Two electrons have ##l_1=1## and ##l_2=3##, what are L and S

[Kara386]

Homework Statement

Two electrons in helium have $l_1=1$ and $l_2=3$. What are the values of $L$ and $S$? From this, deduce the possible values of $J$ and find how many quantum states this excited state of helium can occupy.

Homework Equations

The Attempt at a Solution

For $L$ the allowed values are given by $L= \hbar m$, so for $l_1=1$ $L = hbar$, $0$ and $L=-\hbar$.
For $l_2$,
$L = -3\hbar$, $-2\hbar$, $-\hbar$, $0$, $\hbar$, $2\hbar$ and $L=3\hbar$.

For $S$ the allowed values are again the eigenvalues which are $\hbar m_s$, $m_s$ runs from $-s$ to $s$ in integer steps. Electrons have $s=\frac{1}{2}$, so
$S = -\frac{1}{2}\hbar$ or $S = \frac{1}{2} \hbar$.

I'm not sure how to work out $J$, in my lecture notes it says for a single electron $j = l \pm \frac{1}{2}$ but doesn't say how that changes for multiple electrons. $J=\hbar m_j$ where $m_j$ runs from $-j$ to $j$.

For the last part, I think for every combination of $L$, $J$ and $S$ there are $2J+1$ quantum states. So I think the main question I have is: how can I work out the allowed values of $J$? Can I work them out individually for each electron using $j = l \pm \frac{1}{2}$ and then add them?

 

[kuruman]

It looks like you are confusing L with L_z. When L = 1, L_z can take values $\pm \hbar$ and zero.
It also looks like you are not sure about the rules for adding angular momenta. You are given l₁ = 1 and l₂ = 3. You are asked to find the possible values for L defined as L = l₁ + l₂. Does this look familiar?

 

[Kara386]

It looks like you are confusing L with L_z. When L = 1, L_z can take values $\pm \hbar$ and zero.
It also looks like you are not sure about the rules for adding angular momenta. You are given l₁ = 1 and l₂ = 3. You are asked to find the possible values for L defined as L = l₁ + l₂. Does this look familiar?

Well the only things in my lecture noted are $L^2$ and $L_z$, so I assumed the $L$ I was being asked about was $L_z$. Ok, so I should be using $L = l_1+l_2$? I was also using the definition for $S_z$, does a similar thing apply there? Should I be calculating $S = s_1+s_2$? But I think $s_1$ and $s_2$ can both have two values, $\frac{1}{2}$ and $-\frac{1}{2}$, so would I need to account for that?

 

[kuruman]

Yes, you should be using $L=l_1+l_2$. And yes, a similar thing applies to $S=s_1+s_2$. No, $s_1$ and $s_2$ cannot have the values you suggest. Just like the orbital angular momentum, $s_{1z}= \pm \frac{\hbar}{2}$ and similarly for $s_{2z}$. You need to add angular momenta three times:
1. $L=l_1+l_2$
2. $S=s_1+s_2$
3. $J=L+S$

 

[Kara386]

Yes, you should be using $L=l_1+l_2$. And yes, a similar thing applies to $S=s_1+s_2$. No, $s_1$ and $s_2$ cannot have the values you suggest. Just like the orbital angular momentum, $s_{1z}= \pm \frac{\hbar}{2}$ and similarly for $s_{2z}$. You need to add angular momenta three times:
1. $L=l_1+l_2$
2. $S=s_1+s_2$
3. $J=L+S$

Is $s_{1z}=s_1$? Or are they different, different orientation or something?

 

[kuruman]

They are different. Think of $s_1=\frac{1}{2}$ as a vector. It's a quantum vector the component of which in the z-direction is denoted by $s_{1z}$ and can have only two values, $+\frac{\hbar}{2}$ and $-\frac{\hbar}{2}$. The same holds for $s_2$. You are looking for the sum $S=s_1+s_2$. The sum of two quantum vectors must also be a quantum vector. This means that it can have an integer or half-integer value. Do you remember learning about this? You may wish to look at this Wikipedia entry https://en.wikipedia.org/wiki/Total_angular_momentum_quantum_number.

 

[Kara386]

They are different. Think of $s_1=\frac{1}{2}$ as a vector. It's a quantum vector the component of which in the z-direction is denoted by $s_{1z}$ and can have only two values, $+\frac{\hbar}{2}$ and $-\frac{\hbar}{2}$. The same holds for $s_2$. You are looking for the sum $S=s_1+s_2$. The sum of two quantum vectors must also be a quantum vector. This means that it can have an integer or half-integer value. Do you remember learning about this? You may wish to look at this Wikipedia entry https://en.wikipedia.org/wiki/Total_angular_momentum_quantum_number.

Yes, it definitely rings a bell. So this link seems to suggest that since $S^2 = \hbar^2 s(s+1)$ then I would square root this to find $S$:
https://en.wikipedia.org/wiki/Spin_quantum_number#Electron_spin.
The bit I was looking at is section 3.

$s=\frac{1}{2}$ for electrons so then would I add the values from this equation? I'd get $S = 2\hbar\sqrt{\frac{3}{4}}$. And then it also says this is analagous to how you'd calculate $L$ but if I repeated the same thing for $L$ as for $S$ that would be a different answer to just adding $l_1$ and $l_2$. I'm having trouble working out from different texts what they are using L and S and all these capital and little letters for, because it feels like they're being used for different things, or I'm just getting very confused. And so I can't identify how these equations I'm finding relate to the numbers in my question.

The other possibliity would be $S=1$ and $L=4$...

 

[kuruman]

OK, here are the rules for adding angular momenta. They apply to all angular momentum quantities regardless[/color] of the symbol you use. Say you add $l_1$ and $l_2$ to get $L$. The possible values for $L$ start at $l_1+l_2$ and go down in decrements of 1 all the way to $|l_2-l_1|$. For example, if you were to add l₁ = 4 and l₂=7 to get L, the possible values are L = 11, 10, 9, 8, 7, 6, 5, 4, 3. If you were to add l=4 and s=3/2 to get j, the possible values are j = 11/2, 9/2, 7/2, 5/2. Do you see how it works? If yes, apply it to your problem and consult with post#4 on how to add things.

 

[Kara386]

OK, here are the rules for adding angular momenta. They apply to all angular momentum quantities regardless of the symbol you use. Say you add $l_1$ and $l_2$ to get $L$. The possible values for $L$ start at $l_1+l_2$ and go down in decrements of 1 all the way to $|l_2-l_1|$. For example, if you were to add l₁ = 4 and l₂=7 to get L, the possible values are L = 11, 10, 9, 8, 7, 6, 5, 4, 3. If you were to add l=4 and s=3/2 to get j, the possible values are j = 11/2, 9/2, 7/2, 5/2. Do you see how it works? If yes, apply it to your problem and consult with post#4 on how to add things.

That makes sense, I'll apply that to my problem! I was having real trouble deciphering my lecture notes. Thank you for your help and patience, I really appreciate it! :)