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Hermetian Operators-Simplify

  1. Mar 19, 2008 #1
    Hi,

    So for an harmonic oscillator we need to to find the average value for [tex] x^4[/tex], so [tex] <n|x^4|n> [/tex]. We split it up to [tex] \sum_m |<n|x^2|n>|^2 [/tex] and recognize that only m = n+2, m=n and m = n-2 can be used. We find that

    m=n

    [tex] \frac{\hbar}{2m\omega}<n|\hat{A}\hat{A^\dagger}|n>[/tex]

    m= n+2


    [tex] \frac{\hbar}{2m\omega}<n+2|\hat{A^\dagger}\hat{A^\dagger}|n>[/tex]

    m = n-2

    [tex] \frac{\hbar}{2m\omega}<n-2|\hat{A}\hat{A}|n>[/tex]

    So we can reduce it all to

    [tex] <n|x^4|n> = \frac{1}{4} \hbar^2 \omega^2 (2n+1)^2 + \frac{1}{2} (\frac{\hbar}{m \omega})^2 <n|n>[/tex]


    How I simplify the [tex] <n|n> [/tex].

    Thanks.

    Cheers,

    Biest
     
    Last edited: Mar 19, 2008
  2. jcsd
  3. Mar 19, 2008 #2

    Tom Mattson

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    Staff Emeritus
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    Gold Member

    The basis states are orthonormal, so [itex]<n|m>=\delta_{mn}[/itex].
     
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