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American Journal of Physics

Another look through the Heisenberg microscope

Herbert C. Corben

Published Dec 1, 1979• 1 citation
Publisher

Abstract

Conditions are examined for measurement of the position and momentum of a particle by Compton scattering from various directions. In addition to the usual uncertainty relation that emerges from this analysis, it is shown that, for a given initial frequency, there is generally a critical angular aperture at which the error in position is a minimum.

DOI
10.1119/1.11969

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Conditions are examined for measurement of the position and momentum of a particle by Compton scattering from various directions. In addition to the usual uncertainty relation that emerges from this analysis, it is shown that, for a given initial frequency, there is generally a critical angular aperture at which the error in position is a minimum.
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.
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.
The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.
Position measurements are examined under the assumption that object position and probe position just after the measurement are expressed as a linear combination of positions and just before the measurement. The Heisenberg uncertainty relation between the position uncertainty and momentum disturbance holds when the measurement error ε(xt) for the object position is adopted as the position uncertainty. However, the uncertainty in the measurement result obtained for is the standard deviation of the measurement result, and not the measurement error ε(x0). This difference is due to the reduction of a wave packet. The validity of the linearity assumption is examined in detail.
In single-particle quantum mechanics there is no impediment to a particle's wave function having a significant amplitude for arbitrarily short wavelengths, i.e., for arbitrarily large momenta, so the iconic single-particle uncertainty relation permits a particle's position to be arbitrarily accurately ascertained. Once a single-particle theory is second quantized, however, the physics of imparting a particle's wave function with ever larger momenta eventually encounters stiff competition from the formation of multiparticle states wherein none of the individual particles is characterized by high momentum. In fact, the single-particle uncertainty principle itself is modified by the presence of the expectation value of the particle number operator on its right side. Since the threshold for the high-momentum-diverting formation of additional particles is set by the particle's rest mass, it stands to reason that particles have an irreducible position uncertainty of the order of their Compton wavelength, for which we develop a specific model.
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.