Abstract
The concept of Spin of a particle is introduced and analyzed. The spin eigenvalue problem is discussed and its eigenvalues are derived. The system of a particle characterized only by its spin under the influence of a homogeneous magnetic field is considered. Resonance phenomena in the case of a time-dependent magnetic field (Nuclear Magnetic Resonance) are analyzed.
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Notes
- 1.
Apart from the nucleus that plays no role in the phenomena to be discussed.
- 2.
- 3.
An alternative notation that can be more convenient at times is \(|\uparrow \rangle =|+\rangle \) and \(|\downarrow \rangle =|-\rangle \).
- 4.
With \(\left\{ A,\,B\right\} =AB+BA\) we symbolize the anticommutator of two matrices or operators.
- 5.
\(\mathbf { 1.}\) The only outcomes of the measurement of an observable are its eigenvalues. \(\mathbf { 2.}\) The state of the system before the measurement of an observable A is a superposition of A-eigenstates \(\psi \,=\,\sum _{\alpha }C_{\alpha }\psi _{\alpha }\), where \(|C_{\alpha }|^2\) is the probability of each measured eigenvalue. \(\mathbf { 3.}\) The state of the system after a measurement of A that has given the eigenvalue \(\alpha \) is the corresponding eigenstate \(\psi _{\alpha }\). Any subsequent measurement of A will yield exactly the same eigenvalue \(\alpha \) and the system will continue to occupy the state \(\psi _{\alpha }\).
- 6.
In the case of a nucleus \(e,\,g\), and M stand for the corresponding charge, Lande factor (\(g\ne 2\)), and mass. Note, however, that even electrically neutral particles (e.g., the neutron) can have nonzero magnetic dipole moment due to their spin.
- 7.
\(\omega \) is the so-called Larmor frequency for the electron (\(e<0,\,g\approx 2\)).
- 8.
See also [1].
- 9.
We take \(e>0\) this time, having in mind applications in the case of nuclei.
References
D. Griffiths, Introduction to Quantum Mechanics, 2nd edn. (Cambridge University Press, Cambridge, 2017)
E. Merzbacher, Quantum Mechanics, 3rd edn. (Wiley, New York, 1998)
A. Messiah, Quantum Mechanics (Dover publications, Mineola, 1958). Single-volume reprint of the John Wiley & Sons, New York, two-volume 1958 edition
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Tamvakis, K. (2019). Spin. In: Basic Quantum Mechanics. Undergraduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-22777-7_10
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DOI: https://doi.org/10.1007/978-3-030-22777-7_10
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