In general, an atom can have both orbital angular momentum and spin angular momentum. The total angular momentum is defined to be J =L + S. The total angular momentum is quantized in the same way as L and S. That is, J =√jj(j +1) ℏ, where j is the total angular momentum quantum number. The zcomponent of J is J_z = L_z + S_z = m_j ℏ, where m_j 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 Jz and determine the quantum number mj. Put your results in a 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 and S can combine. Quantum mechanics is different. You’ve now shown that there are only two allowed ways to add these two angular momenta.