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Eigenvalue of the Hamiltonian

  1. Oct 24, 2010 #1
    1. The problem statement, all variables and given/known data
    attachment.php?attachmentid=29387&stc=1&d=1287982405.jpg


    2. Relevant equations



    3. The attempt at a solution
    attachment.php?attachmentid=29388&stc=1&d=1287982405.jpg

    I think I have to use the fact that [a+ , a] = 1 but I don't know where to apply this.
     

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  2. jcsd
  3. Oct 25, 2010 #2

    gabbagabbahey

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    If [itex]\left[\hat{a}^{\dagger},\hat{a}\right]=1[/itex], what is [tex]\left[\hat{b}^{\dagger},\hat{b}\right][/tex]? What are [tex]\left[H,\hat{b}\right][/tex] and [tex]\left[\hat{H},\hat{b}^{\dagger}\right][/tex]?

    What are [tex]\hat{H}\left(\hat{b}|\psi_E\rangle\right)[/tex] and [tex]\hat{H}\left(\hat{b}^{\dagger}|\psi_E\rangle\right)[/tex]....What does that tell you?

    If [tex]E_0\hat{b}^{\dagger}\hat{b}|\psi_E\rangle-\frac{E_1^2}{E_0}|\psi_E\rangle=E|\psi_E\rangle[/tex], what is [tex]\langle\psi_E|\hat{b}^{\dagger}\hat{b}|\psi_E\rangle[/tex]? Note that in any Hilbert space, an inner product is always greater than or equal to zero....what does that tell you about [itex]E[/itex]?
     
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