We construct a discrete non-Hermitian momentum operator, which implements faithfully the non-self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit, with the quarter wave as first nontrivial eigenstate. We show how to construct the corresponding Hermitian Hamiltonian for the infinite well as a concrete example to realize unitary dynamics. The resulting Hilbert space can be decomposed into a physical and unphysical subspace, which are mutually orthogonal. The physical subspace in the continuum limit reproduces that of the continuum theory and we give numerical evidence that the correct probability distributions for momentum and energy are recovered. Published by the American Physical Society 2024We construct a discrete non-Hermitian momentum operator, which implements faithfully the non-self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit, with the quarter wave as first nontrivial eigenstate. We show how to construct the corresponding Hermitian Hamiltonian for the infinite well as a concrete example to realize unitary dynamics. The resulting Hilbert space can be decomposed into a physical and unphysical subspace, which are mutually orthogonal. The physical subspace in the continuum limit reproduces that of the continuum theory and we give numerical evidence that the correct probability distributions for momentum and energy are recovered. Published by the American Physical Society 2024

arXiv:2403.13558Physical Review ResearchAug 12, 20244 citations

Quantum and Classical Momentum Distributions. I. Theory and elementary examples

Hans Jürgen Korsch, B. Schnellhaß

Jan 1, 20000 citations

A Quantum Time Coordinate

Armando Martínez-Pérez, Gabino Torres-Vega

We discuss quantum time states formed with a finite number of energy eigenstates with the purpose of obtaining a time coordinate. These time states are eigenstates of the recently introduced discrete time operator. The coordinate and momentum representations of these time eigenstates resemble classical time curves and become classical at high energies. To illustrate this behavior, we consider the simple example of the particle-in-a-box model. We can follow the quantum-classical transition of the system. Among the many existing solutions for the particle in a box, we use a set which leads to time eigenstates for use as a coordinate system.We discuss quantum time states formed with a finite number of energy eigenstates with the purpose of obtaining a time coordinate. These time states are eigenstates of the recently introduced discrete time operator. The coordinate and momentum representations of these time eigenstates resemble classical time curves and become classical at high energies. To illustrate this behavior, we consider the simple example of the particle-in-a-box model. We can follow the quantum-classical transition of the system. Among the many existing solutions for the particle in a box, we use a set which leads to time eigenstates for use as a coordinate system.

SymmetryFeb 11, 20212 citations

On the momentum of a particle in a box

G. M. Muha

Errors and misrepresentations found in some textbooks.

Journal of Chemical EducationMay 1, 19842 citations

A Floer Theoretic Approach to Energy Eigenstates on one Dimensional Configuration Spaces

Kevin Ruck

In this article we consider two classical problems in Quantum Mechanics, namely the 'particle on a ring' and the 'particle in a box' from the viewpoint of symplectic topology. Interpreting the solutions of the corresponding time independent Schrödinger equation as orbits in a suitably chosen time dependent Hamiltonian system allows us to investigate them using Floer theory. More precisely we extend the definition of Rabinowitz Floer homology to non-autonomous Hamiltonians on $\mathbb{R}^{2n}$ with its standard symplectic structure and show that compactness of the moduli space of J-holomorphic curves still holds. With this homology we are then able to prove existence results for energy $E$ eigenstates on the 'ring' or in the 'box' for a big range of exterior potentials.In this article we consider two classical problems in Quantum Mechanics, namely the 'particle on a ring' and the 'particle in a box' from the viewpoint of symplectic topology. Interpreting the solutions of the corresponding time independent Schrödinger equation as orbits in a suitably chosen time dependent Hamiltonian system allows us to investigate them using Floer theory. More precisely we extend the definition of Rabinowitz Floer homology to non-autonomous Hamiltonians on $\mathbb{R}^{2n}$ with its standard symplectic structure and show that compactness of the moduli space of J-holomorphic curves still holds. With this homology we are then able to prove existence results for energy $E$ eigenstates on the 'ring' or in the 'box' for a big range of exterior potentials.

math.SGCross-listedarXiv:2603.29201Apr 1, 2026

\mathbb{R}^{2n}

with its standard symplectic structure and show that compactness of the moduli space of J-holomorphic curves still holds. With this homology we are then able to prove existence results for energy

E

eigenstates on the 'ring' or in the 'box' for a big range of exterior potentials.