Uncertainty relations for a particle confined in a finite square well potential

Abstract

In order to specify the uncertainty relation for a particle confined in a finite square well potential we derived the standard deviation of momentum Δp and that of position Δx. It is shown that Δp has a peak as a function of the well width and both Δx and Δp Δx have a dip in contrast to the case of the particle confined in an infinite square well potential. We clarified that Δp0 Δx0 of the particle occupying the ground state exists in the finite range as a function of the well width and the potential-barrier height, but a particle confined in an infinite square well potential has a constant Δp0 Δx0. In addition, we found that the values of Δpn Δxn for excited states diverge in a certain range of well widths, where the energy level of the confined particle exceeds the potential barrier.