Using the relativistic quantum stationary Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding oneself on both relativistic and quantum Lagrangians, we construct a Lagrangian of a relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton's law. Then, we plot the relativistic quantum trajectories of a particle moving under the constant and the linear potentials. We establish the existence of nodes and link them to the de Broglie's wavelength.
In this paper, we apply the one dimensional quantum law of motion, that we recently formulated in the context of the trajectory representation of quantum mechanics, to the constant potential, the linear potential and the harmonic oscillator. In the classically allowed regions, we show that to each classical trajectory there is a family of quantum trajectories which all pass through some points constituting nodes and belonging to the classical trajectory. We also discuss the generalization to any potential and give a new definition for de Broglie's wavelength in such a way as to link it with the length separating adjacent nodes. In particular, we show how quantum trajectories have as a limit when ℏ → 0 the classical ones. In the classically forbidden regions, the nodal structure of the trajectories is lost and the particle velocity rapidly diverges.
Using the relativistic quantum stationary Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding oneself on both relativistic and quantum Lagrangians, we construct a Lagrangian of a relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton’s law. Then, we plot the relativistic quantum trajectories of a particle moving under the constant and the linear potentials. We establish the existence of nodes and link them to the de Broglie’s wavelength. PACS: 03.65.Bz; 03.65.Ca; Key words: relativistic quantum law of motion, Lagrangian, relativistic quantum stationary Hamilton-Jacobi equation, relativistic velocities.
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.