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Orthonormal basis

  1. May 24, 2009 #1
    1. The problem statement, all variables and given/known data

    Hey guys.

    http://img39.imageshack.us/img39/2345/27760913.jpg [Broken]

    I need to show that these wave functions are orthonormal.
    I'm a bit confuse, what's i and what's j?
    I mean, do I need to take both of the functions, put them in the integral and to show that the result is the Kronecker delta?
    Can I neglect the exponent for this?

    Thanks a lot.


    2. Relevant equations



    3. The attempt at a solution
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 24, 2009 #2
    i and j are just arbitrary indices. Yes you have to calculate the integrals for different cases. When i=j the exponential cancels (when you take the complex conjugate it changes sign). When i and j aren't equal you'll get some exponential dependence as well but in that case you should get zero anyway.
     
  4. May 24, 2009 #3
    Well, where are i and j in my problem?
    I mean, this is not a series, it's a function.
     
  5. May 24, 2009 #4

    gabbagabbahey

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    You have two wave functions, [itex]\psi_1[/itex] and [itex]\psi_2[/itex], so the indices [itex]i[/itex] and [itex]j[/itex] can each take on the values [itex]1[/itex] and [itex]2[/itex].
     
  6. May 24, 2009 #5
    Yeah, but I don't have i and j inside the functions so how can I come up with the kronecker delta?

    How can I show that if i=j then it's 1 and if i does not equal to j, it's 0 if I don't have i and j?

    Thanks.
     
  7. May 24, 2009 #6

    gabbagabbahey

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    Showing that

    [tex]\int \psi_i \psi_j dx =\delta_{ij}[/tex]

    just means that you need to show:

    [tex]\int \psi_1 \psi_1 dx =\int \psi_2 \psi_2 dx =1[/tex]

    and

    [tex]\int \psi_1 \psi_2 dx=\int \psi_2 \psi_1 dx =0[/tex]
     
  8. May 24, 2009 #7
    Oh, now I get it.

    Thanks a lot.
     
  9. May 25, 2009 #8
    Well, here is the second part of the question

    http://img207.imageshack.us/img207/879/95899388.jpg [Broken]

    I also posted there answer.
    I think they have a mistake, I marked it in the red box.
    Shouldn't it be A^2=1/2 ?
    Am I missing something?

    Thanks a lot.
     
    Last edited by a moderator: May 4, 2017
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