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Understand some quantum numbers in a problem

  • Thread starter fluidistic
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  • #1
fluidistic
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Homework Statement


Consider the following states of the hydrogen atom corresponding to [itex]l=2[/itex] whose quantum numbers corresponding to [itex]L_z[/itex] and [itex]S_z[/itex] are given by [itex]m_l=2[/itex], [itex]m_s=-1/2[/itex] and [itex]m_l=1[/itex], [itex]m_s=1/2[/itex]. What are the possible values for the quantum number j for the states [itex]m_j=3/2[/itex]?


Homework Equations


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

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 "[itex]m_j[/itex]"? And [itex]m_j[/itex] is the quantum number for the total angular momentum of the atom? I don't really understand what it means.
 

Answers and Replies

  • #2
Redbelly98
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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
fluidistic
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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?
Hey redbelly, I'm a bit confused.
Thanks for your description. Isn't [itex]J_z[/itex] equal to [itex]S_z+L_z[/itex]?
When [itex]l=2[/itex], [itex]m_l[/itex] runs from -2 to 2 and [itex]m_s[/itex] runs from [itex]-1/2[/itex] to [itex]1/2[/itex].
I think that [itex]L_z=\frac{\hbar}{2}[/itex] no matter what [itex]l[/itex] is worth. And [itex]S_z[/itex] could be worth [itex]-\hbar /2[/itex] or [itex]\hbar /2[/itex].
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
Redbelly98
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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.

Hey redbelly, I'm a bit confused.
Thanks for your description. Isn't [itex]J_z[/itex] equal to [itex]S_z+L_z[/itex]?
Yes.
When [itex]l=2[/itex], [itex]m_l[/itex] runs from -2 to 2 and [itex]m_s[/itex] runs from [itex]-1/2[/itex] to [itex]1/2[/itex].
Yes.
I think that [itex]L_z=\frac{\hbar}{2}[/itex] no matter what [itex]l[/itex] is worth.
Well, no. Do you really mean Lz here? Lz is the value of the z-component of orbital angular momentum, and equal to [itex]m_l \hbar[/itex].
And [itex]S_z[/itex] could be worth [itex]-\hbar /2[/itex] or [itex]\hbar /2[/itex].
Yes.
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 :)
.​
Hey, I just realized that I did not answer an important question of yours from Post #1:
I don't really understand what is the j. Is it just an index used in the "mj"? And mj is the quantum number for the total angular momentum of the atom?
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
fluidistic
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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 [itex]m_j[/itex] could be either [itex]-5/2[/itex], [itex]-3/2[/itex], [itex]-1/2[/itex], [itex]1/2[/itex], [itex]3/2[/itex] and [itex]5/2[/itex] for [itex]j=5/2[/itex] and [itex]m_j=-3/2[/itex], [itex]-1/2[/itex], [itex]1/2[/itex], [itex]3/2[/itex] for [itex]j=3/2[/itex].
Now I don't know how to determine the value of [itex]m_j[/itex] given the values of [itex]m_l[/itex] and [itex]m_s[/itex]
 
  • #6
fluidistic
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Hmm looking back at the original question, I'd answer that both j=5/2 and j=3/2 can give [itex]m_j=3/2[/itex].
But I didn't use [itex]m_l[/itex] nor [itex]m_s[/itex]. I don't feel like I'm doing the things right.
 
  • #7
Redbelly98
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Hmm looking back at the original question, I'd answer that both j=5/2 and j=3/2 can give [itex]m_j=3/2[/itex].
Yes, I agree.
But I didn't use [itex]m_l[/itex] nor [itex]m_s[/itex]. I don't feel like I'm doing the things right.
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
fluidistic
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Ok Redbelly. You've been so helpful, thank you very much!
 
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