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arXiv:2510.26856 · Journal of Physics A Mathematical and Theoretical

The particle in a box in Koopman–von Neumann mechanics: a Hilbert space representation of classical mechanics

Abhijit Sen, Lev Kaplan

Published Jun 3, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ae777c

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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.
This work is to consolidate current literature on Koopman-von Neumann (KvN) Mechanics into a simple and easy to understand text. KvN Mechanics is a branch of Classical Mechanics that has been recast into the mathematical language of Quantum Mechanics. KvN Mechanics utilizes a Hilbert phase space with operators to calculate the expectation values of observables of interest (expectation values such as position, momentum, etc.) It is an important tool in statistical physics and guide to illuminate the mysterious relationship between Quantum and Classical Physics. A lot of important applications of KvN Mechanics have been developed in the last decade.
The Koopman–von Neumann (KvN) formulation recasts deterministic classical mechanics in a Hilbert-space operator framework analogous to quantum mechanics. In this approach, the Liouville equation is expressed as a Schrödinger-type evolution equation generated by a Hermitian operator, yielding unitary propagation for phase-space-volume-preserving dynamics. We first clarify the relationship among the Koopman, Perron–Frobenius, and Liouville operators through an explicit example. Wilczek's construction shows that the KvN formalism is not only restricted to canonical Hamiltonian mechanics but this can be generalized any first-order classical dynamical system. Wilczek shows how a general autonomous first-order system can be generated by a Hermitian temporal-evolution operator after the introduction of auxiliary conjugate variables. We develop the KvN description of systems driven by Hamiltonian curl forces–position-dependent, non-conservative forces that are non-dissipative and preserve phase-space volume. Finally, we extend the KvN formalism to non-Hamiltonian systems using the Jacobi Last Multiplier and examine its connection to the Boltzmann H-theorem.
A new perspective on quantization is presented in this article, where the Koopman-von Neumann formulation (KvN) and ordinary non-relativistic quantum mechanics (OQM) are viewed as unitarily inequivalent representations of the same underlying structure. This approach to quantization is based on the idea that contextuality, rather than ℏ, is a fundamental aspect of quantum mechanics. The novel view is motivated by the limitations of traditional approaches to quantization together with previous research demonstrating contextuality in KvN. In this representation theoretic approach, both KvN and OQM are considered valid quantum theories separated by a superselection rule.
In a previous work by the author it was shown that the Koopman-von Neumann formulation of classical mechanics (KvN) exhibits quantum interference. As such it was there claimed that KvN ought to be considered as a proper quantum mechanical theory in a similar sense as ordinary non-relativistic quantum mechanics (OQM) is. In this article this claim is made manifest by showing that KvN and OQM can be taken as merely inequivalent representations resulting from one and the same quantization scheme. Of course, as the typical notion of quantization necessitates the canonical commutation relations (CCR) to hold, which they do not in KvN, the concept needs to be properly revised. To justify such a revision the typical reasons for necessitation the CCR, reductionism and quantization as 'turning brackets to commutators', are examined and found insufficient.
Hilbert-space techniques are widely used not only for quantum theory, but also for classical physics. Two important examples are the Koopman-von Neumann (KvN) formulation and the method of ``classical'' wave functions. As this paper explains, these two approaches are conceptually distinct. In particular, the method of classical wave functions was not due to Bernard Koopman and John von Neumann, but was developed independently by a number of later researchers, perhaps first by Mario Schönberg, with key contributions from Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan. The primary goals of this paper are to explain these two approaches, describe the relevant history in detail, and give credit where credit is due.
This paper examines the particle-in-a-box model as a simple but powerful example that highlights the fundamental differences between classical and quantum physics. In classical mechanics, a particle confined between two rigid walls moves with constant velocity and has a uniform probability of being found anywhere in the box; its energy can take any continuous value. In contrast, solving the Schrödinger equation for an infinite potential well yields standing-wave solutions whose wavelengths must satisfy fixed boundary conditions. These constraints produce discrete, quantized energy levels and non-uniform probability distributions, features with no classical analog. Through analytical derivations and numerical simulations using Python, we visualize wave functions, probability densities, and the statistical convergence of random samples toward the theoretical quantum distributions. Comparisons with the classical uniform distribution emphasize how quantum mechanics replaces certainty with probability and continuity with quantization. Finally, we connect the model to real physical systems—such as electrons confined in atoms—to show how this idealized system provides essential insight into the structure and stability of matter.