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Student mathematical library

The energy, the angular momentum, and other examples of observables

Lyudvig Dmitrievich Faddeev, O. Yakubovskiĭ

Published Mar 9, 2009• 0 citations
Publisher
DOI
10.1090/stml/047/13

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More than a century ago, pioneeringworks carried out by Poynting (1909) and other physicists have provided the evidence of the validity of Marxwell equations for an electromagnetic field, showing how a beam of light carries energy and momentum, both in the linear and angular components. In particular the angular momentum of light, related to the generator of rotations in quantum mechanics, has been typically associated with its polarization, and more specifically with its circular polarization components. An optical beam traveling in the positive direction of the z axis that is circularly polarized, carries a z-component angular momentum content σ = ±h per photon, which is positive if the circular polarization is left-handed and negative if it is right-handed. This angular momentum content is not just a formal property, but a very concrete one that can have significant mechanical effects as for example an observable induced rotation by absorption to amaterial particle Beth (1936); Friese et al. (1998)
Synopsis: General operator properties of angular momenta are recalled, and a physical motivation for adding them is given. The product and total angular momentum bases are defined. It is shown how to express one basis in terms of the other, and an explicit example of adding two spin 1/2 angular momenta is given. A further example consists of adding a general orbital angular momentum value to spin 1/2. These considerations are applied to perturbations in the hydrogen atom energy spectrum, where the correct choice for calculations is the total angular momentum basis.