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Functions, operator => eigenfunction, eigenvalue

  1. Sep 24, 2007 #1
    [SOLVED] Functions, operator => eigenfunction, eigenvalue

    1. The problem statement, all variables and given/known data
    Show, that functions
    f1 = A*sin([tex]\theta[/tex])exp[i[tex]\phi[/tex]] and
    f2 = B(3cos[tex]^{2}[/tex]([tex]\theta[/tex]) - 1) A,B - constants
    are eigenfunctions of an operator
    [​IMG]
    and find eigenvalues


    3. The attempt at a solution
    This is what i got for the first function:
    [​IMG]

    The next step is to solve left part of the equation, and than compare it to the right part.

    The question arises, how to solve that equation?

    I tried simplifying left part of an equation in mathcad, and I got
    [​IMG]

    Next question from that part, is if I am doing it right, how to compare those parts, and answer a question - weather this function is an eigenfunction of an operator?

    Thank You in advance, and I am constantly near computer and waiting for suggestions.
     
    Last edited: Sep 25, 2007
  2. jcsd
  3. Sep 24, 2007 #2
    Your eigen operator has partial derivatives wrt to [itex]\theta[/itex] and [itex]\phi[/itex]. When you operate it on your given eigen function, you should get back your original function multiplied by a scaling factor which is your eigenvalue.
     
  4. Sep 24, 2007 #3
    Yes, Thank You, but how do I calculate that? May I ask for instructions on how to calculate that left part of an equation? Is the result I got is correct?
     
  5. Sep 24, 2007 #4

    nrqed

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    Just go ahead and apply the derivatives!! It's that simple. (btw, I don't know what you entered in mathcad but what it gave you is wrong).

    all you have to do is to apply the derivatives
     
  6. Sep 24, 2007 #5

    nrqed

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    For the first term, what you have to calculate is

    [tex] \frac{1}{sin \theta} \frac{\partial}{\partial \theta} ( sin \theta ~\frac{\partial}{ \partial \theta} (A sin \theta} e^{i \phi}}) ) [/tex]
     
  7. Sep 24, 2007 #6
    I will give it a try right know. Thank You.
     
  8. Sep 24, 2007 #7
    So, after proper calculations, the result is
    [​IMG]

    Is this one correct?

    I posted an image of what I am given, and as far as I know, differentiation sign usually is placed before the function?
    I just don't get it - to what parts of an equation do underlined derivatives belong t?
    [​IMG]

    Thank You.
     
    Last edited: Sep 24, 2007
  9. Sep 24, 2007 #8
    The part in brackets is an "operator"... every incomplete differentiation sign (ie, without anything to differentiate) operates on whatever is "multiplied" to it.

    For example, the d/d(theta) is your image also operates on A*sin(theta)*exp(i*phi), because when you open the bracket, it gets to differentiate that term.

    Eg. (the d's are partial)
    [d/dx + d/dy]*x*y^2 = y^2 + 2*y*x
     
  10. Sep 25, 2007 #9
    Just operate it on the function. They are simple partial derivatives, I won't take much time to solve. Just a little patience for the initial steps. I tried your problem and many terms get canceled out and you get the solution correctly. You don't need mathcad to do it.
     
  11. Sep 25, 2007 #10
    Thank You Reshma! I already managed to solve it correctly, and find eigenvalues as well.
    For the case with the first function : a = 2h^2, and for the case with second function b = 6h^2. The problem was initially in understanding how to apply operator properly. Once example have been given, and operator properties reanalyzed - problem got solved in like 10 minutes (with putting it all on a paper as well). Thank You everybody who took part in this one!
     
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