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Angular momentum of a particle in a spherically symmetric potential

  1. Apr 5, 2007 #1
    1. The problem statement, all variables and given/known data
    A particle in a spherically symmetric potential is in a state described by the wavepacked

    [tex] \psi (x,y,z) = C (xy+yz+zx)e^{-alpha r^2} [/tex]

    What is the probability that a measurement of the square of the angular mometum yields zero?
    What is the probability that it yields [tex] 6\hbar^2 [/itex]?
    If the value of l is found to be 2. what are the relative probabilities of m=-2,-1,0,1,2

    2. The attempt at a solution

    i think the first part is simply aking to calculate [itex] <L^2>[/itex]

    but the carteisna coords are throwing me off... Should i convert to spherical polars?? Till now whenever the angular momentum L^2 and Lz were required, they were gotten using
    [tex] \hat{L^2} \psi_{nlm_{l}} = l(l+1) \psi_{nlm_{l}} [/tex]

    really from the spherical harmonics... however conversion to spherical polars doesnt yield any familiar spherical harmonic either.

    can it written in a way that yields familiar spherical harmonics, however??
     
  2. jcsd
  3. Apr 6, 2007 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes. Convert to spherical coordinates. You won't necessarily get a spherical harmonic but you can decompose it into spherical harmonics in the usual way you split a wavefunction relative to an orthonormal basis.
     
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