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 $\ldb=\lcr/\beta$.
No report in the literature has directly described this relation. New constants for particles are presented. One relates to the Compton wavelength, called here the “mass-wave” constant for all particles. The other relates to the deBroglie wavelength, called here the “velocity-wave” constant for a particle. An equation is derived based on these two constants encapsulating a fundamental relation between the matter-states, particle and wave, to the velocity of light. New approaches to the Uncertainty relations are shown. The basic Schrodinger equation is derived from the perspective of a non-dimensional second-order differential equation free of any assumed empirical constants. The resulting time-dependent wave equation for a free particle was then expressed in terms of the particle velocity and deBroglie wavelength.
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.
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.
For the wave-particle duality of microscopic particles,the relationship of the velocity and the velocity of phase and the velocity of group for the microscopic particles is analyzed,it is pointed out that the velocity relating to the frequency and wavelength of two De Broglie expressions is not the velocity of group but the velocity of phase.On the contrary,the velocity in the defining expression of momentum is the real speed of particles,which doesn't equal to the velocity of phase but equals to the velocity of group.
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.