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Homework Help: Tough exponential integral (QM, Variational Principle)

  1. Feb 19, 2012 #1
    1. The problem statement, all variables and given/known data
    http://img4.imageshack.us/img4/224/32665300.png [Broken]

    3. The attempt at a solution
    http://img684.imageshack.us/img684/2920/scan0003xo.jpg [Broken]

    I've uploaded my work so far since its much faster than typing and I'm stuck on the last line trying to solve the integral.
    The first part of the integral is calculable but the second term(containing [itex]V_{0}[/itex]) doesn't evaluate with Maple or Wolfram.
    On the question sheet it gives the hint that [itex]\int dx x^{n} e^{-\alpha x}[/itex] = [itex]\frac{n!}{\alpha^{n+1}}[/itex] which leads me to rearrange the 2nd term to -2[itex]V_{0}[/itex][itex]\alpha[/itex]a*[itex]e^{r\frac{(-1-2\alpha a)}{a}}[/itex]*[itex]r^{-1}[/itex]
    According to the tip, [itex]\int e^{r\frac{(-1-2\alpha a)}{a}}[/itex]*[itex]r^{-1}[/itex] is equal to [itex]\frac{(-1)!}{\frac{(-1-2\alpha a)}{a}^{-1+1}}[/itex] which is just -1.
    This doesn't seem to match up with the given Hamiltonian on the problem set so I'm asking for some extra help on this,
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Feb 19, 2012 #2


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    You need to start over from the beginning. You need to integrate over all three dimensions, not just over r. Also, don't forget to use the correct volume element for spherical coordinates.
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