What is the Nuclear Angular Momentum Quantum Number?

[eprparadox]

Homework Statement

An energy level of a certain isolated atom is split into three components by the
hyperfine interaction coupling of the electronic and nuclear angular momenta. The
quantum number j , specifying the magnitude of the total electronic angular
momentum for the level, has the value j = 3/2 . What is the quantum number i, specifying
the magnitude of the nuclear angular momentum?

Homework Equations

The Attempt at a Solution

I'm confused as to how to incorporate the electronic angular momentum. I'm not sure that I need it. The question mentions that the atom is split into 3 components because of the hyperfine interaction. So to me, that says that whatever energy state the electron was in will be split into 3 more energy states and if this is so then it seems like the nuclear angular momentum should be i = 1 so that each component of i will be -1, 0, 1.

Am I thinking about this correctly?

thanks a lot

 

[kuruman]

Am I thinking about this correctly?

You are not, and yes, you do need the electronic angular momentum. The three observed states are the result of coupling electronic angular momentum J= 3/2 to nuclear spin I to get total spin F. In other words
F = 3/2 + I (following the rules of addition of angular momenta)
Now answer these questions.
1. What must F be so that 2F+1 = 3? (Easy)
2. What must nuclear spin I be so that F = the above answer?

 

[eprparadox]

Hey thanks a lot for your response.

Everything you said makes sense. But I'm actually given the answer to this problem and it's listed as I = 1. I guess I tried to reverse engineer the explanation by having the answer and I'm more confused. I suppose the answer could be wrong. It's from a practice GRE exam.

Could the discrepancy be that the question is asking not for the total angular momentum of the atom and just of the nucleus and so because the hyperfine interaction splits the energy into 3 states we need to look for an I such that 2I+1 = 3?

Thanks a lot.

 

[kuruman]

Everything you said makes sense. But I'm actually given the answer to this problem and it's listed as I = 1. I guess I tried to reverse engineer the explanation by having the answer and I'm more confused. I suppose the answer could be wrong. It's from a practice GRE exam.

The answer is not wrong. You did not follow through with the answers to the questions that I asked you. Anyway, let's reverse engineer. Assume that J = 3/2 and I = 1. What are the possible values of F such that F = J + I?

Could the discrepancy be that the question is asking not for the total angular momentum of the atom and just of the nucleus and so because the hyperfine interaction splits the energy into 3 states we need to look for an I such that 2I+1 = 3?

What discrepancy is this? I don't see one.

 

[eprparadox]

Sorry, I wasn't thorough enough in my last response.

I did go through what you mentioned in your initial post.

1. What must F be so that 2F+1 = 3? (Easy)
This is just F = 1.

2. What must nuclear spin I be so that F = the above answer?
Going back to F = 3/2 + I, then if F = 1, I = -1/2.

Is this correct?

So the answer for the magnitude of I be 1/2?

Thanks for the help.

 

[kuruman]

I suspect you don't know how to add angular momenta. If your plan is to take the GRE, addition of angular momenta is likely to be on it in one form or another. Please review the subjects "addition of angular momenta" an example of which is "L-S coupling". It is more important for you to understand how this works than to get the answer to this particular question.

Basically, when you add two angular momenta in quantum mechanics, you can view this as a vector addition with the constraint that the resultant must a valid quantum angular momentum, i.e. integer or half-integer, and never negative.

An example of addition of angular momenta can be found at

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2

You should also be able to find how this is done in any Modern Physics or Quantum Mechanics textbook.

 

[eprparadox]

I think you're definitely right. I'm going to go back and read this over and hopefully this concept will no longer be troubling.

Thanks a lot, though, for your responses. I really appreciate it. Sorry to have wasted your time.

 

[kuruman]

I think you're definitely right. I'm going to go back and read this over and hopefully this concept will no longer be troubling.

Thanks a lot, though, for your responses. I really appreciate it. Sorry to have wasted your time.

Not a waste of time. If you're still troubled after your review, come back for more help.

 

[theoriginal]

You are not, and yes, you do need the electronic angular momentum. The three observed states are the result of coupling electronic angular momentum J= 3/2 to nuclear spin I to get total spin F. In other words
F = 3/2 + I (following the rules of addition of angular momenta)
Now answer these questions.
1. What must F be so that 2F+1 = 3? (Easy)
2. What must nuclear spin I be so that F = the above answer?

dont we have to make corrections, nuclear and relativistic?

 

[kuruman]

I don't think so. This is a simple question on addition of angular momenta.