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Homework Help: Momentum eigenfunctions with periodic boundary conditions

  1. Mar 13, 2008 #1
    1. The problem statement, all variables and given/known data
    A particle of mass m is confined to move in one dimension. its wavefunction is periodic with period [tex]L\gg 1[/tex] - i.e. periodic boundary conditions are imposed.

    a)Determine the eigenfunctions and eigenvalues of momentum. Normalise the eigenfunctions on the interval [tex][0,L)[/tex]

    b)The wavefunction is the periodic function
    [tex]\psi (x) = { 1-|x-mL}, |x-mL| < 1, m=0, \pm 1, \pm 2,...[/tex]
    [tex]\psi (x) =0[/tex] otherwise
    normalise [tex]\psi (x)[/tex] on the interval [tex][0,L)[/tex] and expand the normalised wavefunction in eigenfunctions of momentum.
    2. Relevant equations

    3. The attempt at a solution
    I understand that periodic boundary conditions imply that [tex]\psi (0) = \psi (L)[/tex]

    Now this is where I'm not sure if I've gone right:

    I let [tex]\psi = e^{ikx}, [/tex] with [tex]k=\frac{2\pi n}{L}[/tex] and [tex]n=1,2,...[/tex]
    Hence I quantised momentum in terms of n. My lecturer hinted that the solution would involve quantising momentum, since we are not considering the wavefunction over all of space, and by letting L go to infinity it would approach continuous values for p, my solution doesn't seem to do this though.

    I can follow the maths through to get the eigenfunctions/values from here, I'm just not sure about my choice of [tex]\psi (x)[/tex]

    In part b I'm not sure how to normalise the wavefunction, I understand it will involve splitting the integral up into sections with different boundaries, but I'm not sure how to tackle a function like this.
  2. jcsd
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