Please help with the following problems.
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.
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 = ...
We use the theory of angular momentum to do the computations needed for this problem