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Discrete Spectrum Proof In 1D

  1. Sep 15, 2014 #1
    1. The problem statement, all variables and given/known data

    Prove that in the 1D case all states corresponding to the discrete spectrum are non-degenerate.

    2. Relevant equations

    [tex]\hat{H}\psi_n=E_n\psi_n[/tex]

    3. The attempt at a solution

    Okay so, what I am stuck on here is that the question is quite broad. I can think of specific cases like the 1D square-well where [itex]E = \frac{n^2\pi^2\hbar^2}{2ma^2}[/itex] which is non-degenerate. But in a more general sense this does not seem so easy to prove. We do know that the eigenvalues in this case are discrete ([itex]E_n[/itex]) and the eigenfunctions are [itex]\psi_n[/itex], however I do not know where to go from here.
     
  2. jcsd
  3. Sep 16, 2014 #2

    BvU

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    So basically what you want to prove is that if ##\hat{H}(\psi_n-\psi_m)=0##, then ...
     
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