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On the Radiation Emitted by a Multipole and its Angular Momentum

Published online by Cambridge University Press:  24 October 2008

W. Heitler
W. Heitler
Affiliation:
Bristol
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Extract

1. By expanding the radiation field in a series of spherical waves the complete expressions for the field emitted by an electric and magnetic 2l-pole are obtained (§§ 1, 2).

2. It is shown that the wave emitted by a (electric or magnetic) 2l-pole has an angular momentum about the z-axis Mz = mU/ν (U = total energy), where m, according to the orientation of the 2l-pole in space, can assume the values −l, −l + 1, …, + l (§ 3). The angular momentum is contained in that region of the field in which the product EH decreases as r−3.

3. By quantizing the waves it is shown that the angular momentum of a light quantum emitted by a 2l-pole behaves like the angular momentum of an electron in a central field of force without spin (commutation relations, etc.) (§ 4). The angular momentum of a single light quantum is an integral multiple of ħ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 1 , January 1936 , pp. 112 - 126
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

† Abraham, M., Phys. Zs. 15 (1914), 914.Google Scholar Compare also Sommerfeld, A., Atomic structure and spectral lines, Appendix 15.Google Scholar

‡ Brinkmann, , Zur Quantenmechanik der Multipolstrahlung, Proefschrift, Utrecht, 1932.Google Scholar

† The expressions for can be found in Handb. d. Phys. xxiv (1), article by Bethe.

† In (11) the potentials are normalized in such a way that the scalar potential ø comprises the static instantaneous part of the field of the multipole only. The effect of retardation is contained in A. The same applies for the remainder of this paper.

‡ The factor C, equ. (8), is the same for both terms occurring in (12).

† A charge system having a 2l-pole moment is represented by a tensor of the lth rank. If the symmetry of the charge system is such that all 2 (l − 1), 2(l − 2), …-pole moments vanish, the tensor has certain symmetries also. It contains then just 2l + 1 independent parameters. A quadripole for instance is described by an ordinary symmetrical tensor with a diagonal sum equal to zero. It contains 5 independent parameters (l = 2) (cf. Brinkmann, loc. cit.).

† The same is true if an infinite plane wave is absorbed by a finite screen. The angular momentum is then contained in the diffraction wave arising from the edge of the screen. The angular momentum transferred to the screen is also given by (52), U representing the total energy absorbed. Cf. Epstein, , Ann. d. Phys. 44 (1914), 593CrossRefGoogle Scholar. The effect seems to have been observed by Beth, R. A., Phys. Rev. 48 (1935), 471CrossRefGoogle Scholar. Cf. also Darwin, C. G., Proc. Roy. Soc. 136 (1932), 36.CrossRefGoogle Scholar

‡ Det. kgl. dansk. Vidensk. Selskab (in the press). I am very much indebted to Prof. Rosenfeld for his kind communication of the results of this paper.

† The sign of is defined so that .

‡ See, for instance, Bethe, , Handb. der Physik, XXIV (1), 555.Google Scholar

§ Ibid. p. 558.

∥ See, for instance, Whittaker, , Modern Analysis, p. 275.Google Scholar