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Homework Help: Finding the expectation value of the angular momentum squared for a wave function

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data

    Consider a hydrogen atom whose wave function at time t=0 is the following superposition of normalised energy eigenfunctions:

    Ψ(r,t=0)=1/3 [2ϕ100(r) -2ϕ321(r) -ϕ430(r) ]

    What is the expectation value of the angular momentum squared?

    2. Relevant equations

    I know that L2 operator is:

    -ℏ2 [1/sinθ d/dθ sinθ d/dθ+1/(sin2 θ) d2/dϕ2 ]

    although I don't think I need to use it.

    I know L2=Lx2+Ly2+Lz2

    3. The attempt at a solution

    I am confused as to how to go about this. I don't think I need to be calculating an integral, as you would do to find the expectation value of, for example, x2 for a wavefunction. I think I need to calculate the number from squaring the coefficients of each part, and adding, but I'm not sure how to incorporate the L2 bit into this?

    I would appreciate any help, I have been puzzling over this for ages now!
     
  2. jcsd
  3. May 3, 2010 #2

    diazona

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    Hopefully you remember that the expectation value of [itex]L^2[/itex] in a state [itex]\vert\psi\rangle[/itex] is
    [tex]\langle\psi\vert L^2\vert\psi\rangle[/tex]
    When you plug in the given wavefunction, what do you get?

    Now, what is the expectation value of [itex]L^2[/itex] in an eigenstate [itex]\vert\psi_{nlm}\rangle[/itex], in terms of the quantum numbers n,l,m?
     
  4. May 4, 2010 #3
    Ok I know that:

    〈H ̂ 〉= <S|H ̂|S>

    which is:

    =sum(a*nam<En|H|Em>)
    =sum(a*nam<En|Em|Em>)
    =sum(a*namEm<En|Em>)
    =sum(a*namEmdeltamn)
    =sum(|am|2 En)
    =<E>

    So am I right in thinking that I just have to do:

    <L2> = sum(|coefficients|2 * L2)

    If so, what do I use for L2?

    Is it l(l+1)hbar2 ?

    Thanks
     
  5. May 4, 2010 #4

    diazona

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    Sounds like you're on the right track.
     
  6. May 4, 2010 #5
    Thank you for getting back to me so quickly.

    I did as above, and got:

    1/3(22l(l+1)hbar2 + 22l(l+1)hbar2 + l(l+1)hbar2)

    Then used the values of l given in the subscript of each eigenfunction, and got an overall answer of 12hbar2. Does that sound about right?

    Thanks again x
     
  7. May 4, 2010 #6

    diazona

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    Homework Helper

    At the beginning, remember that you get a factor of 1/3 from each [itex]\psi[/itex] in
    [tex]\langle\psi\vert L^2 \vert\psi\rangle[/tex]
    Other than that, it seems OK.
     
  8. May 4, 2010 #7
    May I ask why you do not need to use the L^2 operator explicitly? How do you end up with your
    sum(an* am <En|H|Em>) term?
     
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