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Homework Help: Orthogonality of eigenfunctions with continuous eigenvalues

  1. Oct 30, 2008 #1
    1. The problem statement, all variables and given/known data

    With knowledge of the orthogonality conditions for eigenfunctions with discrete eigenvalues, determine the orthonormal set for eigenfunctions with continuous eigenvalues. Use the definition of completeness to show that | a(k) |^2 = 1.

    2. The attempt at a solution

    The first step is:

    http://img55.imageshack.us/img55/5229/81115215vg1.jpg [Broken]
    http://g.imageshack.us/img55/81115215vg1.jpg/1/ [Broken]

    Since the integral is only equal to 0 when k' = k. (The same condition as the kronecker delta.)


    http://img352.imageshack.us/img352/9519/94213081to0.jpg [Broken]
    http://g.imageshack.us/img352/94213081to0.jpg/1/ [Broken]

    After here I get a bit lost, even though I think this is almost the solution. My work differs from the course notes and a QM book I've looked through. They both have an integration with respect to k for the RHS, not x.

    Any pointers would be greatly appreciated.
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Oct 30, 2008 #2


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    no. the right hand side is always equal to zero except when k'=k

    The very first line is incorrect. The RHS should not be integrated over x but rather over k.
    Last edited by a moderator: May 3, 2017
  4. Oct 30, 2008 #3
    Should I integrate the RHS w.r.t. k, multiply both sides by the conjugate of Phi and then integrate both sides over x?
  5. Oct 30, 2008 #4


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    First... get the definition of [itex]\psi(x)[/itex] in terms of a(k) and [itex]\phi_k(x)[/itex] correct.

    Then square psi(x) (which is in terms of a(k) and phi(k,x) now) and integrate over x using the "first step" you mention in the OP.

    Use the fact that psi(x) is normalized to see that a(k) is is normalized.
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