Abstract
Classically, a point charge that circulates on a ring of radius r produces a current \(i = \frac{ev}{2 \pi r}\), which causes a magnetic dipole moment \(\overrightarrow{\mu }= \frac{i}{c} S \overrightarrow{n}\), \(S= \pi r^{2}\), in obvious notation; \(\overrightarrow{\mu }= \frac{e v}{2c} r \overrightarrow{n}= \frac{e}{2c} \overrightarrow{r} \wedge \overrightarrow{v}= \frac{e}{2 m c} \overrightarrow{L}.\).
An electron has radius 0 (as far as we know), a quantized angular momentum (spin) \(\frac{1}{2}\hbar \) and a magnetic moment. Let us discover how.
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- 1.
The Bohr model predicted orbits, and classically there is no other way than a current to produce a magnetic moment; but Nature had a surprise in store.
- 2.
Most nuclei have spins and magnetic moments; the nuclear magnetic moments, however, are negligible compared to the electronic ones. This is related to the proton and neutron masses, which are much heavier than the electron.
- 3.
R.A. Beth Phys. Rev. 5, 115 (1936).
- 4.
The following theory is due to Wolfgang Pauli (1900 Vienna - 1958 Zurich) one of the most important theoreticians of the twentieth century.
- 5.
In general the system has many other degrees of freedom and the full Hilbert space is the tensor product of the spin Hilbert space times the Hilbert space arising from the other degrees of freedom. See Sect. 12.6.
- 6.
Specifically, I refer to Protium \(^{1}H\); the Deuterium \(^{2}H\) nucleus has a neutron and a proton and spin 1, and the unstable Tritium \(^{2}H\) has spin 1/2, too.
- 7.
Isidor Isaac Rabi (1898–1988) was a U.S. Physicist, who received the Nobel Prize in 1944.
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Cini, M. (2018). Spin and Magnetic Field. In: Elements of Classical and Quantum Physics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-71330-4_18
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DOI: https://doi.org/10.1007/978-3-319-71330-4_18
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