Fischer F, Burgarth D, Lonigro D (2026)
Publication Type: Journal article
Publication year: 2026
Book Volume: 10
Article Number: 1985
DOI: 10.22331/q-2026-01-27-1985
When numerically simulating the unitary time evolution of an infinite dimensional quantum system, one is usually led to treat the Hamiltonian H as an “infinite-dimensional matrix” by expressing it in some orthonormal basis of the Hilbert space, and then truncate it to some finite dimensions. However, the solutions of the Schrödinger equations generated by the truncated Hamiltoni ans need not converge, in general, to the solution of the Schrödinger equation corresponding to the actual Hamiltonian. In this paper we demonstrate that, under mild assumptions, they converge to the solution of the Schrödinger equation generated by a specific Hamiltonian which crucially depends on the particular choice of basis: the Friedrichs exten sion of the restriction of H to the space of finite linear combinations of elements of the basis. Importantly, this is generally different from H itself; in all such cases, numerical simulations will unavoidably reproduce the wrong dynamics in the limit, and yet there is no numerical test that can reveal this failure, unless one has the analytical solution to compare with. As a practical demonstration of such results, we consider the quantum par ticle in the box, and we show that, for a wide class of bases—which include as sociated Legendre polynomials as a concrete example—the dynamics generated by the truncated Hamiltonians will always converge to the one corresponding to the particle with Dirichlet boundary conditions, regardless the initial choice of boundary conditions. Other such examples are discussed.
APA:
Fischer, F., Burgarth, D., & Lonigro, D. (2026). Quantum particle in the wrong box (or: the perils of finite-dimensional approximations). Quantum, 10. https://doi.org/10.22331/q-2026-01-27-1985
MLA:
Fischer, Felix, Daniel Burgarth, and Davide Lonigro. "Quantum particle in the wrong box (or: the perils of finite-dimensional approximations)." Quantum 10 (2026).
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