Quantum telegraph equation with given relativistic correction

Abstract

The wave function representing a free particle spreads instantaneously to infinity according to the Schr ̈odinger equation. Thus, even at a very short time t, one faces the problem of having a nonzero probability of finding the particle at an arbitrarily large distance away from the interval where the wave function was initially confined. This paper introduces a correction term to the nonrelativistic Schr ̈odinger equation which solves the instantaneity problem. Instead of analytically continuing the heat equation as is usually done, the term comes from considering the energy-momentum relationship in relativity and then applying quantization. The Schrödinger equation with the correction term have solutions that propagate at a finite speed equal to the speed of light.