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.

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