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arXiv:1806.01271

Compton Scattering of Hermite Gaussian Wave Gamma-Ray

Tomoyuki Maruyama, Takehito Hayakawa. Toshitaka Kajino

Published Jun 4, 2018• 0 citations
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Abstract

We calculate the differential cross sections for Compton scattering of photons described by Hermite Gaussian (HG) wave function in the framework of relativistic quantum mechanics. The HG wave gamma-rays propagating along the z-direction have quantum numbers of nodes of nx and ny in the x- and y-directions, respectively. The calculated differential cross section is symmetric with respect to both the zx- and zy-planes. The nodes whose number is identical with nx and ny appear in the energy spectrum measured in zx- and zy-planes, respectively. These results indicate that it is possible to identify the HG wave photon and its quantum numbers nx and ny by measuring Compton scattering. The present proposed method can be also applied to gamma-ray astronomy.

DOI
10.48550/arxiv.1806.01271

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We calculate the differential cross sections for Compton scattering of photons described by Hermite Gaussian (HG) wave function in the framework of relativistic quantum mechanics. The HG wave gamma-rays propagating along the z-direction have quantum numbers of nodes of nx and ny in the x- and y-directions, respectively. The calculated differential cross section is symmetric with respect to both the zx- and zy-planes. The nodes whose number is identical with nx and ny appear in the energy spectrum measured in zx- and zy-planes, respectively. These results indicate that it is possible to identify the HG wave photon and its quantum numbers nx and ny by measuring Compton scattering. The present proposed method can be also applied to gamma-ray astronomy.
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.
Abstract In this work we analyze the wave–particle aspects of radiation in (incoherent) Compton scattering in the radiation energy range from 2–100 keV. From the calculated cross sections of the scattering from the electron (positron), hydrogen and positronium we construct the interpretation functions (IFs), where our assertion is that the Compton scattering from the free electron (positron) is an established example of the particle behavior of radiation. These IFs estimate the possibility of the interpretation of radiation in terms of waves or particles in an analogy with the analysis carried out in the coherent scattering of light. Based on these IFs we propose a new criterion for the estimation of the validity of the impulse approximation (IA).
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.
A Stokes parameter representation for two-photon systems is developed to calculate cross sections for both entangled and classically correlated mixed states scattering off unpolarized and polarized Compton electrons. The cross section of Compton scattering for pairs of maximally entangled annihilation photons, generated by the disintegration of para-positronium, is compared with classical analogs. An analysis of the symmetrical properties and basis independence of these systems is conducted to elucidate the observed differences. We propose a method to establish an upper bound for identifying correlations influenced by entanglement. Furthermore, we calculate the cross section for Compton scattering of annihilation photons from spin-polarized electrons. A qualitative analysis reveals a contradiction to the contemporary assumption that Compton scattering acts as an entanglement kill switch. These findings contribute to a broader understanding of photon systems and their behavior in various scattering scenarios. Although the primary focus of this article is on Compton scattering, the formalism can be adapted to other interactions, provided that the Mueller matrix representation of the interaction is available. Published by the American Physical Society 2024
In this work we compute, via the quantum potential approach, the Hamiltonian system determined by Hermite–Gaussian beams. Then we show that the integral curves of the Poynting vector, exact optics energy trajectories, conform to a subset of solutions to the corresponding Hamilton equations lying on hyperboloidal surfaces. The geometrical light rays associated with these beams are given by the tangent lines to the integral curves of the Poynting vector at the zeroes of the quantum potential, and the caustic region coincides with the zeroes of quantum potential and quantum force. One of the main contributions of this work is to present the relationship between the physical phase k Φ, the geometrical-optics phase k Φ G , and the quantum potential Q HG in the Hermite–Gaussian beams. Furthermore, note that for any solution to the paraxial wave equation in free space, the tangent lines to the integral curves of the Poynting vector that correspond to the geometric light rays are those that pass through the points where the region determined by zeroes of the quantum potential is tangent to the geometrical caustic determined by the geometric light rays .