Relativity and the Uncertainty Principle

Abstract

A simple ideal experiment to measure simultaneously the position and momentum of a free electron, also its kinetic energy and the instant when this kinetic energy is measured, taking into account the finite velocity of propagation of light as embodied in the Lorentz transformation, is proposed. This experiment is as follows: an observer sends out a light signal which strikes a given free electron and returns along its own track. From measurements made on the outgoing and returning signals the required quantities, and the uncertainties in their measurement, are calculated. In the absence of detailed knowledge concerning the nature of the interaction between the signal and the electron, it is found that for any electron velocity between 0 and $c$, $\ensuremath{\Delta}p\ensuremath{\Delta}x$ and $\ensuremath{\Delta}E\ensuremath{\Delta}t$ have an upper and a lower bound. The upper bound depends only on the initial electron velocity and the lower bound is a function of the electron velocity and the outgoing frequency. These bounds come close together for low electronic velocities and low frequencies, but their separation increases as the initial electron velocity increases becoming infinite in the limit $v=c$. In the limit $c\ensuremath{\rightarrow}\ensuremath{\infty}$ the upper and lower bounds coalesce for all frequencies and all electron velocities giving Heisenberg's results. The possibility, but not the necessity, of a relativistic quantum mechanics with $\ensuremath{\Delta}p\ensuremath{\Delta}x=h$ (and $\ensuremath{\Delta}E\ensuremath{\Delta}t=h$) is, we believe, established by the present results.