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Schrodinger Equation in Cosmological Inertial Frame

Sangwha Yi

Published Dec 28, 2021• 2 citations
Publisher

Abstract

Schrodinger equation is a wave equation. Wave function uses as a probability amplitude in quantum mechanics. We make Schrodinger equation from Klein-Gordon free particle’s wave function in cosmological special theory of relativity.
DOI
10.31237/osf.io/nyt7z

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This chapter introduces the concepts of the relativistic theory to basic problems of quantum physics, in particular, the extension of the Schrodinger equation to a relativistic form. Starting from the fundamental laws of relativity, the chapter evaluates the Klein–Gordon equation and investigates if the equation can be considered a generalization of the non-relativistic Schrodinger equation to the case of the relativistic energy. The chapter shows that the wave function, which is a solution to the Klein–Gordon equation, cannot be connected with the probability wave function. For this reason, the Dirac equation is used to solve the problems faced by the Klein–Gordon equation. The Dirac equation also includes the spin. The advantage of the Dirac equation over the Schrodinger and Klein–Gordon equations is that it naturally includes the spin. In the Schrodinger and Klein–Gordon equations, the spin is not present and has to be added to the wave function manually whenever the wave function determines a system with the spin.
The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.
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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.
As its name suggests, Schrodinger’s equation is the brain child of Schrodinger and was first disclosed in 1926 in a paper entitled An Undulatory Theory of the Mechanics of Atoms and Molecules . Schrodinger begins by considering the dynamics of a particle moving through a force field. The Schrodinger’s equation is arrived at in an alternative heuristic path modeled after Haken. The Schrodinger’s equation is a wave equation that incorporates classical particle concepts, and classical wave function concepts, in its derivation. In other words, a heuristic approach to Schrodinger’s equation utilizes Max Planck’s quantum energy, classical kinetic energy of a free particle, and the classical “wave functions of ordinary wave optics.” Schrodinger’s equation is capable of explaining all atomic phenomena except those involving magnetism and relativity.