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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

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    <L_z^2>=(+1+0+9)/14=5/7.
    Thus, you should get Delta L_z=3/7.
     
  4. Feb 21, 2007 #3

    dextercioby

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    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....


    also

    [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

    dextercioby

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    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 ...
    https://www.physicsforums.com/showthread.php?t=157392
     
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