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Map energy eigenstates to cartesian unit vectors - Harmonic Osillator

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Evaluate the matrix elements
    [tex]x_{nn'} = \left<n\left|x\right|n'\right>[/tex]
    and
    [tex]p_{nn'} = \left<n\left|p\right|n'\right>[/tex]
    and map the energy eigenstates
    [tex]\left|n\right>[/tex]
    to Cartesian unit vectors.

    2. Relevant equations

    [tex] x = \sqrt{\frac{\hbar}{2m \omega}}\left(a+a^{\dagger}\right) [/tex]
    [tex] p = -i \sqrt{\frac{\hbar m\omega}{2}}\left(a-a^{\dagger}\right) [/tex]

    3. The attempt at a solution

    I have

    [tex] x_{nn'} = \sqrt{\frac{\hbar}{2m \omega}}\left(\sqrt{n'}\left<n|n'-1\right>+\sqrt{n'+1}\left<n|n'+1\right>\right) [/tex]

    and

    [tex] p_{nn'} = -i \sqrt{\frac{\hbar m\omega}{2}}\left(\sqrt{n'}\left<n|n'-1\right>-\sqrt{n'+1}\left<n|n'+1\right>\right) [/tex]

    But I'm confused with the second part of the question. For example, I believe that mapping a state vector into position space would mean to get the projection of the state vector in position space meaning that one has to take the inner product <x|ψ> = ψ(x). But I don't see how i can do this with Cartesian unit vectors.
     
  2. jcsd
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