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Wave Function Spherical Coordinates Probabilities

  1. Nov 16, 2008 #1
    1. The problem statement, all variables and given/known data

    A system's wave function has the form

    [tex]\psi(r, \theta, \phi) = f(t, \theta)cos\phi[/tex]

    With what probability will measurement of [tex]L_z[/tex] yield the value m = 1?

    2. Relevant equations

    [tex]L_z|\ell, m> = m|\ell, m>[/tex]

    3. The attempt at a solution

    I feel like there may be a typo, in that that "t" should be "r" in the wave function. Is there a general expression for [tex]\psi_{n,\ell,m}[/tex] that I should know/use?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 16, 2008 #2
    use the postulate of quantum mechanics. it says when you measure something, you'll get one of its eigenvalues with probabilities equal to the |coefficient|^2 of the corresponding eigenstates.
     
  4. Nov 17, 2008 #3
    What is the proper eigenvalue equation to use?
     
  5. Nov 17, 2008 #4
    The one you displayed under "2. Relevant equations"!
     
  6. Nov 17, 2008 #5
    So the probability L_z will give m=1 comes from:

    <Psi | L_z | Psi> = m

    Then, the probability of getting 1 is 1^2=1? Likewise, the probability of getting 0 is 0? I'm pretty sure I'm wrong here, can someone please correct me?
     
  7. Nov 17, 2008 #6
    [tex]<\Psi | L_z | \Psi>[/tex] is the expectation (i.e., average) value of L_z.

    The probabilities to find specific values m for L_z follow from writing your wavefunction in the form

    [tex]|\Psi>=\sum_m c_m |l,m>[/tex]

    as in your special case you only have 1 value of l in the superposition.
    In this case the probability to find the value "m" is given by

    [tex]P_m=|c_m|^2[/tex]

    More generally, you'd write

    [tex]|\Psi>=\sum_m \sum_l a_{lm}|l,m>[/tex]

    and the probabilty to find the result "m" would be given by a sum over l:

    [tex]P_m=\sum_l |a_{lm}|^2[/tex]


    (And use that the wavefunction should be normalized properly!)
     
  8. Nov 17, 2008 #7
    Why is there only one value of l in my case? I'm assuming I need to use some separation of variables in order to put the wave function in the appropriate form? I think I'm having a hard time seeing what I'm supposed to do with that wave function that was given to me. And that "t" is throwing me off...
     
  9. Nov 17, 2008 #8
    Can anybody give me a clue here? I'm preparing for a test tomorrow, so I'd really, really appreciate the help!
     
  10. Nov 18, 2008 #9

    malawi_glenn

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    brooke, use the spherical harmonics, you should have a table of them. Then you rewrite f(t,theta*cos(phi) as a function of them. Then use the fact that L_Z is an eigenoperator on those functions.

    "First few spherical harmonics"
    http://en.wikipedia.org/wiki/Spherical_harmonics

    Now you study where the phi depandance is on these functions, and you might want to rewrite your cos(phi) to exponentials using eulers formula..http://en.wikipedia.org/wiki/Euler's_formula
     
  11. Nov 18, 2008 #10

    malawi_glenn

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    So for clarification.

    i) the Spherical Harmonics is a complete basis for angular functions, you can express any function dep. on anges as a sum of those sphreical harmonics, compare with Fourier series.

    ii) The spherical harmonics is eigenfunctions to L_z operator with eigenvalue m.

    iii) You don't need to know the values of L in this problem, just look at the phi dependence of the function you are given and have a look at the spherical harmonics on the page i gave you or in your book. You will find that only two possible spherical harmonics exists.

    iv) Rewrite your phi-dependence in terms of exponentials
     
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