Photon momentum, quantum light theory

We can also try to deduce the radiation formula, not as above from the pure wave standpoint by quantisation of the cavity radiation, but from the standpoint of the theory of light quanta, that is to say, of a corpuscular theory. For this we must therefore develop the statistics of the light-quantum gas, and the obvious suggestion is to apply the methods of the classical Boltzmann statistics, as in the kinetic theory of gases the quantum hypothesis, introduced by Planck in his treatment of cavity radiation by the wave method, is of course taken care of from the first in the present case, in virtue of the fact that we are dealing with light quanta, that is, with particles (photons) with energy hv and momentum Av/c. It turns out, however, that the attempt to deduce Planck s radiation law on these lines also fails, as we proceed to explain. 

Classical electromagnetics and Boltzmann statistics, respectively, lead to explicit forms -of Eq. (8.29) for A - oc and A -> 0. However, both theories fail to provide the explicit form for an arbitrary A. Extensive research for this explicit form eventually led Planck to the discovery of quantum mechanics, which explains radiation in terms of particles (photons) traveling with the speed of light. The energy and momentum associated with each photon, respectively, are... 

Experiments by MilUken in 1908 soon confirmed Einstein s predictions. In 1921, A.H. Compton succeeded in determining the motion of a photon and an electron both before and after a collision between them. He found that both behaved like material bodies in that both kinetic energy and momentum were conserved in the collision. The photoelectric effect and the Compton effect, then, seemed to demand a return to the corpuscular theory of light. The reconciliation of these apparently contradictory experiments has been accomplished only since about 1930 with the development of quantum electrodynamics, a comprehensive theory that Includes both wave and particle properties of photons. Thus, the theory of light propagation is best described by an electromeignetic wave theory while the Interaction of a photon with matter is better described as a corpuscular phenomenon. 

The transformation of the time-dependent function Pj into a momentum operator is consistent with Einstein s description of light in terms of particles (photons), each of which has momentum hv (Sect. 1.6 and Box 2.3). We can interpret the quantum number rij in Eq. (3.50) either as the particular excited state occupied by oscillator j, or as the number of photons with frequency Vj. The oscillating electric and magnetic fields associated with a photon can stiU be described by Eqs. (3.44) and (3.45) if the amplitude factor is scaled appropriately. However, we will be less concerned with the spatial properties of photon wavefunctions themselves than with the matrix elements of the position operator Q. These matrix elements play a central role in the quantum theory of absorption and emission, as we ll discuss in Chap. 5. 

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