Advanced Undergraduate Quantum Mechanics pp 465-496 | Cite as

# Fine Structure of the Hydrogen Spectra and Zeeman Effect

## Abstract

You might still have a vague recollection of me mentioning the spin–orbit coupling in Sect. 9.5.1, where I introduced the tensor product of spin and orbital spaces as a means to construct vectors representing both orbital and spin components of a quantum state (if you do not remember that, you would do yourself a favor by going back and rereading that part of the book). More specifically, the issue of spin–orbit coupling came up in the discussion of generic vectors in the tensor product space, which could be presented as a superposition of basis vectors, in which different spin states are paired with different orbital components. Such states can be called spin–orbit coupled because the orbital properties of a system in such a state can be changed by affecting its spin and vice versa. However, practically, such states can only be realized in systems with actual spin–orbit interaction contributing a special term containing a combination of spin and orbital operators to their energy and, correspondingly, quantum Hamiltonian. This interaction is quite common. It appears in many ordinary systems, such as atoms or semiconductors, and is responsible for a number of important phenomena. In atoms it gives rise to the spectral features known in the early days of quantum mechanics, while in semiconductors it brings about the relatively recently discovered effects allowing, for instance, to use the spin to control electron spatial flow. Combining spin and orbital phenomena in such nontrivial situations is never a simple task, even if merely because it doubles the number of equations that must be solved. At the same time, the phenomena resulting from the spin–orbit interaction are way too important to be simply ignored and shall be discussed even if you only start getting comfortable with intricacies of the quantum description of the world. Therefore, in this section, I am giving you a chance to learn about some aspects of the spin–orbit interaction in a relatively non-threatening environment by considering, again, a simple model of a single electron in a hydrogen-like atom.