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Quick Dirac Notation question

  1. Jan 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Find <lz> using [itex]\Psi[/itex], where [itex]\Psi[/itex]=(Y11+cY1-1)/(1+c^2)).

    Ylm are spherical harmonics, and <lz> is the angular momentum operator in the z direction.


    2. Relevant equations

    <lz> Ylm = [STRIKE]h[/STRIKE]mYlm

    3. The attempt at a solution

    The brackets around <lz> are throwing me off. This isn't defined in my book, but am I just supposed to apply the above equation to [itex]\Psi[/itex]?

    So <lz> = [STRIKE]h[/STRIKE]m(Y11+cY1-1)/(1+c^2)?

    Also what would m be?
     
  2. jcsd
  3. Jan 3, 2012 #2
    The m is the eigenvalue of the lz operator for the spherical harmonic in question, i.e. the m in Ylm, i.e. +1 or -1 in your problem.

    The <> notation surrounding an operator is implicitly the expectation value with respect to some given wavefunction:

    <lz> = <[itex]\Psi[/itex]| lz |[itex]\Psi[/itex]>

    I think your "relevant equation" should perhaps read

    lz Ylm = [itex]\hbar[/itex]m Ylm

    i.e. the operator is "naked" when it acts on the spherical harmonic. However, the equation you wrote is not exactly incorrect... it's just that <lz> is simply a number, not an operator, and the number is just [itex]\hbar[/itex]m provided [itex]\Psi[/itex] = Ylm. That's subtly different from the equation I wrote, which indicates that the operator acting the wavefunction gives you a multiple of the wavefunction. Does that make sense?


    This is a pretty straightforward problem once you get the notation. The point is that the wave function is just a linear superposition of two eigenfunctions of the angular momentum operator, so the expectation is a linear function of the eigenvalues. But you have to do the algebra to get the right answer. And by "do the algebra" I mean to write down the expectation value, in which the wave function ψ shows up in both the bra and the ket form, in integral form, and make sure you understand how the orthonormality of the spherical harmonics makes the resulting integral "simple"....
     
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