Coherence and Quantum Optics pp 241-257 | Cite as

# Debye Representation and Multipole Expansion of the Quantized Free Electromagnetic Field

## Abstract

Quite often in classical electrodynamics, a problem is simplified, considerably, by choosing an appropriate representation for the field. Well-known examples are the Cauchy initial-value problem and the Dirichlet or Neumann boundary-value problem. For the Cauchy problem an ideal representation for the field is a plane-wave expansion [1]; this being so because the plane-wave fields form a complete set of solutions to the wave equation into which the Cauchy data can be easily expanded. For the case of Dirichlet or Neumann boundary conditions given on an infinite plane surface, together with Sommerfeld’s radiation condition at infinity, a plane-wave expansion is again appropriate, although in this case it is necessary to include evanescent (inhomogeneous) plane waves in the expansion [2], Plane-wave expansions are not always the most appropriate representation to use, however. For example, with Dirichlet or Neumann conditions prescribed on the surface of a sphere, it is desirable to expand the field into a multipole expansion since the multipole fields form a complete, orthogonal set of functions on such surfaces [3].

## Keywords

Multipole Moment Multipole Expansion Angular Momentum Operator Debye Representation Magnetic Field Operator## Preview

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## References

- 1.R. Courant and D. Hilbert,
*Methods of Mathematical Physics*, (Interscience Publishers, New York, 1962) Vol.II.,Chap.III,§5.MATHGoogle Scholar - 2.J.A. Stratton,
*Electromagnetic Theory*, (McGraw-Hill, New York, 1941)§6.7.MATHGoogle Scholar - 3.See, for example, reference 2, §7.11.Google Scholar
- 4.W. Heitler,
*The Quantum Theory of Radiation*,(Oxford University Press, London, 1954 ) Chap. II.MATHGoogle Scholar - 5.See, for example, J.M. Blatt and V.F. Weisskopf,
*Theoretical Nuclear Physics*, John Wiley and Sons, New York, 1952, App. B.See where also, A.M. Messiah,*Quantum Mechanics*, (John Wiley and Sons, 1966 ) Vol. II, Chap. X XI.MATHGoogle Scholar - 6.For a treatment of the Debye representation of the classical field see: T.J. I’a. Bromwich, Phil. Mag., Ser 6,
*38*(1919) 143; C.H. Wilcox,J. Math. Mech.,*6*,(1957), 167. T. e relationship of this representation to that of the radial Hertz vector representation has been established by: A. Sommerfeld, contribution in P. Frank and R.V. Mises,*Riemann-Weber’s Differential-gleichungen der Mathematischer Physik*,(Dover Publications, 1961); A. Nisbet, Physica,*21*(1955), 799. See also, C.J. Bouwkamp and H.B.G. Casimir,Physica,*20*(1954) 539.CrossRefGoogle Scholar - 7.As, for example, done in B.W. Shore and D.H. Menzel,
*Principles of Atomic Spectra*, (John Wiley and Sons, New York, 1968)Chap.X.Google Scholar - 8.A.J. Devaney,
*A New Theory of the Debye Representation of Classical and Quantized Electromagnetic Fields*, Ph.D. Thesis, University of Rochester, 1971. The classical theory will also appear in a future paper by Devaney and E.Wolf.Google Scholar - 9.E.T. Whittaker, Math. Ann., 57(1902), 333. See also, F. Rohrlich,
*Classical Charged Particles*, (Addison-Wesley, Reading, Mass., 1965) §4.3.MathSciNetCrossRefGoogle Scholar - 10.We use the definitions of Y
_{ℓ}^{m}(α,β) and j_{ℓ}(ω/c r) that are given in A. Messiah,*Quantum Mechanics*, (North-Holland Publishing Co., Amsterdam) Vol.I. App. B II and B.IV, (third printing 1965).Google Scholar - 11.See reference 8, §3.3.Google Scholar
- 12.Definitions of the vector spherical harmonics Y
_{ℓ}^{m}(α,β) vary from author to author, but are, essentially, equivalent to the one used here. For a treatment of vector spherical harmonics see: E.L. Hill, Amer. Jour. Phys.,*22*(1954) 211.ADSMATHCrossRefGoogle Scholar - 13.See, for example, reference 7.Google Scholar
- 14.C. Wilcox (see reference 6) also used Hodge’s theorem in a study of the Debye representation. His use of the theorem differs completely from that given here, however.Google Scholar
- 15.Here we assume that the operator a(ω/c s) possesses the necessary continuity properties required in Hodge’s theorem. This assumption seems justifiable since, in a coherent state representation, the (diagonal) matrix elements of this operator are of the form of a classical plane-wave amplitude and, in the classical case, the plane-wave amplitudes are generally entire analytic functions of ω/c s (cf. reference 8).Google Scholar
- 16.See, for example, the paper by Wilcox. The Green function is derived also in R. Courant and D. Hilbert, Methods of Mathematical Physics, (Interscience Publishers, New York, 1953 ) Vol. I., 378.Google Scholar
- 17.For a derivation of the commutators given in (4.8) and (4.9) the reader is referred to reference 8, App. VIII.Google Scholar