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Annals of Physics

Revisiting completeness of the eigenstates of a Hermitian operator for a particle in 1-D: An alternative proof, an alternative description, and the possibility of quantum measurement of the momentum in a box

Shyamal Biswas, Kalyan Brata Chatterjee

Published Sep 17, 2026• 0 citations
Publisher
DOI
10.1016/j.aop.2026.170732

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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 2024