Probability of Presence Versus $\psi(x,t)^* \psi(x, t)$

TL;DR

This paper challenges the traditional interpretation of the wave function's magnitude as a probability density, proposing a new correction term derived from a Lagrangian that impacts particle models and reveals novel effects like negative effective mass.

Contribution

It introduces a new correction to the probability density interpretation derived from a Lagrangian, enabling the development of modified kinetic models for particles and fields.

Findings

Derived a relativistic correction proportional to gradient terms.

Found negative effective mass at high energies.

Observed localization effects driven by energy.

Abstract

Postulating the identification of $\psi^*(x, t) \psi(x,t)$ with a physical probability density is unsatisfactory conceptually and overly limited practically. For electrons, there is a simple, calculable relativistic correction proportional to $\nabla \psi^* \cdot \nabla \psi$. 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 {\it 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. \end{abstract}