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2.11 Derive the expressions (2.198) and (2.199) for the positive-frequency part of the quantized magnetic fields from the corresponding expressions (2.187) and (2.197) for the electric fields using Faraday's law. The calculation becomes qui te simple, once you notice that the positive-frequency components (2.187) and (2.197) of the electric
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{(Xn . E out ) (X3 . E out ) = 8n 3
{[1 + {[1 + I'
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(Eout ) = {Bout} =
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{[1 + cos2 OJ sin2 (k z z - 8) + sin2 Ocos2 (kzz - 8)}, {[1 + cos2 OJ cos2 (kzz - 8) + sin2 Osin2 (kzz - 8)},
where E,B = Et + E,8 and B,B = Bt + B,8' Then substitute these expressions in (2.202) using (2.201) and derive (2.203).
2.13 Use the Euler-Maclaurin fonnula, as was done in Section 2.5.1, to show that (2.206) yields (2.207). 2.14 Using (2.175), show that the multiple reflections between the plates transmitted to the outside exactly cancel the reflection of the incident wave by the plate at z = l, when kz = mr / 1 and to ---* O. In other words, at the cavity resonances there is no reflected wave, just as Michael Berry suggested [50].
Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box Sergio M. Dutra Copyright 2005 John Wiley & Sons, Inc.
The alternative free tasting: First quantization of light and the photon's wavefunction
We shall begin by developing an extreme light quantum theory, and forget for the time all connection with Maxwell equations. . .. OUf present problem is to find the wave equation for the de Broglie waves of the quantum; the theory of the field we shall then obtain by a suitable (second) quantization of the de Broglie amplitudes. -Julius Robert Oppenheimer [472]
Welcome to our free tasting of alternative wines. So far we have presented the traditional route to QED. From the classical Maxwell equations, canonical quantization takes us straight to many-body (i.e., second-quantized) quantum electrodynamics. This is quite a shortcut on the usual route taken in elementary quantum mechanics of nonrelativistic point particles, where one first has to construct a single-particle first-quantized theory (e.g., SchrMinger equation) from which to build the manybody second-quantized theory. Why not follow the ordinary longer route from first to second quantization for electromagnetic radiation as well The main problem is that first quantization is really good only in nonrelativistic quantum mechanics. Pair creation out of the vacuum implies that we always have a many-body problem in the relativistic case. Now, unlike matter, there is no nonrelativistic limit for radiation. The photon has no rest mass and it always travels at the speed of light. There is no inertial frame where a photon is at rest. As a consequence, any interaction with matter, even at very small energies, can lead to the creation of photons and hence the need for a many-body theory. To create an electron-positron pair, for example, one needs at least twice the rest energy of the electron (Le., 2mc2 , where m is the rest mass of the electron). Even though m is very small (about 9.1 x 10- 31 kg) as c is
THE PHOTON'S WAVEFUNCT/ON
large, 2mc2 is still a respectable amount of energy: about 1 MeV. Compare this to 1 eV, the energy of a photon of visible light produced in a typical spontaneous-emission event when an excited atom decays to its ground state. But in this chapter, as in 2, we are not looking at the interaction with matter yet. What about a first-quantized theory of light without sources, then There is still a problem. Another consequence of being a truly relativistic particle with zero rest mass is that the photon cannot be localized with infinite precision: There is no position operator for the photon!' This means that it impossible to define a proper probability density for finding a photon at a given point in space, making the idea of a photon wavefunction in configuration space meaningless. So the first quantization of light is plagued with problems. Still, this is an idea to which people return over and over again because of its appeal and the subtleness of its problems. It is very tempting, for instance, to imagine that the wavefunction of the photon would be the classical electromagnetic field itself. In this chapter we look into all these issues in more detail. In the next section we discuss the problem of measuring the position and momentum of a particle in the relativistic regime. We will see that the momentum, but not the position, can be determined with arbitrary precision just as in nonrelativistic quantum mechanics. There is a general group theory treatment by Newton and Wigner [464] about the localizability of elementary systems which predicts that there is no position operator for the photon. We do not present this treatment here because even though it is very general, it is also very abstract and mathematical. The curious reader can have a look at the references recommended at the end of the chapter. We chose instead to base our discussion on a concrete and specific case: What would happen if the photon had a rest mass In the spirit of Oppenheimer's "extreme quantum theory of light" [472], we pretend that we know nothing about the classical limit of the theory and trace the reverse route to that of the usual quantization approaches: We start from a quantum wave equation and go from there to its classical limit. The key difference between our extreme quantum theory oflight and Oppenheimer's is that we do not assume that the photon's rest mass vanishes. Our derivation of the photon's wave equation uses Lanczos's generalization of Dirac's method2 to get from the Klein-Gordon equation to an equation that is first order in time. This leads us to Proca equations. The problems with a configuration space wavefunction become evident. Then we show how, when the rest mass does not vanish, the problems disappear in the nonrelativistic limit. But as the underlying logic of the present-day physical theory as well as experimental evidence seems to suggest that the photon has no rest mass, we take the limit of vanishing rest mass and write down a first-quantized theory whose classical limit is Maxwell equations. Finally, in the last section, we construct a second-quantized theory, starting from the first-quantized one. We show then that the second-quantized theory coincides with the usual quantum electrodynamics derived in 2 through canonical quantization.
I I mean. in the sense of the Newton-Wigner-Wightman group theory approach [464. 643]. 2That is the method Dirac developed to derive the Dirac equation for the e1eclron.