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Suppose you know that an electron is in l = 1 state. As usual we refer the total angular momentum by J = L + S.
(a) What does | j = 3/2, j2 = 3/2> state correspond to in terms of the states in the product basis, | l2, s2 >?
(b) By acting J_ on this state find | j = 3/2, j2 = 1/2 >
(c) In the product basis, what are the states with j2 = 1/2?
(d) You already found the combination of the part (c) states which correspond to j = 3/2. Now, find the state with j = 1/2, using the fact that it must be orthogonal to the j = 3/2 state.

#### Solution Preview

We have:

L+|l,m> = sqrt[ l(l+1) - m(m+1)] |l,m+1>

L-|l,m> = sqrt[ l(l+1) - m(m-1)] |l,m-1>

The highest jz state for j = 3/2 is:

|j=3/2,jz = 3/2> = |s=1/2,sz = 1/2>|l=1,lz = 1>

Applying J- = L- + S- on both sides gives:

J-|j = 3/2,jz = 3/2> = (L- + S-) |s=1/2,sz = 1/2>|l=1,lz = 1> ----->

sqrt(3)|j = 3/2, jz = 1/2> = L-|s=1/2,sz = 1/2>|l=1,lz = 1> + S-|s=1/2,sz = ...

#### Solution Summary

We use the theory of angular momentum to do the computations needed for this problem

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## Addition of Angular Momentum in a Helium Atom

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.

See attachment for better symbol representation:
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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