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arXiv:quant-ph/0511259

Comment on the quantum nature of angular momentum using a coupled-boson representation

Ilki Kim, GERALD J. IAFRATE

Published Nov 29, 2005• 0 citations
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Abstract

A simple approach for understanding the quantum nature of angular momentum and its reduction to the classical limit is presented based on Schwinger's coupled-boson representation. This approach leads to a straightforward explanation of why the square of the angular momentum in quantum mechanics is given by j(j+1) instead of just j^2, where j is the angular momentum quantum number.

DOI
10.48550/arxiv.quant-ph/0511259

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A simple approach for understanding the quantum nature of angular momentum and its reduction to the classical limit is presented based on Schwinger's coupled-boson representation. This approach leads to a straightforward explanation of why the square of the angular momentum in quantum mechanics is given by j(j+1) instead of just j^2, where j is the angular momentum quantum number.
A simple method is given for finding the values of the total angular momentum J for configurations of equivalent electrons in jj coupling. It is also shown how a knowledge of both the LS and jj terms can contribute to the student’s understanding of the role of different representations in elementary quantum mechanics. Because the principles and the computations required are elementary, this material is particularly suitable for presentation to undergraduates and to graduate students taking a first course in quantum mechanics.
Abstract The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.