We show that the Schr\"odinger equation can be derived assuming the Galilean covariance of a generic wave equation and the validity of the de Broglie's wave-particle duality hypothesis. We also obtain from this set of assumptions the transformation law for the wave function under a Galilean boost and prove that complex wave functions are unavoidable for a consistent description of a physical system. The extension to the relativistic domain of the above analysis is also provided. We show that Lorentz covariance and wave-particle duality are consistent with two different transformation laws for the wave function under a Lorentz boost. This leads to two different wave equations, namely, the Klein-Gordon equation and the Lorentz covariant Schr\"odinger equation.
We show that the Schr"odinger equation can be derived assuming the Galilean covariance of a generic wave equation and the validity of the de Broglie's wave-particle duality hypothesis. We also obtain from this set of assumptions the transformation law for the wave function under a Galilean boost and prove that complex wave functions are unavoidable for a consistent description of a physical system. The extension to the relativistic domain of the above analysis is also provided. We show that Lorentz covariance and wave-particle duality are consistent with two different transformation laws for the wave function under a Lorentz boost. This leads to two different wave equations, namely, the Klein-Gordon equation and the Lorentz covariant Schr"odinger equation.
A hypothesis is put forward according to which the formulas of Relativity are a consequence of the laws of quantum physics and may be applied only to quantum objects. According to the wave - particle duality quantum bodies are simultaneously the waves and considered in the present research with a use of the wave equation and requirement of its Lorentz‘s invariance. The property of invariance with respect to the Lorentz’s transform is analysed in hyperbolic coordinates in an area of complex numbers. An examination showed that formulas of relativistic kinematics can be obtained based on the condition of equality of the amplitudes and phases of the wave, observed from different inertial frames. There is no need to introduce the concept of a unified entity of space-time, but the return to the idea of aether and matter waves de Broglie occurs. The physical model becomes borrowed partly from quantum field theory. It is assumed that the moving particle binds to a wave of the aether disturbance, and only this wave acts on the other particle like an electromagnetic field pushes an electrical charge. The model of colliding elastic balls becomes inapplicable, but equations of relativity mechanics remain the same, however, with a new interpretation. The mass of a particle in all coordinate systems is constant, in exchange the expressions of energy and momentum alter relative the classical case, include the property of aether – coefficient c of the wave equation, and acquire relativistic forms.
In this study, both the Klein-Gordon equation and time-dependent Schrodinger equation are directly derived from De Broglie's (\lamda=h/p). Besides that, a new formula of the Klein-Gordon equation was obtained by using two equations, i.e., the classical wave function and relativistic total energy relation [[hw] = [(h^2 k^2)/2m] + V]. Moreover, a new equation of time-dependent Schrodinger (TDSE) in three-dimension 3D was derived and evidently obtained. Thus, the previous result corresponds to (TDSE) in 3D calculated using two equations, (i.e., Einstein energy and Newton's second law of mechanics).
In general, it is well known that the Schrödinger equation is not compatible with special relativity theory to a necessary extent.Thus far, there are already several trials to formulate versions of the Schrödinger equation to ensure compatibility with special relativity theory, the Klein-Gordon-equation or the Dirac equation are some of these attempts.Material and Methods: In this paper, Einstein's relativistic energy momentum relation is re-analyzed, a normalized relativistic energy momentum relation is derived.The derived normalized relativistic energy momentum relation together with the known Schrödinger equation is used as a starting point to establish a wave equation consistent with special relativity theory.Results: In this publication, based on Einstein's relativistic energy momentum relation, the historical problem of the "particle-wave-duality" is solved.Furthermore, a special relativity theory consistent wave equation is derived.
The covariance of the Schr\"odinger equation under Galilei boosts and the compatibility of nonrelativistic quantum mechanics with Einstein's equivalence principle have been constrained for so long to the existence of a superselection rule which would prevent a quantum particle from being found in superposition states of different masses. In an effort to avoid this expedient, and thus allow nonrelativistic quantum mechanics to account for unstable particles, recent works have suggested that the usual Galilean transformations are inconsistent with the nonrelativistic limit implied by the Lorentz transformation. Here we approach the issue in a fundamentally different way. Using a formalism of unitary transformations and employing quantum reference frames rather than immaterial coordinate systems, we show that the Schr\"odinger equation, although form variant, is fully compatible with the aforementioned principles of relativity.
It is known that the Schroedinger equation is not covariant under Galilei boosts, unless the phase of its solutions are shifted simultaneously. It is argued that the phase shift is not a coordinate transformation, because it depends on the mass of the Schroedinger particle. The phase shift also cannot be derived from low speed Lorentz boost. It is proposed to extend the Galilei boost with two terms of order v^2/c^2 to avoid these issues and to guarantee covariance of the Schroedinger kinetic energy and momentum. The extensions imply that proper time and relativity of simultaneity are essential features of Schroedinger quantum mechanics.
The proposed paper presents the unobserved inadequacies in de Broglie's concepts of wave-particle duality and matter waves in the year 1923. The commonly admitted quantum energy or frequency expression hν=γmc2 is shown to be inappropriate for matter waves and is acceptable only for photons. The superluminal phase velocity expression c2/\u{psion}, for matter waves, is investigated in detail and is also reported to be inadequate. The rectifications in the inadequate concepts of de Broglie's theory and refinements in the analogy implementation between light waves and matter waves are presented, which provides the modified frequency and phase velocity expression for matter waves. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. In accordance with the proposed concepts, a wave-particle duality picture is presented which elucidates the questions coupled with the wave-particle duality, existing in the literature. Consequently, particle type nature is shown to be a characteristic of waves only, independent from the presence of matter. The modifications introduced in the frequency expression for matter waves leads to variation in the wave function expression for a freely moving particle and its energy operators, with appropriate justifications. A new relation between the Kinetic energy and Momentum of the moving body is proposed and is subsequently applied to introduce novel General and Relativistic Quantum Mechanical Wave Equations. Applications of these equations in bound state quantum mechanical systems provide the information regarding particle's general and relativistic behavior in such systems. Moreover, the proposed wave equations can also be transformed into Schrödinger's and Dirac's equations. The interrelation of Schrödinger's, Dirac's and proposed equations with the universal wave equation is also presented.