This paper analyzes how the existence of electron spin changes the equation for the probability current density in the quantum-mechanical continuity equation. A spinful electron moving in a potential energy field experiences the spin-orbit interaction, and that additional term in the time-dependent Schrödinger equation places an additional spin-dependent term in the probability current density. Further, making an analogy with classical magnetostatics hints that there may be an additional magnetization current contribution. This contribution seems not to be derivable from a non-relativistic time-dependent Schrödinger equation, but there is a procedure described in the quantum mechanics textbook by Landau and Lifschitz to obtain it. We utilize and extend their procedure to obtain this magnetization term, which also gives a second derivation of the spin-orbit term. We conclude with an evaluation of these terms for the ground state of the hydrogen atom with spin-orbit interaction. The magnetization contribution is generally the larger one, except very near an atomic nucleus.

## REFERENCES

The terms “paramagnetic” and “diamagnetic” for two of the terms in the electric current density in Eq. (47) have been used for a long time.^{11} The distinction being made has nothing to do with spin, so an explanation can be given on the basis of a Hamiltonian for a spinless particle. The textbook by Cohen-Tannoudji *et al.* (Ref. 5, p. 828), for example, gives an explanation of the origin of these names. Their discussion begins from a Hamiltonian similar to our Eq. (33) but without the term proportional to $S$, and with a vector potential that describes a uniform magnetic field $A(x)=B\xd7x/2$. They expand the square in Eq. (33) and obtain several terms. One of the terms is proportional to the orbital angular momentum $L$, and it describes the orientational energy of the orbital magnetic moment (proportional to $L$) in the field. The lowest value of this energy occurs with this moment parallel to the field, so this term describes paramagnetism. It can be shown (see Ref. 3, Appendix A) that the expectation value of the orbital angular momentum is related to the probability current density by $\u27e8L\u27e9=\u222bd3x\u2032x\u2032\xd7jP$, so this relation shows the appropriateness of the name “paramagnetic current density.” Another term obtained in expanding the first term of Eq. (33) is proportional to $A2$. It describes the Lenz Law effect on the electron motion from applying the field, which induces a magnetic moment opposite to the field, and that is diamagnetism. That energy is analogously related to the “diamagnetic current density.”

*mc*

^{2}term can be removed by a time-dependent change of phase of the wave function.

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