalisation constant A? I guess I have to integrate from 0 to infinity and do a change of variable ... but then I get an ugly integral with the Bessel function divided by r

Second question: how do I estimate the first energies giving values to g and a? Should I seek the zeros of the bessel function?

Thanks in advance

 

[castlemaster]

Hi,

I think I see question 2 now.
The Bessel functions are only finite at the origin when the order $\nu$ is a positive integer. Then I only have to be sure K is big enough for the Bessel functions to have 3 zeros, that's it bigger than 5.1356 which is the first zero of J₂ . This gives me a relation between a and g.
Then the energies are compute for $\nu$ = 0,1 and 2

For the first question I think there is a series expansion of the bessel functions from where I can take the constant A.

Regards