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Wavefunction in the energy representation

  1. May 16, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\psi(x)=\frac{3}{5}\chi_{1}(x)+\frac{4}{5}\chi_{3}(x)[/tex]

    Both [tex]\chi_{1}(x) \chi_{3}(x)[/tex] are normalized energy eigenfunctions of the ground and second excited states respectivley. I need to find the 'wavefunction in the energy representation'


    3. The attempt at a solution

    I can find the expectation value of the energy but what is the wavefunction in the energy representation?
     
  2. jcsd
  3. May 16, 2009 #2
    It looks like it is given in the energy representation already. Except I'd write it as:

    [tex]\left|\psi\right> = \frac{3}{5}\left|E_1\right>+\frac{4}{5}\left|E_3\right>[/tex]
     
  4. May 16, 2009 #3
    I'm not so sure. I think what they are after is [tex]\psi(E)[/tex] i.e. a function of energy. I suppose in analogy with [tex]\psi(x) = \left<x|\psi\right>[/tex] it would be [tex]\psi(E) = \left<E|\psi\right>[/tex]. Completeness of states could be of help. I suspect you should get delta functions.
     
  5. May 16, 2009 #4

    diazona

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    I wouldn't have thought of that, but yeah, that might be it.
     
  6. May 16, 2009 #5

    Matterwave

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    My guess at what they want: (I think it's the same as what Phsopher said, but possibly a bit more intuitive)

    By completeness, you can represent any wave-function:

    [tex]\Psi (x) = \sum_{n=1}^{\infty} c_n\psi_n(x)[/tex]

    Where the [tex]\psi_n[/tex] represent energy eigenfunctions. The energy representation of the wave function would probably then just be the coefficients [tex]c_n[/tex] at different energies. The probabilities of finding a specific energy would be [tex]|c_n|^2[/tex]. This is a bit weird, because it's not any real function that I can think of (dirac deltas can't really be squared properly...as far as I know...so I don't think ).

    I'm sort of stumped on this one too!

    EDIT:

    Oh, I just thought, maybe instead of using Dirac Deltas, you can use Kronecker deltas...

    Perhaps:

    [tex]\Psi(E) = \frac{3}{5}\delta_{E, E_1} + \frac{4}{5}\delta_{E, E_3}[/tex]

    Not 100% on this though.
     
    Last edited: May 16, 2009
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