Abstract
The Angular Momentum eigenvalue problem is solved and the angular momentum eigenvalues are determined. The angular momentum eigenfunctions in the position representation (Spherical Harmonics) are derived and their properties analyzed.
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Notes
- 1.
- 2.
Note that the Completeness relation is
$$\sum _{ \ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Y_{\ell m}(\Omega )\,Y_{\ell m}^*(\Omega ')\,=\,\delta (\cos \theta -\cos \theta ')\delta (\phi -\phi ')\,.$$.
- 3.
The Legendre Polynomials are a complete orthonormal set of functions in the interval \([-1,1]\). Note the orthonormality relation \(\int _{-1}^{+1}dx\,P_{\ell }(x)P_{\ell '}(x)=\frac{2\delta _{\ell \ell '}}{(2\ell +1)}\).
References
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
W. Greiner, B. Müller, Quantum Mechanics Symmetries, 2nd edn. (Springer, Berlin, 1992)
D. Griffiths, Introduction to Quantum Mechanics, 2nd edn. (Cambridge University Press, Cambridge, 2017)
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Tamvakis, K. (2019). Eigenstates of the Angular Momentum. In: Basic Quantum Mechanics. Undergraduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-22777-7_9
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DOI: https://doi.org/10.1007/978-3-030-22777-7_9
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