The Particle in a Box in Koopman--von Neumann Mechanics: A Hilbert Space representation of Classical Mechanics

Abstract

This paper revisits the textbook 'particle in a box', but from the point of view of Koopman-von Neumann (KvN) mechanics. KvN mechanics is a way to describe \emph{classical} dynamics in a Hilbert space. That simple fact changes the usual expectation: hard walls do \emph{not} force energy quantization here. We show, in a clear and physical way, why a KvN particle confined between two ideal walls still has a continuous range of energies. With the correct wall condition, one that captures ordinary elastic reflection rather than 'vanishing at the boundary,' the KvN description naturally produces spatial confinement without discrete energy levels. Beyond establishing this result, we also clean up common misunderstandings: for example, treating the KvN wavefunction like a quantum probability amplitude in position alone leads to the wrong boundary picture and, with it, the wrong conclusion about quantization.