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Addition of Angular Momentum in a Helium Atom

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Hello, I have attached a homework problem I need help with as a Picture file. With my exam only a day away, I'm unfortunately stuck trying to get to the solutions to these problems before I can fully attempt them myself, so that I can study them for the exam and get as much preparation possible. There were seven total, but I have finished three of them myself. This is the third of four I need help with in order to have time to study. Thank you for your assistance.

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Consider a Helium atom with two electrons. Suppose you know that one of the electrons is in the l1=3 state, while the other is in the l2 = 2 state. What are the possible values of l1z and l2z? So how many different quantum states describing the orbital angular momentum configuration of the two electrons are possible? Suppose L = L1 + L2 represents the total orbital angular momentum. What are the possible values of l, the quantum number associated with the total orbital angular momentum of the two electrons. For each possible value of l, list the possible values of lz, the total z-component of the orbital angular momentum. Show that counting the states in this l, lz basis agrees with that of the product basis.

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I'm going to go beyond what the problem requires you to do here, as you have an exam on this subject tomorrow and the theory of angular momentum requires you to know more (not just the answer of the problems but also why the answer is what it is from first principles), so this problem should have been quite easy. When studying, I always make sure that I can reproduce the entire theory on a black piece of paper with no books or other sources around (although I'm not a student anymore, when preparing to teach a new subject I'll still study the subject as if I had to do an exam, just to make sure 'm completely on top of the subject). So, let's start from the beginning, we'll derive the theory from the commutation relations:

[Lx, Ly] = i Lz (1)

where I use natural units so that hbar = 1. You can, of course, derive (1) from the commutation relation between position and momentum and using the definition of angular momentum.

The others are obtained by cyclically permuting the indices:

[Ly, Lz] = i Lx (2)

[Lz, Lx] = i Ly (3)

Note that that (2) and (3) follow from (1) due to rotational invariance. You are free to relabel the x,y and z axis by cyclically permuting them but then you need to apply a rotation to make them point in the directions as indicated by the labels before you relabeld them. Since the laws of nature should be invariant under rotations, this won't affect the commutation relation.

It's useful to know the following formula for manipulating commutators:

[A B, C] = A [B ,C] + [A, C] B

This is very simple to prove:

[A B, C] = ABC - CAB = A(BC - CB) + ACB - CAB = A(BC - CB) + (AC - CA)B = A [B,C] + [A,C] B

The operator L^2 = Lx^2 + Ly^2 + Lz^2 commutes with Lz. Let's see if we can derive this fact:

[L^2, Lz] = [Lx^2 + Ly^2 + Lz^2, Lz] = [Lx^2, Lz] + [Ly^2, Lz] + [Lz^2, Lz] = [Lx^2, Lz] + [Ly^2, Lz]

[Lx^2, Lz] = Lx [Lx,Lz] + [Lx,Lz] Lx = ...

Solution Summary

This in-depth solution of 1500 words explains the principles of angular momentum and also the structure of a Helium momentum in great detail. All formula derivations are shown in a stepwise manner with clear explanations.