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

TL;DR

The paper probes whether spatial confinement alone, within the Koopman–von Neumann (KvN) Hilbert-space formulation of classical mechanics, induces energy quantization. Using the Liouville generator $\hat{L}$ and two representations, it shows that enforcing elastic reflection at the walls implies a $Q$-parity boundary that preserves probability current and self-adjointness, leading to a spectrum organized into continuous bands $E_{n,\kappa} = \frac{\hbar^2}{m}\frac{n\pi}{L}\kappa$ with $n\in\mathbb{N}$ and $\kappa\in\mathbb{R}$; thus, spatial confinement discretizes the $q$-motion (through $k_n = n\pi/L$) but does not quantize energy since the dual coordinate $Q$ remains unbounded. This demonstrates that energy quantization is inherently quantum, arising from noncommuting observables, not merely from Hilbert-space structure, and it clarifies boundary-condition roles and common KvN misconceptions. The work also links KvN to phase-space formulations via a κ-Wigner framework, showing how classical Liouville dynamics emerge in the $\kappa\to 0$ limit and how interference is absent in KvN, reinforcing the classical nature of KvN dynamics.

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.