Quantum Mechanics Solution Manual 

© Leon van Dommelen 

4.3.4.2 Solution hyddb
Question:
Check from the conditions
that , , , and are the only states of the form that have energy . (Of course, all their combinations, like 2p and 2p, have energy too, but they are not simply of the form , but combinations of the basic
solutions , , , and .)
Answer:
Since the energy is given to be , you have 2. The azimuthal quantum number must be a smaller nonnegative integer, so it can only be 0 or 1. In case 0, the absolute value of the magnetic quantum number cannot be more than zero, allowing only 0. That is the state. In the case that 1, the absolute value of can be up to one, allowing 1, 0, and 1.