In general, an atom can have both orbital angular momentum and spin angular momentum. The total angular momentum is defined to be J(vector) = L(vector) + S(vector). The total angular momentum is quantized in the same way as L(vector) and S(vector). That is, J = √j(j +1)h, where j is the total angular momentum quantum number. The z-component of J(vector) is J_z = L_z + S_z = m_jh, where mj goes in integer steps from -j to +j. Consider a hydrogen atom in a p state, with l = 1.

a. L_z has three possible values and S_z has two. List all possible combinations of L_z and S_z. For each, compute J_z and determine the quantum number m_j. Put your results in table.

b. The number of values of J_z that you found in part a is too many to go with a single value of j. But you should be able to divide the values of J_z into two groups that correspond to two values of j. What are the allowed values of j? Explain. In a classical atom, there would be no restrictions on how the two angular momenta L(vector) and S(vector) can combine. Quantum mechanics is different. You’ve now shown that there are only two allowed ways to add these two angular momenta.