How Do You Calculate Quantum Numbers for a Two-Electron System?

[dq1]

Homework Statement

(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 $$\psi= r cos \theta exp(r/2a)$$ has parity
quantum number -1.

Homework Equations

$$J=L+S \\ j=l \pm s \\ L^2 = l(l+1) \\ P \psi = e^{i\theta}\psi$$

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?
$$L^2 = l(l+1) \\ L_1=\sqrt{20} = \pm 4.47 \\ L_2=\sqrt{6} = \pm 2.449 \\ L = -6.919, -2.021, 2.021, 6.919$$
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:

$$L^2 = l(l+1)$$

which should probably be $$L^2 = l(l+1)\hbar^2$$

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
$$\sqrt{6} \ \hbar$$

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]

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. in the same direction or opposite direction)?

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 $$\pm 1/2$$ from each one for j?

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

 

[Redbelly98]

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

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

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.

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

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

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

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