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MATTER International Journal of Science and Technology

CLASSICAL VS. QUANTUM BEHAVIOR IN THE PARTICLE-IN-A-BOX MODEL

Yinuo Guo

Published Jun 17, 2026• 0 citations

Abstract

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.

DOI: 10.20319/mijst.2026.12.3754

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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.

In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The wave functions for the hydrogen atom are then guessed based on the historical solutions.

In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The wave functions for the hydrogen atom are then guessed based on the historical solutions.

In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The wave functions for the hydrogen atom are then guessed based on the historical solutions.

In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The wave functions for the hydrogen atom are then guessed based on the historical solutions.

This paper is illustrated about the behavior of quantum mechanics looked like the that of classical mechanics when the principle quantum number approaches large values. This phenomenon was explained based on the energy quantization and probability density. When the energy levels of Bohr hydrogen atom for large quantum states are determined, these energy values are continuous values. It has been stated that the one- dimensional particle in a box wave function and probability density by changing the quantum numbers. The probabilities of finding the particle in a box for two sections are calculated with the aid of computer programming. It is found that the calculated probability density results depend not only on the quantum number but also on particle’s location.