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Physical Review

The Conservation of Energy and Momentum in Compton Scattering

William G. Cross, Norman F. Ramsey

Published Dec 15, 1950• 28 citations
Publisher

Abstract

An experiment similar to that of Bothe and Maier-Leibnitz and others has been carried out to verify that in Compton scattering of 2.62-Mev quanta the scattered quantum and recoil electron are emitted without delay and in the relative directions demanded by the conservation laws. Better angular resolution and statistics and fewer experimental uncertainties than in the previous experiments were obtained. The results are in agreement with simultaneity and the conservation of energy and momentum.

DOI
10.1103/physrev.80.929

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An experiment similar to that of Bothe and Maier-Leibnitz and others has been carried out to verify that in Compton scattering of 2.62-Mev quanta the scattered quantum and recoil electron are emitted without delay and in the relative directions demanded by the conservation laws. Better angular resolution and statistics and fewer experimental uncertainties than in the previous experiments were obtained. The results are in agreement with simultaneity and the conservation of energy and momentum.
The Bothe-Geiger experiment has been repeated under improved conditions of time resolution, obtained by using stillbene scintillation counters. The recoil electron and scattered photon are emitted together within a time interval of less than 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ second. The electron and photon are also projected without a delay larger than 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ second after arrival of a quantum. The familiar concepts used in discussions of the Compton effect are shown to be valid.
Bohr et al. once believed that both the conservation of energy and the conservation of momentum in a microscopic process are statistically conserved. And then Bothe et al. verified that the laws of the conservations of total energy and total momentum of an electron and an X-ray photon before and after a single collision are strictly valid by using the Compton scattering experiment, and such results also denied statistical conservation. There are expressions about strict conservations of total energy and total momentum of the system in Compton scattering experiment without quantum mechanical concepts in the introductions of some quantum mechanics books. Different from this, the aim of this paper is attempt to express those with quantum mechanical concepts. There is no expression about the strict conservation and the statistical conservation of physical quantities in quantum mechanics books. Our viewpoints are that the total energy and the total momentum of an isolated system are strict conservation by exchanging energy and momentum within the system in the evolution of the system state, and the momentum and kinetic energy (an interaction energy is not all belong to anyone subsystem) of a subsystem are statistical conservation. The “strict conservation” in journal articles is different from ours. A recent paper by the corresponding author of this paper and his coauthors (Int.J Theor. Phys. , 2020, 59 : 229) used the terms of strict conservation and statistical conservation (we offer more precise statements in this paper, and these statements have some differences with that of a conserved physical quantity) , and offered the expression ways of the strict conservation of total momentum for an isolated composite system and the statistical conservation of momentum for a subsystem in terms of the concepts of entangled states and eigenvalues. In this paper, we try to use these concepts and mixed states to express the strict conservations of total energy and total momentum of the system before and after a Compton scattering, and also explain the statistical conservations of energy (no interaction after scattering) and momentum of a subsystem. Similarly, we can express the strict conservations of some physical quantities of an isolated composite system and the statistical conservations of some physical quantities of a subsystem with quantum mechanical concepts.
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.
Compton scattering figures prominently in the introductory chapters of textbooks on quantum mechanics as providing evidence for the ‘‘particulate’’ nature of light. It also arises in discussions of ‘‘Heisenberg’s microscope,’’ in which it is shown that attempts to measure the position and the momentum of an electron simultaneously are governed by the uncertainty principle. However, recent developments in Compton scattering have not yet found their way into the textbooks. In this paper we review the history of the Compton effect and describe briefly some of the modern developments in x-ray and electron Compton scattering and their application to the study of electron momentum densities which we believe to have a pedagogical value. Finally, we return to Heisenberg’s microscope, and show how it has now become a reality. A complete analysis for all scattering angles depends crucially on the different expressions for the resolution and the depth of focus of the microscope.
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.