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 classical dynamics in a Hilbert space. That simple fact changes the usual expectation: hard walls do 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 clarify a common root of confusion that can arise when one attempts to solve this problem: treating the KvN wavefunction as though it were a quantum probability amplitude in position space alone leads to the wrong boundary picture and, consequently, to the wrong conclusion about quantization.