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The eigenvalues of electromagnetic angular momentum

Published online by Cambridge University Press:  24 October 2008

M. H. L. Pryce
M. H. L. Pryce
Affiliation:
Trinity College
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Extract

It is shown that the eigenvalues of the angular momentum of the electromagnetic field containing a number of charged particles, apart from spin angular momentum, are only the integral multiples of ħ. This is shown by using cylindrical polar variables, and taking a particular choice of the vector potential in which the radial component is zero, defined explicitly in terms of the magnetic field strengths. By expanding in terms of the Fourier functions einθ, the angular momentum is separated out into terms independent of one another, each taking on only integral values in units of ħ.

The arguments all apply equally well to a modified field theory such as that of Born and Infeld.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 4 , December 1936 , pp. 614 - 621
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

† Born, M. and Infeld, L., “On the quantization of the new field theory. II”, Proc. Roy. Soc. A, 150 (1935), 141.CrossRefGoogle Scholar

‡ Pryce, M. H. L., “On the new field theory. II”, Proc. Roy. Soc. A (in the Press)Google Scholar; quoted as A.

† The field strengths D and B at the same point commute. This happens because the commutation brackets for different points contain the derivative of a delta function, which vanishes at the point zero. The order of the factors is therefore immaterial. That this is so may be verified at every stage of the calculation.

‡ See Pauli, , Handbuch der Physik, 24 (1), (1933), 2nd edition, p. 233Google Scholar, equation 77 and p. 261, equation 261.

§ This can also be written symbolically:

† is the same as δ(x − x′) δ(y − y′) δ(z − z′)

‡ The integrals over the boundary at infinity are supposed to vanish. If this assumption were not made, the integral in the angular momentum would not be convergent. It can be justified by restricting ourselves to states in which mean field tends to zero at infinity sufficiently rapidly, or by introducing a periodicity condition, or equivalently a closed space.