Spin Orbit Interaction

dq1

1. The problem statement, all variables and given/known data
(a) Consider a system composed of two electrons with orbital angular momentum
quantum numbers l_1 = 4 and l_2 = 2.
Give all the possible values of
(i) the total orbital angular momentum quantum number L,
(ii) the total angular momentum quantum number J. [8]
(b) Explain what is meant by the parity of an atomic or nuclear state. Show that
the state described by the wave-function [tex] $\psi= r cos \theta exp(r/2a) $ [/tex] has parity
quantum number -1.


2. Relevant equations
[tex]
$ J=L+S $\\
$ j=l \pm s $\\
$ L^2 = l(l+1) $\\
$ P \psi = e^{i\theta}\psi $
[/tex]


3. The attempt at a solution

I know this is probably extremely easy but I've been given no examples and I keep getting myself in a muddle. Are the answers for L and J suppose to come out as non integers?
[tex]
$ L^2 = l(l+1) $\\
$ L_1=\sqrt{20} = \pm 4.47 $ \\
$ L_2=\sqrt{6} = \pm 2.449 $\\
L = -6.919, -2.021, 2.021, 6.919
[/tex]
Are the negative values valid?

Redbelly98

(i) Start by thinking about two cases:
1. the angular momentum vectors of the two electrons point in the same direction
2. they point in opposite directions

What would be l (that's a lower-case "L") for the system in these two situations?
Are any other values of l possible?

dq1

l would be 6, 2, -2, -4

?

Redbelly98

Okay, except that l just takes on positive values or zero, so it would be 6 and 2.
Next, what other values could it have, given that the two vectors need not be aligned (i.e. not in the same direction or opposite direction)?

Note:
which should probably be [tex]L^2 = l(l+1)\hbar^2 [/tex]

This isn't really needed here. L refers to the magnitude of the actual angular momentum. But it is much more common to refer to angular momentum simply by the quantum number, l. So for example, if l=2, we just say the orbital angular momentum is 2, rather than the actual value of
[tex]\sqrt{6} \ \hbar[/tex]

When I read your questions, it seems they really want the quantum numbers. These will be integers (or, if spin is included, then possibly half-integers).

dq1

Sorry I don't understand, are you saying there are more values for l (lower L)?

Presumably when I have all the l's I just [tex]\pm 1/2 [/tex] from each one for j?

For b. I understand I have to multiple it by [tex]$ e^{i\theta}$[/tex] do I need to convert the cos to terms of [tex]$ e^{i\theta}$[/tex]

Redbelly98

You probably should review the rules for adding two angular momenta together. It should be explained in your textbook or class lectures.

Yes. So far, we have just found the maximum and minimum values, the ones we get if the two L's (vectors) point in the same direction (maximum, 4+2=6) or directly opposite (minimum, 4-2=2).

If the two L's are at some angle to each other, l will be somewhere in between 2 and 6.

This one is more complicated. If there were just 1 electron, then yes you'd [tex]\pm 1/2 [/tex] since one electron has a spin of 1/2. But in this case you need to [tex]\pm[/tex] the combined spin of the two electrons.

I'm not sure, it has been a while since I worked in this area.