Probability of presence versus ψ∗(x,t)ψ(x,t)

Abstract

Postulating the identification of ${\psi }^{\ast }(x,t)\psi (x,t)$ with a physical probability density is unsatisfactory conceptually and overly limited practically. For electrons, there is a simple, calculable relativistic correction proportional to $\nabla {\psi }^{\ast }\cdot \nabla \psi$. In particular, zeroes of the wave function do not indicate vanishing probability density of presence. We derive a correction of this kind from a Lagrangian, in a form suitable for wide generalization and use in effective field theories. Thus we define a large new class of candidate models for (quasi-)particles and fields, featuring modified kinetic terms. We solve for the stationary states and energy spectrum in some representative problems, finding striking effects including the emergence of negative effective mass at high energy and of localization by energy.