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

Angular momentum

  1. Feb 20, 2007 #1
    1. The problem statement, all variables and given/known data
    For the spherical solution of the Schrodinger equation in spherical coordinates given the superposition of spherical harmonic functions

    [tex] \frac{1}{\sqrt{14}} (Y_{1,-1}+ 2Y_{1,0}+3Y_{1,1}) [/tex]

    calculate [itex] <\hat{L_{z}}> [/itex] and [itex] \Delta L_{z} [/itex]

    2. The attempt at a solution

    now from my textbook (brehm and mullin)
    [tex] <\hat{L_{z}}> = \hbar m_{l} [/tex]
    [tex] <\hat{L_{z}}> = \frac{\hbar}{14} (-1 + 4(0) +9(1)) = \frac{8}{14} \hbar = \frac{4}{7} \hbar [/tex]

    while [tex] <L_{z}^2> = (\hbar m_{l})^2 [/tex]

    this implies that the uncertainty in the Z component of the angular momentum [itex] \Delta L_{z} =0 [/itex]

    but i was marked wrong in my assignment for this...

    am i missing something

    is there a difference between [itex] <\hat{L_{z}}> [/itex] and [itex] <L_{z}> [/tex]?

    thanks in advance for any input
  2. jcsd
  3. Feb 20, 2007 #2

    Meir Achuz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Thus, you should get Delta L_z=3/7.
  4. Feb 21, 2007 #3


    User Avatar
    Science Advisor
    Homework Helper

    How did you get that

    [tex] \langle L_{z}^2\rangle = (\hbar m_{l})^2 [/tex] ?
  5. Feb 21, 2007 #4
    my textbook says so....


    [tex] <L_{z}^2>=\int \Psi^{*}_{nlm}\left(\frac{\hbar}{i}\frac{\partial}{\partial\phi}\right)\left(\frac{\hbar}{i}\frac{\partial}{\partial\phi}\right)\Psi_{nlm} d\tau = (\hbar m)^2 [/tex]
    Last edited: Feb 21, 2007
  6. Feb 22, 2007 #5


    User Avatar
    Science Advisor
    Homework Helper

    Yes, but in your case the state is no longer [itex] \langle r, \theta, \varphi|n, l, m \rangle [/itex] , but a linear combination of spherical harmonics. So blindly using a fomula in the book is a wrong decision...
  7. Feb 22, 2007 #6
    ... im not sure how to proceed then...

    do i 'prove' it???

    thanks for the help so far...

    but could you look at this thread of mine... its in more of ugent need ...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook