Understand some quantum numbers in a problem

1. Nov 9, 2011

fluidistic

1. The problem statement, all variables and given/known data
Consider the following states of the hydrogen atom corresponding to $l=2$ whose quantum numbers corresponding to $L_z$ and $S_z$ are given by $m_l=2$, $m_s=-1/2$ and $m_l=1$, $m_s=1/2$. What are the possible values for the quantum number j for the states $m_j=3/2$?

2. Relevant equations
This is where the problem lies. In my class notes, I noted that $m_j=m_l+m_s$ but for the second cases this makes no sense.
I also have noted $j=l+s, l+s-1,...,|l-s|$.

3. The attempt at a solution
I tried to find some information on j in hyperphysics and wikipedia but I'm still stuck. I don't really understand what is the j. Is it just an index used in the "$m_j$"? And $m_j$ is the quantum number for the total angular momentum of the atom? I don't really understand what it means.

2. Nov 10, 2011

Redbelly98

Staff Emeritus
j is the index for the total angular momentum. mj gives the z-component of j, i.e. it's the quantum number corresponding to Jz.

If you're still stuck, I would make a list of:
• Possible values of j, given that l=2
• For each j, what are the possible values of mj?

3. Nov 10, 2011

fluidistic

Hey redbelly, I'm a bit confused.
Thanks for your description. Isn't $J_z$ equal to $S_z+L_z$?
When $l=2$, $m_l$ runs from -2 to 2 and $m_s$ runs from $-1/2$ to $1/2$.
I think that $L_z=\frac{\hbar}{2}$ no matter what $l$ is worth. And $S_z$ could be worth $-\hbar /2$ or $\hbar /2$.
Hmm I'm sure I'm wrong, I need some sleep I think. I'm getting back to it right after breakfast. Feel free to correct me meanwhile :)

4. Nov 11, 2011

Redbelly98

Staff Emeritus
I am now remembering how confusing it was for me to get a handle on a lot of quantum mechanics concepts when I was first learning them.

Yes.
Yes.
Well, no. Do you really mean Lz here? Lz is the value of the z-component of orbital angular momentum, and equal to $m_l \hbar$.
Yes.
.​
Hey, I just realized that I did not answer an important question of yours from Post #1:
No, not really.

Since the orbital and spin angular momenta represent the same type of physical quantity -- namely, angular momentum -- they can be added together to get a total angular momentum. This total angular momentum is denoted by J, which is the vector sum of orbital and spin, L+S. The quantum number associated with J is j (not mj as you said).

By straightforward vector addition, the magnitude of J must be from |L-S| (minimum) to |L+S| (maximum). It's quantum number j takes on values from |l-s| to |l+s|, in increments of 1.

Just like we do for the other angular momentum quantities L and S, we can talk about the z-component of J, which we call Jz. The quantum number associated with Jz is mj, which takes on values from -j to +j in increments of 1. Note that mj works the same way that ml and ms do with respect to l and s.

5. Nov 11, 2011

fluidistic

Ok thank you very much for your last post, I've learned much from it.
What I've done:
j is either 5/2 or 3/2 in both cases.
This gives me $m_j$ could be either $-5/2$, $-3/2$, $-1/2$, $1/2$, $3/2$ and $5/2$ for $j=5/2$ and $m_j=-3/2$, $-1/2$, $1/2$, $3/2$ for $j=3/2$.
Now I don't know how to determine the value of $m_j$ given the values of $m_l$ and $m_s$

6. Nov 12, 2011

fluidistic

Hmm looking back at the original question, I'd answer that both j=5/2 and j=3/2 can give $m_j=3/2$.
But I didn't use $m_l$ nor $m_s$. I don't feel like I'm doing the things right.

7. Nov 13, 2011

Redbelly98

Staff Emeritus
Yes, I agree.
You're answer is good. ml and ms were used in the actual problem statement, to come up with mj (=ml and ms) = 3/2.

Just like the z-components of vectors are added to calculate the z-component of a vector sum, ml and ms add up to mj.

8. Nov 13, 2011

fluidistic

Ok Redbelly. You've been so helpful, thank you very much!