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.
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.
In 1924, Bohr, Kramers and Slater tried to introduce into microphysics conservation principles that hold only on the average. This attempt was abandoned in the light of the Compton-Simon experiment (and its later refinements). Since that time, except for a moment of doubt in 1936, it has been thought that the classical conservation laws (for energy and momentum) hold in quantum theory for each individual interaction, in a way that yields the classical exchange-and-balance of momentum (or energy) familiar from the laws of elastic collisions. It has been thought, that is, that in each individual “collision” what one part of the total system loses in linear momentum (say) another part gains, so as to maintain the same total amount afterwards as before. To those familiar with discussions of the interpretation of quantum theory, however, it will be apparent that the very concepts needed to express this idea of an elastic collision are generally not admitted in the theory. For one needs the idea that both before and after collision the total system has a well-defined value for the conserved quantity and, moreover, that the various component parts of the system have well-defined values for those quantities out of which the conserved one is composed. But in the case of spin, for example, it is customary to say that although total spin may be conserved in certain interactions one cannot attribute values to the separate components of spin (in various directions) because the associated operators do not commute. (Thus the whole sum is not even the sum of its parts.) Moreover, one does not generally refer to the values of quantities except in eigenstates. Hence, in general, one would not refer to the value of the conserved quantity before and after interaction, except for interactions initiated in eigenstates.
The recent two papers have successively proposed the concepts and more precise expressions of strict conservation or statistical conservation of physical quantities, which are different from the expression of a conserved quantity in quantum mechanics. The Hamiltonian operator of the system discussed in this paper are independent of time, so it is commutative with the operators of an angular momentum itself, its component and its square, so that all these quantities are conserved quantities. In an isolated composite system, one of the angular momentum components of the total system are strictly conserved, while the angular momentum components of the subsystems can only be conserved statistically, both of which can be expressed in terms of eigenvalues and eigenfunctions. Generally, angular momentum itself does not have eigenvalues and eigenfunctions, so it cannot be expressed as strict conservation in terms of these concepts. Although there are eigenvalues and eigenfunctions for the square of angular momentum, they cannot express strict conservation either. And they can only be expressed as statistical conservation.
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.
At the beginning of the century doubts were growing about the whole edifice of classical physics, including the laws of mechanics in their Lagrangian and Hamiltonian forms, of statistical thermodynamics, as formulated by Boltzmann and Gibbs, and of optics, namely Maxwell’s and Fresnel’s theory. Einstein proposed that electromagnetic radiation exists in discrete quanta whilst propagating in space as well as during the emission or absorption processes. Scattering by free electrons is a common interaction connected with the absorption of electromagnetic radiation. If the electromagnetic energy is large compared with the ionization energy, the atomic electron can be treated as free. The characteristic features of the scattered radiation have been explained by Compton, on the basis of an elastic collision of a photon with a free electron, where both energy and momentum are conserved. The concept of the matter wavelength holds for both relativistic and non-relativistic particles, provided that the appropriate form of momentum is used.
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.