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Variation method (Quantum Mech.)

  1. Apr 11, 2008 #1

    T-7

    User Avatar

    Hi,

    Here's the problem:

    1. The problem statement, all variables and given/known data

    Quantum particle moving in 1D. Potential energy function is [tex]V(x) = C|x|^{3}[/tex]. Using the variational method, find an approx. ground-state wave function for the particle.

    3. The attempt at a solution

    Using [tex]\psi = Ae^{-ax^{2}}[/tex], I find that

    [tex]A = \left(\frac{2a}{\pi}\right)^{1/4}[/tex]

    Then,

    [tex]<H> = <T> + <V> = -\frac{\hbar^2}{2m}\frac{d^{2}}{dx^{2}}+C|x^{3}|[/tex]

    [tex]<T> = -\frac{\hbar^2}{2m}A^{2}\left( -2a\int_{-\infty}^{+\infty}e^{-2ax^{2}} dx + 4a^{2}\int_{-\infty}^{+\infty}x^{2}e^{-2ax^{2}} dx \right)[/tex]

    which I work out to be

    [tex]<T> = \frac{\hbar^{2}a}{2m}[/tex]

    For the potential energy component, I state that

    [tex]<V> = CA^{2}\int_{-\infty}^{+\infty} e^{-ax^{2}} |x|^{3} e^{-ax^{2}} dx = 2CA^{2} \int_{0}^{+\infty} x^{3}e^{-2ax^{2}} dx[/tex]

    which I find to be

    [tex]<V> = \frac{CA^{2}}{4a^{2}} = ... = \frac{C}{2a\sqrt{2a\pi}}[/tex]

    And thus

    [tex]<H> = \frac{\hbar^{2}a}{2m} + \frac{C}{2a\sqrt{2a\pi}}[/tex]

    Minimising (diff. H wrt. a and set to zero) I obtain

    [tex]\left(\frac{3Cm}{2\hbar^{2}\sqrt{2\pi}}\right)^{2/5}[/tex],

    but this (substituted back into the expression for <H> above) produces a rather ugly looking expression, which makes me suspicious that I have made a mistake somewhere down the line.

    Can anyone see where (or if) I am going wrong?

    Cheers!
     
    Last edited: Apr 11, 2008
  2. jcsd
  3. Apr 11, 2008 #2
    Dr Hooley intended you to do the assessed work on your own. ;)
     
  4. Apr 11, 2008 #3

    T-7

    User Avatar

    Whilst we are intended to work through the exercises ourselves, as far as I know we're encouraged to discuss all the problems, and the lecturer has been willing to take questions himself on any of the set problems. I've not reproduced my working in detail here for anyone to copy, nor am I asking for detailed working in reply. It's against this forum's policy to do your homework for you.

    However, if you're under the impression that 'starred' Qs can't be discussed at all, I'll ask my tutor for clarification. Consider the thread closed.
     
    Last edited: Apr 11, 2008
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