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Commutator relations in simple harmonic oscillator

  1. Jan 31, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that, [tex] [a, \hat H] = \hbar\omega, [a^+, \hat H] = -\hbar\omega [/tex]

    2. Relevant equations

    For the SHO Hamiltonian [tex] \hat H = \hbar\omega(a^+a - \frac{\ 1 }{2}) [/tex] with [tex][a^+, a] = 1 [/tex]

    [a, b] = -[b, a]

    3. The attempt at a solution

    I have tried the following:

    [tex] [a, \hat H] = a\hat H - \hat Ha = \hbar\omega ( (aa^+a - \frac{\ 1 }{2}a) - (a^+a - \frac{\ 1 }{2})a )
    = \hbar\omega (aa^+a -a^+aa) = \hbar\omega [a, a^+] a = - \hbar\omega a [/tex]

    And this is nothing like [tex] \hbar\omega [/tex] I am supposed to get. Could anyone point out where I have gone wrong?
    Last edited: Jan 31, 2009
  2. jcsd
  3. Jan 31, 2009 #2


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    Homework Helper

    I got that result too:

    [tex][a, \hat H] = \hbar \omega [a, a^\dagger a] = \hbar\omega\left( a^\dagger [a, a] + [a, a^\dagger] a \right) = \hbar\omega( 0 - a ) = - \hbar \omega a[/tex]

    which agrees perfectly well with the expressions given on Wikipedia.

    So having three independent sources with the same result, I suppose that your question is incorrect?
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