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.
The Compton effect is commonly cited as a demonstration of the particle feature of light, while the wave nature of matter has been proposed by de Broglie and demonstrated by Davisson and Germer with the Bragg diffraction of electron beams. In this investigation, we present an entirely different interpretation of the Compton effect based on the postulates of de Broglie and on an interaction between electromagnetic and matter waves. The speeds of interacting electrons in the Compton scattering are quite fast and its mechanism relies heavily on the mass variation. Thus, based on this wave interpretation, the Compton effect can be viewed as a further demonstration of the postulates of de Broglie for high-speed particles. In addition to the scattered wave, a direct radiation depending on the mass variation is predicted, which provides a means to test the wave interpretation.
We will here show that there is one more relativistic wave equation rooted in the relativistic energy Compton moment relation, which should not be confused with the standard relativistic energy momentum relation. The standard momentum is, from a quantum perspective, rooted in the de Broglie wave. The de Broglie wave is not mathematically defined for rest-mass particles and has strange properties such as converging to infinity when the particle is almost at rest. As mentioned in the previous paper, the de Broglie wave is likely just a mathematical derivative of the true matter wave, which we have good reasons to think is the Compton wave. A wave equation that satisfies the relativistic energy Compton momentum relation will, in addition automatically satisfy the standard relativistic energy momentum relation, so there is no conflict between these two relations. The new one related to Compton is just the deeper reality and help explain why it gives simpler and more elegant relativistic wave equations, which are likely easier to interpret in terms of the physical reality.
The Compton wavelength is conventionally introduced as the rest-mass scale $\lc=h/(mc)$, whereas moving particles are normally assigned the de Broglie wavelength $\ldb=h/p$. Haug recently introduced and subsequently refined a frame-dependent relativistic Compton wavelength, $\lcr=h/(\gamma mc)$, with reduced form $\lbarcr=\hbar/(\gamma mc)$. For a massive free particle with
p=γmv, the exact identity $\ldb=(c/v)\lcr=\lcr/\beta$ follows immediately. This article examines the consequences of taking $\lcr$, rather than $\ldb$, as the primary wavelength parameter for single-particle relativistic wave mechanics. The distinction is conceptually sharp at rest: $\lcr$ is finite and equals $\lc$ at
v=0, while
h/p is not defined by ordinary division at
p=0, although its limiting spatial wavelength is infinite and its wave number
k=p/ℏ is well defined and vanishes. We show that free-particle phase relations, Schr\"odinger eigenstates, box quantisation, diffraction conditions, uncertainty relations, and the plane-wave sectors of the Klein--Gordon and Dirac equations can all be written using $\lbarcr$ and
β. In this parameterisation, $E=\hbar c/\lbarcr$ and $p=\hbar\beta/\lbarcr$, so the de Broglie wavelength is a derived spatial period generated by the velocity-dependent spatial gradient of a Compton-scale spacetime phase. The reformulation is algebraically equivalent to standard momentum-space quantum mechanics, but it exposes an unequal domain and an overlooked organisation of four-momentum. The quantity
E/c=γmc, already present as the temporal component of four-momentum, is precisely Haug's Compton momentum
pC=h/λC,r. Every finite de Broglie wavelength of a massive on-shell particle can be represented by the relativistic Compton wavelength and velocity through $\ldb=\lcr/\beta$.
We demonstrate that the Compton wavelength mathematically corresponds exactly to the photon wavelength of rest mass energy. On the other hand, the de Broglie wavelength is not defined for a rest-mass particle, but if the particle is nearly at rest, then the de Broglie wavelength approaches infinity, and the corresponding photon wavelength of the rest-mass energy is then this length times
cv again, that is it approaches zero when
v approaches zero. Our analysis indicates that the de Broglie wavelength appears to be a pure mathematical derivative of the Compton wavelength. Everything that can be expressed with the de Broglie wavelength can essentially be expressed by the Compton wavelength. We also demonstrate how spectral lines from atoms and chemical elements are linked to the Compton wavelength of the electron and that the Rydberg constant is not needed. Furthermore, we demonstrate that the Compton frequency is embedded in the Schrodinger equation, the Dirac equation, and the Klein-Gordon equation, where the Planck constant actually cancels out, and the de Broglie wavelength is not present in these equations. The Compton frequency seems to be linked to the quantization in quantum mechanics rather than the Planck constant. Additionally, we discuss recent literature that shows a remarkably simple but overlooked way to quantize Newton's and General Relativity theories, as well as other gravity theories, and also how to link them to the Planck scale. This, once again, leads to the conclusion that the Compton wavelength and Compton frequency are related to the quantization of matter and, thereby, the quantization of gravity. In addition, the Planck length plays a crucial role in quantum gravity, as demonstrated. Viewing physics through the de Broglie wavelength is like looking at the world through a distorted lens; switch to the Compton wavelength, and the distortion is removed, allowing us to see simplicity and clarity even in complex phenomena such as quantum gravity.
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.