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Find the eigenvalues of the Hamiltonian - Harmonic Oscillator

  1. Nov 2, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the eigenvalues of the following Hamiltonian.

    [itex]Ĥ = ħwâ^{†}â + \alpha(â + â^{†}) , \alpha \in |R[/itex]


    2. Relevant equations

    [itex]â|\phi_{n}>=\sqrt{n}|\phi_{n-1}>[/itex]
    [itex]â^{†}|\phi_{n}>=\sqrt{n+1}|\phi_{n+1}>[/itex]

    3. The attempt at a solution

    By applying the Hamiltonian to a random state n I get:

    [itex]Ĥ |\phi_{n}> = E_{n}|\phi_{n}>[/itex]
    [itex]Ĥ |\phi_{n}>= ħwâ^{†}â|\phi_{n}> + \alpha(â|\phi_{n}> + â^{†}|\phi_{n}>) [/itex]
    [itex]Ĥ |\phi_{n}>= ħw\sqrt{n}\sqrt{n}|\phi_{n}> + \alpha(\sqrt{n}|\phi_{n-1}> + \sqrt{n+1}|\phi_{n+1}> )[/itex]
    [itex]E_{n} |\phi_{n}> = ħwn + \alpha(\sqrt{n}|\phi_{n-1}> + \sqrt{n+1}|\phi_{n+1}>) [/itex]

    This is where my problem arrives. I don't know how to prove that

    [itex]\alpha(\sqrt{n}|\phi_{n-1}> + \sqrt{n+1}|\phi_{n+1}>) = 0[/itex]

    Any help would be highly appreciated!
    Thanks.
     
  2. jcsd
  3. Nov 2, 2013 #2

    fzero

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    That linear combination doesn't vanish and the states ##|n\rangle## are not eigenstates of that Hamiltonian. The eigenstates will be infinite linear combinations of the ##| n \rangle##. However, constructing these eigenstates is certainly not the easiest way to compute the eigenvalues of this operator. I would suggest defining a new operator ## b = a + c##, where ##c## is a number to be determined by requiring that ##\hat{H} = \hbar \omega b^\dagger b + C ##, where ##C## is another constant. Using the commutation relations for ##b,b^\dagger##, you should be able to compute the eigenvalues in the same way as for the regular harmonic oscillator.
     
  4. Nov 3, 2013 #3
    And how can I find the operator b?
     
  5. Nov 3, 2013 #4

    fzero

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    You solve the equation

    $$\hbar \omega (a+c)^\dagger (a+c) + C = \hbar \omega a^\dagger a + \alpha (a + a^\dagger)$$

    for ##c## and ##C##. This is a linear equation, since the ##a^\dagger a## terms cancel.
     
  6. Nov 6, 2013 #5
    I just realized I forgot to thank you! Accept my apologies.

    Daniel
     
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