Using the elementary axioms of special relativity and quantum mechanics we construct a wave equation which generalizes the Schrodinger equation. We also solve the general second and some higher order differential equations.
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.
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.
Abstract The fundamentals of a quasi-relativistic wave equation, whose solutions match the Schrödinger results for slow-moving particles but are also valid when the particle moves at relativistic speeds, are discussed. This quasi-relativistic wave equation is then used for examining some interesting quantum problems where the introduction of relativistic considerations may produce remarkable consequences. We argue in favor of the academic use of this equation, for introducing students to the implications of the special theory of relativity in introductory quantum mechanics courses.
The solution of the Schrödinger equation for a physical system in quantum mechanics is of great importance, because the knowledge of wave- function and energy spectrum contain all possible information about the physical properties of a system. In this paper, we have give solution of the Schrödinger equation in three dimensional curved space with Hulthen potential on the positive constant curvature. Then we achieve the wave-function and energy spectrum for the Hulthen potential. In order to solve the corresponding Schrödinger equation, we use of Nikiforov-Uvarov (N.U) method [1]. The N.U method is based on solving the second- order linear differential equations by reducing to a generalized equation of hypergeometric type.