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Solutions of the Maxwell equations and photon wave functions. (English) Zbl 1198.81096
The Maxwell equations of electrodynamics are re-written in the representation of a six-component field. In this representation, the Maxwell field equations become similar to a single Dirac equation but with six components. The symmetry properties of this Dirac-like equation are then thoroughly analyzed. Solutions of this equation are explicitly worked out. These solutions are obtained without recursion to the (unphysical) vector potential. In this way, the ambiguities associated to gauge invariance are avoided. A complete set of eigenfunctions of the Hamiltonian of the system as well as of the linear and angular momentum are calculated. These eigenfunctions are further classified into “transverse” (associated to electrostatic interactions) and “longitudinal” (associated to radiation field) states. A suitable sum of these eigenvalues provides a Green function for the Maxwell equations in this six-component form. It is also shown that, for a wavepacket, the expectation value of the Hamiltonian is exactly \(\hbar\omega_0\) (with \(\omega_0\) frequency associated with the wavevector of the packet) as expected for a photon.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V10 Electromagnetic interaction; quantum electrodynamics
78A25 Electromagnetic theory (general)
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References:
[1] Migdall, A.; Dowling, J., J. mod. opt., 51, 1265-1266, (2004)
[2] Bialynicki-Birula, I., Prog. opt., 36, 245-294, (1996)
[3] Scully, M.O.; Zubairy, M.S., Quantum optics, (1997), Cambridge University Press Cambridge
[4] Keller, O., Phys. rep., 411, 1-232, (2005)
[5] Oppenheimer, J.R., Phys. rev., 38, 725-746, (1931)
[6] Bargmann, V.; Wigner, E.P., Proc. natl. acad. sci. USA, 34, 211-223, (1948)
[7] Weinberg, S., Phys. rev., 133, B1318-B1332, (1964)
[8] Weinberg, S., Phys. rev., 134, B882-B896, (1964)
[9] Zumino, B., J. math. phys., 1, 1-7, (1960)
[10] DeWitt, B.S., Phys. rev., 125, 2189-2191, (1962)
[11] Mandelstam, S., Ann. phys. (N.Y.), 19, 1-24, (1962)
[12] Aharonov, Y.; Bohm, D., Phys. rev., 115, 485-491, (1959)
[13] Jackson, J.D.; Okun, L.B., Rev. mod. phys., 73, 663-680, (2001)
[14] Bialynicki-Birula, I., Acta phys. Pol. A, 86, 97-116, (1994)
[15] Wigner, E.P., Group theory and its application to the quantum mechanics of atomic spectra, (1959), Academic Press New York · Zbl 0085.37905
[16] Mignani, R.; Recami, E.; Baldo, M., Lett. nuovo cimento, 11, 568-572, (1974)
[17] Jackson, J.D., Classical electrodynamics, (1999), John Wiley & Sons New York · Zbl 0114.42903
[18] Weber, H., Die partiellen differential-gleichungen der mathematischen physik nach riemann’s vorlesungen, (1901), Friedrich Vieweg und Sohn Braunschweig, vol. 2
[19] Silberstein, L., Ann. phys. (Leipzig), 327, 579-586, (1907)
[20] Silberstein, L., Ann. phys. (Leipzig), 329, 783-784, (1907)
[21] Inagaki, T., Phys. rev. A, 49, 2839-2843, (1994)
[22] Dragoman, D., J. opt. soc. am. B, 24, 922-927, (2007)
[23] Wang, Z.-Y.; Xiong, C.-D.; Qiu, Q., Phys. rev. A, 80, 032118, (2009)
[24] Mohr, P.J., Phys. rev. lett., 40, 854-856, (1978)
[25] Dunford, R.W.; Gemmell, D.S.; Jung, M.; Kanter, E.P.; Berry, H.G.; Livingston, A.E.; Cheng, S.; Curtis, L.J., Phys. rev. lett., 79, 3359-3362, (1997)
[26] Gel’fand, I.M.; Shilov, G.E., Generalized functions, (1964), Academic Press New York, vol. 1 · Zbl 0115.33101
[27] Corben, H.C.; Schwinger, J., Phys. rev., 58, 953-968, (1940)
[28] Edmonds, A.R., Angular momentum in quantum mechanics, (1960), Princeton University Press Princeton · Zbl 0079.42204
[29] Jacob, M.; Wick, G.C., Ann. phys. (N.Y.), 7, 404-428, (1959)
[30] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1965), Dover Publications New York · Zbl 0515.33001
[31] Mohr, P.J., Ann. phys. (N.Y.), 88, 26-51, (1974)
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