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Homework Help: Energy levels of exponential potential

  1. Jun 12, 2012 #1
    1. The problem statement, all variables and given/known data

    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

    2. Relevant equations

    Time independent Schrodinger equation (SE)

    3. 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 [itex]\sigma[/itex] = 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

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

    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
    Last edited: Jun 12, 2012
  2. jcsd
  3. Jun 12, 2012 #2

    I think I see question 2 now.
    The Bessel functions are only finite at the origin when the order [itex]\nu[/itex] 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 [itex]\nu[/itex] = 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.

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