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Physical Review

Compton Scattering of an Intense Photon Beam

Lee M. Frantz

Published Sep 6, 1965• 51 citations
Publisher

Abstract

The fully quantum-mechanical description of the Compton scattering of a photon beam is shown to be equivalent under certain conditions to a semiclassical description, thereby confirming the prediction by Brown and Kibble and by Goldman of an intensity-dependent increment to the frequency shift. If the incident beam is in a coherent state, the equivalence is exact under all conditions. If it contains a definite number of photons, the equivalence is approximate, and requires many photons and convergence of the expansion in powers of the photon density. The demonstration is based upon the explicit use of wave packets to introduce the boundary conditions, and the equivalence is shown to hold in the sense that the transition probabilities, but not the transition amplitudes, are the same in the quantum and semiclassical treatments. The apparent failure of energy-momentum conservation implied by the incremental shift is seen to be accounted for by the energy-momentum uncertainty of the ensemble of localized wave packets even in the monochromatic limit. It is proved in the Appendix that the coherent states are the only kind for which the equivalence is exact.

DOI
10.1103/physrev.139.b1326

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•Jan 1, 2018•0 citations
The fully quantum-mechanical description of the Compton scattering of a photon beam is shown to be equivalent under certain conditions to a semiclassical description, thereby confirming the prediction by Brown and Kibble and by Goldman of an intensity-dependent increment to the frequency shift. If the incident beam is in a coherent state, the equivalence is exact under all conditions. If it contains a definite number of photons, the equivalence is approximate, and requires many photons and convergence of the expansion in powers of the photon density. The demonstration is based upon the explicit use of wave packets to introduce the boundary conditions, and the equivalence is shown to hold in the sense that the transition probabilities, but not the transition amplitudes, are the same in the quantum and semiclassical treatments. The apparent failure of energy-momentum conservation implied by the incremental shift is seen to be accounted for by the energy-momentum uncertainty of the ensemble of localized wave packets even in the monochromatic limit. It is proved in the Appendix that the coherent states are the only kind for which the equivalence is exact.
The quantum theory of radiation, formulated in wave-mechanical language but in terms of running waves, is shown to lead under suitable conditions to the classical theory of emission by a particle as a first approximation. Non-relativistic mechanics is used, and the limitations of the method are brought out in terms of classical theory. The characteristic quantum phenomenon of the Compton effect requires that the Compton recoil shall exceed the spread in momentum of the particle, and so that the wave-packet shall greatly exceed in diameter the wave-length of the incident radiation; this condition is realizable on any scale of magnitude. For quasi-classical motion just the reverse is required; furthermore, if the motion is to last for a number of waveperiods, the wave-length must greatly exceed hmc\frac{h}{\mathrm{mc}}mch​ (m=massoftheparticlem=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{particle}m=massoftheparticle). The classical packet-motion arises from combinations between quantum-states of the particle-field system; this fact suggests that the photon is not a fundamental constituent of the field but only one possible aspect of its action. The infinite "zero-energy" associated apparently even with the lowest energy-state of the field implies no electromagnetic field intensities at any point. Whereas for problems on energy or momentum it is customary to represent an incident wave-train by a single typical quantum state, to obtain motional phenomena of classical type a more complete expression such as that given in the paper must be used. The physically empty field likewise corresponds to a mixture of quantum states but without correlation among their amplitudes.
Scattering of a plane electromagnetic wave by free and bound wave packets is analyzed by semiclassical radiation theory. It is shown that the theory gives the correct answer to the question of radiation intensity in the photoelectric effect and to the correlation problem in Compton scattering. The expression for the intensity of the scattered radiation differs from the cross section which is derived from the model based on the particle nature of the electromagnetic field. The meaning of this difference is discussed. Low frequency spectrum of the scattered radiation on a bound charge is obtained.
We consider the description of the quantum dynamics of a two-level system interacting with a single Glauber coherent mode of the radiation field. An exact solution (within the rotating wave and electric dipole approximations) is provided for both the time-independent and time-dependent Green’s functions for the coupled matter–radiation system. The atom (molecular) transition probabilities are shown to be Poisson averages of the corresponding semiclassical problem when the latter is likewise treated in the rotating wave approximation. At low radiation intensities, the quantum and semiclassical results differ, but at high intensities, they become the same when the average photon number n0 in the field satisfies the condition n01/2≫1. A similar behavior is displayed for the expectation value of the electric field of the coherent mode in the presence of the atom. The spectrum of the radiation, resonantly scattered by the atom, is shown to coincide with the results of Mollow.
Many quantum phenomena, traditionally described by quantum electrodynamics, can be calculated within the framework of so-called semiclassical theory, in which atoms are described by the wave equations of quantum mechanics (Schrödinger, Dirac, etc.), while light is described by classical electrodynamics without quantization of the radiation. These phenomena include the photoelectric effect, the Compton effect, the Lamb shift, radiative effects, spontaneous emission, the Hanbury Brown and Twiss effect, etc. In this paper, I show that the double-slit experiment can also be calculated in detail in terms of semiclassical theory if we take into account the discrete (atomic) structure of matter. I show that the results of a semiclassical simulation of the “linear” double-slit experiment coincide with predictions of wave theory only for low-intensity light and for short exposure time, while stricter dependences exist for long-term exposure. The semiclassical theory is used for calculation of the “nonlinear” double-slit experiment with an intense laser beam in which multiphoton and tunnel excitation of atoms on a photographic plate occurs. The Born rule for light is derived from the semiclassical theory.
Correspondence theorem for frequencies.---It is well known that the frequency emitted by a hydrogen atom as it falls from one of its quantized states to another may be expressed as a mean value of the frequency of motion of the electron (or overtone thereof) when averaged in the proper manner over orbits intermediate between the initial and final states. In the present paper it is proved that, similarly, the Compton shift is a properly taken mean of the classical Doppler shift. The quantum frequency actually scattered is thus a properly taken average of the frequency which would be scattered on the classical theory as the electron is accelerated from its state of rest to its final recoil condition.Correspondence principle for intensities.---In like manner, the amount of light scattered in various directions may be determined if it is assumed that the intensity in the quantum theory equals a proper average of the intensities scattered according to the classical theory. A comparison is made with observed data on the scattering of $\ensuremath{\gamma}$-rays.The characteristic feature of the present paper is that the corresponding classical electron is assumed to have the same direction of motion as the scattered quantum, whereas an actual classical electron would from symmetry recoil straight forward in the direction of the incident beam, as in Compton's and Woo's theories of intensities. This new point of view eliminates the difficulty of a constant correction-factor which has been encountered by Compton and Woo in their explanation of intensity relations.
The Compton scattering is an elastic collision of the x-ray or gamma photons with free electrons (or loosely bound valence shell electrons). This scattering is demonstrated in Fig. 1. In this communication we are combining Heisenberg's uncertainty principle and the Compton scattering. Note, that some of the expressions (including their derivations) of this communication can be found in modern physics textbooks. Figure 1. Schematic representation of Compton scattering, in which an incoming photon scatters from an electron at rest. The electron of the rest mass m e has the rest energy of E r = m e c 2. The Compton scattered electron (hereinafter the Compton electron) posses a relativistic kinetic energy, the relativistic speed υ e and the linear momentum p e = m e υ e. Before scattering, let the photon moves along the positive x-axis having an energy E 0 = hν 0 , frequency ν 0 , wavelength λ 0 and linear momentum p 0 = hν 0 /c, where h (= 6.63 × 10-34 J s) is the Planck constant and c (= 2.99792×10 8 m s-1) is the speed of light. After scattering, the photon moves at an angle θ off the x-axis with an energy E = hν, frequency ν, wavelength λ and momentum p = hν/c. The conservation of linear momentum of the electron after the collision yields, p e 2 = p 0 2 + p 2-2p 0 pcosθ.