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Eigenfunction concept

  1. Sep 28, 2008 #1
    I am a seond year Quantum Chemistry student. I am having a hard time understanding these concepts. I was wondering I can get help in this concept.

    How can it be demonstrate mathematically in the Hamiltonian operator that the function
    φ(x) = A sin(2x) + B cos(2x)

    is an eigenfunction of the Hamiltonian operator:
    H=-h^2 d^2
    2m dx^2

    What is the eigenvalue equal to?
     
  2. jcsd
  3. Sep 28, 2008 #2

    olgranpappy

    User Avatar
    Homework Helper

    Yikes. That is very difficult to read. You should try and learn some TeX. It is good for the soul, and for the typesetting. for example:

    [tex]
    H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\;.
    [/tex]

    I believe that If you click on the above equation it will show you the TeX source that I used to write the equation in such a pretty manner.

    Anyways. In order to do what you want you need to take the derivative of
    [tex]
    A\sin(2 x)+B\cos(2 x)
    [/tex]
    twice.

    what do you get?

    Then multiply by
    [tex]
    -\frac{\hbar^2}{2m}\;.
    [/tex]

    That is what the symbols on the right hand side of the equation for [itex]H[/itex] are instructing you to do.

    What is the end result?

    Is it proportional to
    [tex]
    A\sin(2 x)+B\cos(2 x)
    [/tex]
    ?

    What is the proportionality constant?
     
  4. Sep 29, 2008 #3
    In case you are not aware of eigenfunctions:
    Eigenvalue -- Wolfram Mathworld
     
  5. Sep 30, 2008 #4
    I found the two derivatives and I found that the function φ(x) = A sin(2x) + B cos(2x)
    is an eigenfunction of the Hamiltonian operator:

    H=-h2 d2
    2m dx2

    and the proportionality constant is
    2h2
    m

    Can someone confirm with me if this is right or what did I do wrong?

    Thanks so much for all the help.
     
  6. Oct 1, 2008 #5

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Yes, you have the correct eigenvalue (proportionality constant).
     
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