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American Journal of Physics

Compton scattering and Heisenberg’s microscope revisited

Brian Gerard Williams

Published May 1, 1984• 6 citations
Publisher

Abstract

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.

DOI
10.1119/1.13648

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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.
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.
The creation of quantum mechanics in the period 1924–1928 restored logical consistency to its rightful place in theoretical physics. The study of interaction between light and electrons provided most of the important clues in the development of the quantum theory. This chapter shows that in the interference experiments the statistical distribution of quanta is correctly given by the intensity of the wave field as computed from classical electromagnetic theory. It also shows that under special circumstances the classical theory also makes contact with some of the empirical facts in Compton's experiment. The chapter discusses a famous example due to W. Heisenberg, in which Compton scattered γ-rays are brought to focus by a microscope. It provides the uncertainty relations for the dynamical variables of a particle from the conservation laws and the properties of photons. Because the motion of charged particles can be deduced from their attendant fields, it is clear that fields cannot be measured to arbitrary precision.
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.
Abstract Heisenberg introduced his famous uncertainty relations in a seminal 1927 paper entitled The Physical Content of Quantum Kinematics and Mechanics . He motivated his arguments with a gedanken experiment, a gamma ray microscope to measure the position of a particle. A primary result was that, due to the quantum nature of light, there is an inherent uncertainty in the determinations of the particle’s position and momentum dictated by an indeterminacy relation, δ q δ p ∼ h . Heisenberg offered this demonstration as ‘a direct physical interpretation of the [quantum mechanical] equation pq − qp = − i ℏ ’ but considered the indeterminacy relation to be much more than this. He also argued that it implies limitations on the very meanings of position and momentum and emphasised that these limitations are the source of the statistical character of quantum mechanics. In addition, Heisenberg hoped but was unable to demonstrate that the laws of quantum mechanics could be derived directly from the uncertainty relation. In this paper, we revisit Heisenberg’s microscope and argue that the Schrödinger equation for a free particle does indeed follow from the indeterminacy relation together with reasonable statistical assumptions.