The Schrödinger Equation and Negative Energies

Abstract

Abstract It is known that there is no room for anti-particles within the Schrödinger regime in quantum mechanics. In this article, we derive a (non-relativistic) Schrödinger-like wave equation for a spin- 1 / 2 $1/2$ free particle in 3 + 1 space-time dimensions, which includes both positive- and negative-energy eigenstates. We show that, under minimal interactions, this equation is invariant under 𝒫 𝒯 $\mathcal{P}\mathcal{T}$ and 𝒞 discrete symmetries. An immediate consequence of this is that the particle exhibits Zitterbewegung (‘trembling motion’), which arises from the interference of positive- and negative-energy wave function components.