# Can total angular momentum j be negative?

1. Mar 10, 2015

### skate_nerd

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. Mar 10, 2015

### TSny

$j$ 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 $l = 0$

3. Mar 10, 2015

### skate_nerd

I appreciate the response! Cheers

4. Mar 11, 2015

### 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 $j_2$.

5. Mar 11, 2015

### TSny

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 $l = 0$.