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European Journal of Physics

The Compton effect-Schrodinger's treatment

Janez Strnad

Published Oct 1, 1986• 6 citations
Publisher

Abstract

The Compton formula is obtained in a semiclassical approximation due to Schrodinger. X-rays are treated within classical electrodynamics and electrons within quantum mechanics. The superposition of the wavefunctions of the incident and recoiling electron sets up a moving periodic probability density. Incident X-rays are specularly reflected from this pattern. The results are transformed back into the original rest frame of the electron by the Doppler shift equation. Finally, some remarks pertaining to the philosophy of physics are added.

DOI
10.1088/0143-0807/7/4/001

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The Compton formula is obtained in a semiclassical approximation due to Schrodinger. X-rays are treated within classical electrodynamics and electrons within quantum mechanics. The superposition of the wavefunctions of the incident and recoiling electron sets up a moving periodic probability density. Incident X-rays are specularly reflected from this pattern. The results are transformed back into the original rest frame of the electron by the Doppler shift equation. Finally, some remarks pertaining to the philosophy of physics are added.
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 equation is normally developed using the photon model, but it is also capable of derivation based on a classical wave model of light, and there is a considerable history of such derivations, including one by Compton himself. A simple derivation in the laboratory frame based on waves scattered by a statistical aggregate of recoil electrons is developed in this paper, providing a bridge between microscopic events and macroscopic observation, and implications of coexisting wave and particle model derivations of the Compton effect are discussed. The parallel derivations based on contrasting models can also furnish an instructive exercise for students.
Chapter I The origins of quantum physics I The Compton effect — a strange dichotomy in the nature of electromagnetic radiation In a series of experiments conducted between 1919 and 1923, Arthur Holly Compton (Figure 1.1) investigated the scattering of monochromatic (single-wavelength) X-rays from graphite targets. He found that the radiation scattered at an angle 0 to the incident beam contained, in addition to radiation of the same wavelength as the incident radiation, a second component with a considerably longer wavelength. The existence of this second component in the scattered radiation could not be explained by the well-established theory of the scattering of electromagnetic waves by electrons. Compton considered instead the process illustrated in Figure 1.2, in which a particle of electromagnetic radiation, collides with a slow moving electron, which then recoils absorbing some of the X-ray particle’s energy. Treating the process as a collision between particles, and using only the (relativistic) conservation laws of energy and momentum, Compton was able to account for the effect completely. However, at the same time, in the same experiment, Compton was using interference effects, depending wholly on the wave model, in order to determine the wavelength of the scattered radiation. So here was an experiment which apparently required the simultaneous use of both the wave and particle models of electromagnetic radiation for the interpretation of the results! The existence of this strange dichotomy in the nature of electromagnetic radiation was just one indication of the need for a radical revision of views regarding the physical world.
The undergraduate courses on modern physics generally consider the particle interpretation of Compton effect. Motivated by a student's solution in an undergraduate examination on modern physics here we consider the wave characteristics of recoiled electrons in the Compton effect. The de Broglie wavelength, wave and clock frequency of the scattered electrons are expressed in terms of the wavelength and the frequency of the incident and the scattered photons respectively using the familiar particle interpretation of the Compton effect, where initially the electron is at rest and its spin is ignored. Both non-relativistic and relativistic cases are considered. Numerical values of de Broglie wavelength, wave and clock frequency of the scattered electron are calculated for an incident photon energy that was used in the original experiment of Compton as a function of the scattering angle of the recoiled electron. Considering the relativistic effects which are however insignificant for the de Broglie wavelength of the recoiled electron under these conditions, the minimum value obtained is in the range of X-rays. The non-relativistic de Broglie wave frequency obtained by neglecting the rest mass of the electron leads to an underestimation of its value. The implications of de Broglie wavelength and clock frequency for Compton scattering experiments are briefly discussed and possible extensions of the obtained mathematical formulations are indicated. The results are useful for understanding the wave-particle duality of the recoiled electron in the context of the Compton effect.
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.
Traditional textbook explanations of the Compton effect treat the photon electron interaction as a particle collision. This explanation is a pedagogical disaster, implying that sometimes interactions are particle-like whereas quantum mechanics always demands that they be wave-like; a photon wavefunction evolves according to a wave equation until its collapse at measurement. If this is so why then does the classical radiation wave equation fail to predict the Compton effect? We address these issues and propose a clearer explanation.