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Express wave function in spherical harmonics

  1. Jan 27, 2014 #1
    1. Problem:
    I have a wave function ψ(r) = (x + y + z)*f(r) and want to find the expectation values of L2 and Lz. It is suggested that I first change the wave function to spherical coordinates, then put that in terms of spherical harmonics of the form Yl,m.

    2. Relevant equations:
    Spherical harmonics and conversions from Cartesian to spherical coordinates

    3. Attempt at solution:
    So I know how to express the wave function in spherical coordinates (which should be ψ(r) = r*f(r)*(sinΘcosø + sinΘsinø + cosΘ). I am having a hard time going from this to spherical harmonics though (I am sort of new to this). I can see from a table of spherical harmonics that Y1,0 does have a cosΘ, but how would I get the terms that have Θ and ø? (I can see that there are some spherical harmonics that have e, but this would give me cosø + isinø, so it doesn't look like that would get me what I want.)
     
  2. jcsd
  3. Jan 28, 2014 #2

    CAF123

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    Gold Member

    Hi z2394,
    The wavefunction can be written rf(r)sinθ[cosø + sinø] + rf(r)cosθ. As you said, the second term here can be written in terms of a single spherical harmonic. To get the first term as a linear combination of spherical harmonics, try expressing cosø and sinø in their complex exponential forms.
     
  4. Jan 28, 2014 #3
    So I see how to do that now, and I get ψ(r) = r*f(r)*([itex]\sqrt{2{\pi}/3}([/itex]-Y1,1+Y1,-1) - (1/i)[itex]\sqrt{2{\pi}/3}[/itex](Y1,1 + Y1,-1) + [itex]\sqrt{4{\pi}/3}[/itex]Y1,0). But now how do I use this to get <L2> and <Lz>? I know to do <ψ| L2 |ψ> and <ψ| Lz |ψ>, but does L2 acting on ψ here still give me the l(l+1)h2 eigenvalue, and Lz the mh eigenvalue, (where h is actually the reduced Planck constant), when acting on this ψ?
     
  5. Jan 29, 2014 #4

    CAF123

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    Gold Member

    When L2 and Lz act on the spherical harmonics, then the eigenvalue returned is l(l+1)h2 and mh respectively. First though, you will need to normalize ψ. Introduce some normalization constant, say A, and solve for A using the fact that the spherical harmonics are a complete set of orthonormal functions.
     
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