Abstract
The general problem of the addition of two independent angular momenta is considered. The arising Clebsch-Gordan coefficients are analyzed and general formulas for them are derived.
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Notes
- 1.
Symbolized for notational economy as \(|m_1;m_2\rangle \).
- 2.
- 3.
The proof that \([\hat{J}^2,\,\hat{J}_z]=0\) proceeds as in the case of the addition of two spins.
- 4.
An alternative equivalent notation to \(|j_1,j_2,m_1,m_2\rangle \) is \(|j_1,m_1;j_2,m_2\rangle \).
- 5.
For notational compactness, we may drop the commas as \(\langle j_1,j_2,m_1,m_2|j_1,j_2,j,m\rangle =\langle j_1j_2m_1m_2|j_1j_2jm\rangle \).
- 6.
Note that they can be taken to be real.
- 7.
A particle of the same mass and spin as the electron but opposite electric charge.
References
W. Greiner, B. Müller, Quantum Mechanics: Symmetries, 2nd edn. (Springer, Berlin, 1992)
A. Messiah, Quantum Mechanics. Dover publications, single-volume reprint of the Wiley, New York, two-volume 1958 edn
S. Weinberg, Lectures on Quantum Mechanics (Cambridge University Press, Cambridge, 2015)
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Tamvakis, K. (2019). Addition of Angular Momenta. In: Basic Quantum Mechanics. Undergraduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-22777-7_11
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DOI: https://doi.org/10.1007/978-3-030-22777-7_11
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