Addition of angular momentum

Problem:

You have a system of two electrons whose orbital quantum numbers are l1 = 2 and l2 = 4 respectively.
(a)  Find the possible values of l (total orbital angular momentum quantum number) for the system.
(b)  Find the possible values of s (total spin angular momentum quantum number) for the system.
(c)  Find the possible values of j (total angular momentum quantum number) for the system

Solution:

Problem:

A hydrogen atom is known to be in a state characterized the quantum numbers n = 3, l = 2.
(a)    Give the allowed values of j.
(b)    For each of the allowed values of j, calculate the square of the magnitude of the total angular momentum.

Solution:

Problem:

Consider a deuterium atom (composed of a nucleus of spin 1 and an electron).  The electronic angular momentum is J = L + S, where L is the orbital angular momentum of the electron and S is its spin.  The total angular momentum of the atom is F = J + I, where I is the nuclear spin.  The eigenvalues of J2 and F2 are j(j + 1)ħ2 and f(f + 1)ħ2 respectively.
(a)  What are the possible values of the quantum numbers j and f for a deuterium atom in the 1s ground state?
(b)  What are the possible values of the quantum numbers j and f for a deuterium atom in the 2p excited state state?

Solution:

Problem:

We are to add two angular momenta characterized by the quantum numbers j1 = 2 and j2 = 1.
(a)  What are the possible values for j?
(b)  Express all eigenkets |j1, j2; j, m> = |2, 1; 1, m>  in terms of  |j1, j2; m1, m2>.
(c)  Express the ket  |j1, j2; m1, m2> =|2, 1; 0, 0> in terms of  |j1, j2; j, m>.
(d)  What are the expectation values of J1z and J2z in the state |j1, j2; j, m> = |2, 1; 1, 1>?

Solution:

Problem:

A particle of spin 3/2, at rest in the laboratory, disintegrates into two particles, one of spin ½ and one of spin 0.
(a)  What values are possible for the relative orbital angular momentum of the two particles?  Show that there is only one possible value if the parity of the relative orbital state is fixed.
(b)  Assume the decaying particle is initially in the eigenstate of Sz with eigenvalue mħ.  Is it possible to determine the parity of the final state by measuring the probabilities of finding the spin ½ particle either in the |+> or |-> state?

Solution: