Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    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


    User Avatar
    Science Advisor
    Homework Helper

    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"

    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


    User Avatar
    Science Advisor
    Homework Helper

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook