Angular momentum quantum numbers

[BeauGeste]

For angular momentum quantum numbers j, l, and s must it be true that m_s, m_l < m_j?
It would seem that it is true because I assume that m_s +m_l = m_j, but I have not actually seen that written down anywhere and am curious.
Thanks.

 

[Parlyne]

No. It is not generally true. Remember that m tells you the component of angular momentum along some certain direction. This means that it can be either positive or negative. Consider, then, the case where m_s = -\frac{1}{2}. It is hopefully clear that m_l = m_j + \frac{1}{2}.

 

[BeauGeste]

ok, I think I meant the magnitude of the m's. i.e.
|m_s|, |m_l| \leq |m_j|.
For example, take the situation where m_j = -1/2, j=3/2, s=1/2. The orbital and spin angular momenta magnetic numbers can add to this for two cases:
1. m_s = -1/2, m_l = 0.
2. m_s = 1/2, m_l = -1.
I would argue from above that case 2 is not a viable option because |m_l| \nleq |m_j|.
What do you think?

 

[Parlyne]

If you're using magnitudes, you can't have negative numbers. However, even if you want to compare the magnitudes of m_s, m_l, and m_j, you'll find that there are states where |m_j| is smaller than either |m_s| or |m_l|. All this requires is that the spin and orbital angular momenta have their z components in opposite directions. This is something we should expect from normal vector analysis (i.e. it has nothing to do with quantum mechanics specifically). If I add two vectors which have projections in opposite directions along the z-axis, I should expect that the magnitude of the z component of the resultant vector must be smaller than at least that of one of the two vectors I added; and, it may be smaller than both.

 

[BeauGeste]

darn, I wrote that wrong. Hopefully this makes my question more clear:

if j=3/2 then possible m_j values are -3/2 to 3/2 by ones. if j=3/2 and we're dealing with an electron then l = 1 and s = 1/2. m_l values are 1,0, and -1.
If m_j = - 3/2 (+3/2) then of course m_l = -1 (+1) and m_s = -1/2 (+1/2) respectively. Those are the only possibilities to form m_j for the maximum values of m_j.
Now when m_j = \pm 1/2 it seems ambiguous as to what m_{l,s} are. For instance:
1. m_s = -1/2, m_l = 0
2. m_s = 1/2, m_l = -1
both give m_j = -1/2.

Is that ok or is one of them not correct?

 

[jtbell]

1. m_s = -1/2, m_l = 0
2. m_s = 1/2, m_l = -1
both give m_j = -1/2.

The state with m_j = -1/2 is a linear combination of the two states with (m_s = -1/2, m_l = 0) and (m_s = +1/2, m_l = -1). The coefficients of the linear combination are called Clebsch-Gordan coefficients:

http://farside.ph.utexas.edu/teaching/qm/lectures/node47.html

To relate the notation on that page to your example, substitute l for its j_1, s for its j_2, m_l for its m_1, m_s for its m_2, and finally m_j for its m.

 

[Parlyne]

darn, I wrote that wrong. Hopefully this makes my question more clear:

if j=3/2 then possible m_j values are -3/2 to 3/2 by ones. if j=3/2 and we're dealing with an electron then l = 1 and s = 1/2. m_l values are 1,0, and -1.
If m_j = - 3/2 (+3/2) then of course m_l = -1 (+1) and m_s = -1/2 (+1/2) respectively. Those are the only possibilities to form m_j for the maximum values of m_j.
Now when m_j = \pm 1/2 it seems ambiguous as to what m_{l,s} are. For instance:
1. m_s = -1/2, m_l = 0
2. m_s = 1/2, m_l = -1
both give m_j = -1/2.

Is that ok or is one of them not correct?

Adding to what jtbell said, you should also be aware that there are j=3/2 states arising from l=2,\ s=1/2. Here, there are 10 possible |m_l, m_s\!\!> states; but, only 8 of them give allowed values of m_j.