Abstract
Electromagnetism is the most important interaction for the study of atoms, molecules and materials. It determines most of the potentials or perturbation operators V which are studied in practical applications of quantum mechanics, and it also serves as a basic example for the implementation of other, more complicated interactions in quantum mechanics. Therefore the primary objective of the current chapter is to understand how electromagnetic fields are introduced in the Schrödinger equation.
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- 1.
The canonial momentum \(\langle \boldsymbol{p}\rangle (t) = -\mathrm{i}\hslash \int \!d^{3}\boldsymbol{x}\,\Psi ^{+}(\boldsymbol{x},t)\boldsymbol{\nabla }\Psi (\boldsymbol{x},t) = m(d\langle \boldsymbol{x}\rangle (t)/dt) + q\langle \boldsymbol{A}(\boldsymbol{x},t)\rangle\) is also generically not conserved, except if the particle moves in a spatially homogeneous electric field \(\boldsymbol{E}(t) = -d\boldsymbol{A}(t)/dt\), e.g. in a plate capacitor. However, note that this is an artifact of the gauge \(\Phi = 0\).
- 2.
Recall that | S nm | 2 is a true transition probability only if the initial and final state are discrete, while otherwise it enters into decay rates or cross sections.
- 3.
W. Kuhn, Z. Phys. 33, 408 (1925); F. Reiche, W. Thomas, Z. Phys. 34, 510 (1925).
- 4.
If the wave functions are N-particle wave functions and the potential V is the corresponding sum of dipole operators, the number on the right hand side of the sum rules becomes N.
- 5.
H. Bethe, Annalen Phys. 397, 325 (1930).
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Dick, R. (2016). Coupling to Electromagnetic Fields. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_15
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DOI: https://doi.org/10.1007/978-3-319-25675-7_15
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