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Express a wave function as a combination of spherical harmonics

  1. Oct 9, 2012 #1
    1. The problem statement, all variables and given/known data

    An electron in a hydrogen atom is in a state described by the wave function:

    ψ(r,θ,φ)=R(r)[cos(θ)+e(1+cos(θ))]

    What is the probability that measurement of L2 will give 6ℏ2 and measurement of Lz will give ℏ?

    2. Relevant equations

    The spherical harmonics

    3. The attempt at a solution

    I know that I need to express the part in (θ,φ) as a linear combination of spherical harmonics. Then I would normalize my wave function, and the coefficients would lead me to the probability I need.

    My problem is to express ψ in terms of the spherical harmonics Ylm. Is there a general method to do so? Thanks.
     
  2. jcsd
  3. Oct 12, 2012 #2
    The way I would do it is by looking at [itex]\mathbf{L}^2|l \mspace{8mu} m> = \hbar^2 l(l+1) |l\mspace{8mu} m> [/itex] and [itex]\mathbf{L}_z |l \mspace{8mu}m> = \hbar m |l\mspace{8mu} m>[/itex]. And looking at a table of spherical harmonics. There is no need to normalize them since they are already normalized.
     
  4. Oct 12, 2012 #3

    vela

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    You could find the projection of the angular part onto the state ##\vert l\ m \rangle## by calculating ##\langle l\ m \vert \psi\rangle##. Seems to be kind of a pain though. I'd just futz around with the spherical harmonics to find a linear combination that works. For example, the ##e^{i\phi}## part has to come from an m=1 state.
     
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