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Potential and kinetic energies in Quantum Oscillator

  1. Jun 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Problem 9. Evaluate the matrix elements [tex]\langle n + \nu|x^2|n\rangle[/tex] and [tex]\langle n + \nu|p^2|n\rangle[/tex] in
    the harmonic oscillator basis, for [tex] \nu = 1, 2, 3, 4[/tex] :
    1. Using the closure property and the matrix elements.
    2. Applying the operators [tex] x^22[/tex] and [tex] p^2[/tex] , expressed in terms of the [tex] a+, a[/tex] on the eigenstates.
    3. Find the ratio [tex] \langle n + \nu|K|n\rangle/\langle n + \nu|V|n\rangle[/tex] [tex] (\nu = 0, \pm2)[/tex] between the kinetic
    and the potential energy matrix elements. Justify the differences in sign
    on quantum mechanical grounds.

    2. Relevant equations

    [tex]H=\frac{p^2}{2m}+\frac{m\omega ^2}{2}x^2[/tex]

    3. The attempt at a solution

    [tex] \frac{\langle n|K|n \rangle}{\langle n|V|n \rangle}=-\frac{\langle n\pm 2|K|n \rangle}{\langle n\pm 2|V|n \rangle}=1[/tex]
    ...but I cannot justify the difference in sign on quantum mechanical grounds!!!
  2. jcsd
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