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Developing a Lorentz Invariant form of the Schrodinger Equation

Declan Traill

Published Jan 1, 2019• 0 citations
Publisher

Abstract

The Schrodinger equation is not Lorentz Invariant, so it cannot be applied to the wave functions of moving particles. However, the Classical Wave Equation is Lorentz Invariant and is also satisfied by particle wave functions. By using this knowledge, a Lorentz Invariant form of the Schrodinger equation can be developed that can be applied to the wave functions of moving particles.

DOI
10.13140/rg.2.2.32087.11680

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The Schrodinger equation is not Lorentz Invariant, so it cannot be applied to the wave functions of moving particles. However, the Classical Wave Equation is Lorentz Invariant and is also satisfied by particle wave functions. By using this knowledge, a Lorentz Invariant form of the Schrodinger equation can be developed that can be applied to the wave functions of moving particles.
Abstract Using the concept of distributions, the square root of the operator - Δ + m2 is taken in a mathematically well defined way for one component wave functions. A new representation of proper Lorentz transformations for one component wave functions makes it possible to construct a relativistic quantum mechanics for spin 0, comprising a Lorentz invariant wave equation, a scalar product, and a positive definite density satisfying, together with a current, a continuity equation, and coupling of scalar and vector potentials. Some interesting consequences of the theory concerning the concept of particle trajectory and velocity of propagation of the probability amplitude are discussed in detail. As an example of practical application a perturbation theory for discrete states is set up.
It is not possible to use the Schrödinger equation to describe the motion of a particle with relativistic effects since it is not invariant relative to the Lorentz transformation. However, the attempt to generalize quantum mechanics in the relativistic domain on the basis of the one particle theory is somehow difficult. So we find the appropriate equations on the basis of the relativistic invariance. To obtain the non-relativistic equations, we take the limit as the speed of light tends to infinity. This obviously recovers the Schrödinger equation. Under this heading, we extend the non-relativistic Schrödinger equation in the description of particles with speeds comparable with that of light. This extension should be consistent with the Lorentz transformation [ 12 ]. The characteristic feature of the relativistic wave equations is the spin which provides a useful gauge in the applicability of particular equations that describe particular kinds of particles. We have two types of relativistic equations, i.e., the equation with spin-zero due to Schrödinger and spin-half due to Dirac that describes the electron.
We consider the two basic theories of the standard physics: (1) the theory of relativity, as a four-dimensional time-space description of our universe, with the invariance of the time-space interval which leads to the Lorentz transformation, and (2) quantum mechanics, based on the Schrödinger equation, with a solution called wave function. However, we find that this equation is very unsatisfactory, describing fully unlocalized free particles, any initially localized particle rapidly spreading in space, in disagreement with the fundamental Hamilton equations. In agreement with the Hamilton equations, we describe a quantum particle by propagation wave functions in the coordinate and momentum spaces, with the Lagrangian in the time dependent phase instead of the Hamiltonian coming from the Schrödinger equation. With the relativistic Lagrangian, we obtain wave functions describing invariant distributions propagating in space.
We develop a class of traveling-envelope solutions of Schrödinger equation for a free particle whose amplitude is moving with constant group velocity while keeping its shape undistorted. We show that solution with arbitrary finite group velocity is obtained by Lorentz boosting the solution with vanishing group velocity, if the quantum average energy EEE and momentum ppp are related to the rest-mass mmm of the particle by Einstein formula E2/c2−p2=m2c2E^2/c^2-p^2=m^2c^2E2/c2−p2=m2c2. The wave function is spatially localized with finite-size support which is decreasing as the rest-mass and/or group velocity are increased. For a particle with vanishing rest-mass yet finite momentum, we show that the group and phase velocities are equal to the velocity of light and the wavelength is given by Einstein another formula λP=h/pλ_P=h/pλP​=h/p.
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.