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Can total angular momentum j be negative?

  1. Mar 10, 2015 #1
    1. The problem statement, all variables and given/known data
    I'm just stuck on one part of a larger problem. I need to find the range of total angular momentum values for an electron in a j-j coupling scheme.

    2. Relevant equations
    j= l + and - 1/2

    3. The attempt at a solution
    The electrons here are in a 5d 6s configuration. So for the second electron, l=0. This means j for the second electron is 0 plus and minus 1/2, so -1/2 and +1/2. This formula for j is what my book says to use with j-j coupling, but it seems to imply that j can be negative, and if that were the case, couldnt then J be a complex number? (Recall J=root(l(l+1))*hbar)
    Just a little stumped here, and I want to get this right so I don't screw up the rest of the problem. Thanks for any hints
     
  2. jcsd
  3. Mar 10, 2015 #2

    TSny

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    ##j## cannot be negative. The ##l = 0## case is a little special. You only get ##j = l + 1/2## in this case. ##j = l - 1/2## is ignored when ##l = 0##
     
  4. Mar 10, 2015 #3
    I appreciate the response! Cheers
     
  5. Mar 11, 2015 #4

    DrClaude

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    Staff: Mentor

    I would disagree that ##l=0## is a special case. When summing angular momenta ##\hat{j}_1## and ##\hat{j}_2## into ##\hat{J} = \hat{j}_1 + \hat{j}_2##, the quantum number ##J## can take the values
    $$
    J = \left| j_1 - j_2 \right|, \left| j_1 - j_2 \right| + 1, \ldots, j_1 + j_2
    $$
    The absolute value prevents ##J## from being negative, whatever the relative values of ##j_1## and ##j_2##.
     
  6. Mar 11, 2015 #5

    TSny

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    Yes, you are right.

    From the OP it appears that the textbook might have written ##j = l \pm \frac{1}{2}## when combining the orbital and spin angular momentum of a single electron. Hopefully it was made clear that this doesn't hold for ##l = 0##.
     
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