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Homework Help: Question about wavenumbers and determining ?

  1. Jan 15, 2007 #1
    1. The problem statement, all variables and given/known data
    The normalized wavefunctions for a particle confined to move on a circle are w(o) = sqrt(1/2pi) e^-imo where m = 0, ±1, ±2, ±3... and o is between 0 and 2pi. Determine o

    w = psi
    o = the o with the vertical line in the middle

    2. Relevant equations
    Not sure what would be relevant in this case. I've tried e^-ix = cos x - i sin x

    3. The attempt at a solution
    I've tried to use the euler method, but I'm not sure how to solve everything to get the o on one side to solve. The answer according to the teacher is pi, but I have no idea how to get there. How do I do this?
    Last edited: Jan 15, 2007
  2. jcsd
  3. Jan 15, 2007 #2


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    Homework Helper

    We have

    [tex]\psi_m (\phi ) = \frac1{\sqrt{2\pi}}e^{-im\phi},\qquad\mbox{ where } m=0,\pm 1,\pm 2,\ldots\mbox{ and }0\leq \phi \leq\2\pi[/tex]​

    There might be something missing from your question: specifically, we are to determine [tex]\phi[/tex] under what condition? It is not normalization, for the condition that wavefunctions [tex]\psi_m (\phi)[/tex] are normalized is already met. The following proves this:

    In this case, the wavefunctions [tex]\psi_m (\phi)[/tex] are normalized if, and only if

    [tex]\int_{0}^{2\pi}\left|\psi_m (\phi)\right|^2 d\phi =1.[/tex]​

    Indeed this is already so since

    [tex]\left|\psi_m (\phi)\right|^2 = \psi_m (\phi)\psi_m^* (\phi) = \left(\frac1{\sqrt{2\pi}}e^{-im\phi}\right)\left(\frac1{\sqrt{2\pi}}e^{im\phi}\right) =\frac1{2\pi} [/tex]​

    where [tex]\psi_m^* (\phi)[/tex] denotes the complex conjugate of [tex]\psi_m (\phi)[/tex] and hence we see that

    [tex]\int_{0}^{2\pi}\left|\psi_m (\phi)\right|^2 d\phi =\int_{0}^{2\pi}\frac1{2\pi} d\phi=1.[/tex]​

    Thus the condition of normaliztion is already met.
  4. Jan 15, 2007 #3
    I think it wants to know the expectation value for phi. The problem has phi like <phi>.
  5. Jan 16, 2007 #4


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    So how do you compute the expectation value of [itex] \varphi [/itex] ?

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