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.
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Notes
- 1.
You might need to refresh your memory of classical electrodynamics at this point using the Internet or one of the available undergraduate textbooks on electrodynamics.
- 2.
Llewellyn Hilleth Thomas was a British physicist who eventually moved to the USA, where he held a professorial position in Ohio State University, was a member of the Watson Scientific Computing Laboratory at Columbia University, and was the IBM First Fellow in the Watson Research Center. His last position was at North Carolina State University.
- 3.
Charles Galton Darwin was an English physicist, another grandson of Charles Darwin, the author of the evolution theory. He became a director of the National Physics Laboratory in 1938 and remained in this position through the World War II participating in the Manhattan Project, where he was responsible for coordinating American, British, and Canadian efforts.
- 4.
If it is not obvious for you, here is the proof: assume that \(\hat {A}\left |q\right \rangle =a_{q}\left |q\right \rangle \). Then \(\hat {A}\hat {A}\left |q\right \rangle =a_{q}\hat {A}\left |q\right \rangle =a_{q}^{2}\left |q\right \rangle \).
- 5.
Willis Lamb was an American experimental physicist who made significant contribution to quantum electrodynamics and the field of quantum measurements. He holds professorial positions at the University of Oxford and Yale, Columbia and Stanford Universities, and the University of Arizona.
- 6.
Hans Bethe was a German-born physicist who immigrated to the USA in 1935 (only 2 years later than Einstein) and became a professor at Cornell University where he worked till his death in 2005. He won the 1967 Nobel Prize in Physics for his work on the theory explaining the formation of chemical elements due to nuclear reaction within stars. During the war, he headed theoretical efforts within the Manhattan Project and played a critical role in calculating the critical mass of the weapons. After the war he was active in efforts to outlaw testing of nuclear weapons convincing Kennedy and Nixon administrations to sign the Partial Nuclear Test Ban Treaty (1963) and the Anti-Ballistic Missile Treaty (1972). He also made important contribution in solid-state physics.
- 7.
Gosh, I am repeating myself, aren’t I? I said exactly the same words in Chap. 9, but well, it is all for your benefit.
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Deych, L.I. (2018). Fine Structure of the Hydrogen Spectra and Zeeman Effect. In: Advanced Undergraduate Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-71550-6_14
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