A class of traveling-envelope solutions of free Schr\"odinger equation generated by Lorentz transformation

Abstract

We develop a class of traveling-envelope solutions of Schr\"odinger 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 $E$ and momentum $p$ are related to the rest-mass $m$ of the particle by Einstein formula $E^2/c^2-p^2=m^2c^2$. 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 $\lambda_P=h/p$.