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Infinite Potential Well

  1. Jul 12, 2005 #1
    Sorry for all the questions - I tend to save them till I'm done with assignments:

    Here's the question:
    Consider a particle of mass 'm' in a one-dimensional infinite potential well of width 'a'
    [tex]
    V (x) = \left\{\begin{array}{c} 0 \ \ \ if \ \ \ 0 \leq x \leq a \\ \infty \ \ \ otherwise
    [/tex]
    The particle is subject to a perturbation of the form:
    [tex]
    \omega (x) = a \omega_0 \delta \left(x - \frac{a}{2} \right)
    [/tex]
    Where 'a' is a real constant with dimension of energy. Calculate the changes in the energy level of the particle in the first order of [itex] \omega_0 [/itex]

    I just need some help starting off at this point. Can anyone suggest how to begin?
     
  2. jcsd
  3. Jul 13, 2005 #2

    Galileo

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    What does first-order pertubation theory say?
     
  4. Jul 13, 2005 #3
    Okay, I think I've got it. Does this look correct?
    ANS:
    I'm looking for first-order correction to the nth eigenvalue - so I need to solve this:
    [tex]
    E_n^1 = \left< \psi_n^0 | H' | \psi_n^0 \right>
    [/tex]
    Where
    [tex]
    \psi_n^0 (x) = \sqrt{ \frac{2}{a} } sin \left( \frac{n \pi x}{a} \right)
    [/tex]
    and
    [tex]
    H' = a \omega_0 \ \delta \left( x - \frac{a}{2} \right)
    [/tex]
    Substituting and solving gives:
    [tex]
    E_n^1 = \left< \psi_n^0 | H' | \psi_n^0 \right> = 2 \omega_0 \int_0^a sin^2 \left( \frac{n \pi x}{a} \right) \delta \left( x - \frac{a}{2} \right) dx
    [/tex]
    [tex]
    = 2 \omega_0 sin^2 \left( \frac{n \pi}{2} \right)
    [/tex]
    [tex]
    = \left\{\begin{array}{c} 2 \omega_0 \ \ \ if \ \ \ n = "odd" \\ 0 \ \ \ if \ \ \ n = "even"
    [/tex]
    Do I "choose" only non-zero answers then, or is the array the complete answer? Thanks.
     
  5. Jul 13, 2005 #4

    Gokul43201

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    Gold Member

    Looks right ! And no, you don't throw away the zero-terms. In fact, you should be asking yourself if it makes sense for the perturbation to have no effect on half the spectrum (the eigenvalues for even n).
     
  6. Jul 13, 2005 #5

    Gokul43201

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    Found this java applet from ZapperZ's link :

    http://www.quantum-physics.polytechnique.fr/en/pages/p0204.html

    It shows you the solutions to the SE for a double well. You can play with the potentials to essentially mimic your problem. Make a (infinitesimally) thin, high wall in the middle, and see what happens when you just barely increase its width : only the odd eigenvalues move, as predicted.
     
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