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Why the magnetic moment is zero for occupied levels?

  1. Jun 11, 2013 #1
    I can not get convinced why for energy level, filled with an even number of electrons, the net magnetic moment is zero!
    If we have a level with the quantum number l, we have two electrons with opposite spins in it. Also magnetic moment is proportional to Jtotal where [itex]J_t= J_1+J_2[/itex](1and 2 refer to electrons). Moreover [itex]J=L+S[/itex]. If magnetic moment were to be zero, the [itex]J_t[/itex] must be zero. How can we show that?
     
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  3. Jun 11, 2013 #2

    Bill_K

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    Isn't the correct statement that the magnetic moment is zero only for an atom whose subshells are completely filled. That is, electrons occupy all m values for a given L, so that Σm = 0
     
  4. Jun 11, 2013 #3
    OK, But if so(Σm = 0) we can only deduce that the z-component of L is zero not the L itself!
     
  5. Jun 11, 2013 #4

    Bill_K

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    Along every z-axis. A closed subshell is spherically symmetric, i.e. its total L is zero.
     
  6. Jun 11, 2013 #5
    Thank you very much. It just remains another question:
    By the above statement, do you mean that in all the spherically symmetric systems, If we have a z-component, then we have x and y component? In this case, the uncertainty principle would be invalid!
     
  7. Jun 11, 2013 #6
    Well if you have 0 total angular momentum which way is the J vector pointing? If you can tell which way it's pointing, how do you know it's projection on the (x,y,z) basis?

    Granted the quantum angular momentum vector really doesn't "point" in a specific direction when if is non-zero, but you see my point.
     
  8. Jun 12, 2013 #7

    Bill_K

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    No, of course not, that is not at all what I said. A closed subshell contains 2(2l+1) electrons, an electron pair occupying every possible value of m. The wavefunction is a Slater determinant, and is spherically symmetric, i.e. invariant under rotations. Not just the potential is spherically symmetric, the total wavefunction is spherically symmetric. This means the total L value is zero.

    EDIT: Are you thinking that [Lx, Ly] = iħ Lz is an example of the uncertainty principle? The uncertainty principle gives the commutator of variables that are canonical conjugates, which Lx and Ly are not.
     
    Last edited: Jun 12, 2013
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