Abstract
Chapter 6 applies the material of the previous chapters to some particular topics, specifically the Wigner–Eckart theorem, selection rules, and gamma matrices and Dirac bilinears. We begin by discussing the perennially confusing concepts of vector operators and spherical tensors, and then unify them using the notion of a representation operator. We then use this framework to derive a generalized selection rule, from which the various quantum-mechanical selection rules can be derived, and we also discuss the Wigner–Eckart theorem. We conclude by showing that Dirac’s famous gamma matrices can be understood in terms of representation operators, which then immediately gives the transformation properties of the ‘Dirac bilinears’ of QED.
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Notes
- 1.
Warning: In other texts, different normalization and indexing conventions are used for the specific basis vectors whose transformation properties the spherical tensors are supposed to mimic. In most quantum mechanics texts, for instance, the components of a spherical tensor are supposed to mimic the transformation properties of the kets |l,m〉 which have different labeling and normalization conventions than our v k , even though they are essentially the same thing.
- 2.
The funny looking normalization factors and index convention come from our normalization and indexing convention for the v k s, which, as noted above, differ from the conventions in the physics literature.
References
J.J. Sakurai, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, Reading, 1994
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Jeevanjee, N. (2011). The Wigner–Eckart Theorem and Other Applications. In: An Introduction to Tensors and Group Theory for Physicists. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4715-5_6
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DOI: https://doi.org/10.1007/978-0-8176-4715-5_6
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4714-8
Online ISBN: 978-0-8176-4715-5
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