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Quantum particle in the wrong box (or: the perils of finite-dimensional approximations)

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Researchers Felix Fischer, Daniel Burgarth, and Davide Lonigro reveal that finite-dimensional approximations of infinite quantum systems often produce incorrect dynamics, even when simulations appear to converge. The study shows truncated Hamiltonians converge to a basis-dependent Friedrichs extension rather than the true Hamiltonian, meaning simulations may unknowingly model the wrong quantum system. Using a particle-in-a-box model, they demonstrate that common bases (like Legendre polynomials) force convergence to Dirichlet boundary conditions regardless of initial settings. The team identifies the limiting dynamics when the Hamiltonian’s ground state energy is finite, providing a way to predict and correct these errors by redesigning approximation schemes. This work exposes a fundamental flaw in quantum simulations, offering mathematical tools to detect and fix discrepancies that would otherwise remain hidden without analytical solutions.

Quantum particle in the wrong box (or: the perils of finite-dimensional approximations)

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AbstractWhen 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 Hamiltonians 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 extension 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 particle in the box, and we show that, for a wide class of bases (which include associated 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.Featured image: When truncating an unbounded Hamiltonian $H$ in some orthogonal basis, information about the domain of the Hamiltonian $\cal{D}(H)$ is usually lost.Popular summaryNumerical simulations of quantum dynamics approximate infinite-dimensional systems by truncating the Hamiltonian to a large finite matrix in a chosen basis. This can lead to misleading results: even when simulations converge, they may do so to a different quantum dynamics than the one originally intended, with the limit depending on the chosen basis. The main result of this work is that this limiting dynamics can be identified whenever the ground state energy of the Hamiltonian in question is finite. We show that truncations generically converge to the dynamics generated by a specific self-adjoint extension of the Hamiltonian, determined by the chosen basis. This insight provides a constructive way to predict and ultimately fix these failures: once the selected extension is known, one can redesign the approximation scheme to recover the desired physical dynamics. We illustrate these ideas with the example of a quantum particle in a one-dimensional box. For a wide class of bases, the truncated dynamics converges to that corresponding to Dirichlet boundary conditions, independently of the boundary conditions one starts from. The main result of this work is that this limiting dynamics can be identified whenevert the ground state energy of the Hamiltonian in question is finite. We show that truncations generically converge to the dynamics generated by a specific self-adjoint extension of the Hamiltonian, determined by the chosen basis. This insight provides a constructive way to predict and ultimately fix these failures: once the selected extension is known, one can redesign the approximation scheme to recover the desired physical dynamics. We illustrate these ideas with the example of a quantum particle in a one-dimensional box. For a wide class of bases, the truncated dynamics converges to that corresponding to Dirichlet boundary conditions, independently of the boundary conditions one starts from.► BibTeX data@article{Fischer2026quantumparticlein, doi = {10.22331/q-2026-01-27-1985}, url = {https://doi.org/10.22331/q-2026-01-27-1985}, title = {Quantum particle in the wrong box (or: the perils of finite-dimensional approximations)}, author = {Fischer, Felix and Burgarth, Daniel and Lonigro, Davide}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1985}, month = jan, year = {2026} }► References [1] D. J. Tannor. ``Introduction to Quantum Mechanics: A Time-Dependent Perspective''.

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Proceedings of the National Academy of Sciences 41, 645–647 (1955). https://doi.org/10.1073/pnas.41.9.645Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-27 11:35:52: Could not fetch cited-by data for 10.22331/q-2026-01-27-1985 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-27 11:35:52: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWhen 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 Hamiltonians 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 extension 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 particle in the box, and we show that, for a wide class of bases (which include associated 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.Featured image: When truncating an unbounded Hamiltonian $H$ in some orthogonal basis, information about the domain of the Hamiltonian $\cal{D}(H)$ is usually lost.Popular summaryNumerical simulations of quantum dynamics approximate infinite-dimensional systems by truncating the Hamiltonian to a large finite matrix in a chosen basis. This can lead to misleading results: even when simulations converge, they may do so to a different quantum dynamics than the one originally intended, with the limit depending on the chosen basis. The main result of this work is that this limiting dynamics can be identified whenever the ground state energy of the Hamiltonian in question is finite. We show that truncations generically converge to the dynamics generated by a specific self-adjoint extension of the Hamiltonian, determined by the chosen basis. This insight provides a constructive way to predict and ultimately fix these failures: once the selected extension is known, one can redesign the approximation scheme to recover the desired physical dynamics. We illustrate these ideas with the example of a quantum particle in a one-dimensional box. For a wide class of bases, the truncated dynamics converges to that corresponding to Dirichlet boundary conditions, independently of the boundary conditions one starts from. The main result of this work is that this limiting dynamics can be identified whenevert the ground state energy of the Hamiltonian in question is finite. We show that truncations generically converge to the dynamics generated by a specific self-adjoint extension of the Hamiltonian, determined by the chosen basis. This insight provides a constructive way to predict and ultimately fix these failures: once the selected extension is known, one can redesign the approximation scheme to recover the desired physical dynamics. We illustrate these ideas with the example of a quantum particle in a one-dimensional box. For a wide class of bases, the truncated dynamics converges to that corresponding to Dirichlet boundary conditions, independently of the boundary conditions one starts from.► BibTeX data@article{Fischer2026quantumparticlein, doi = {10.22331/q-2026-01-27-1985}, url = {https://doi.org/10.22331/q-2026-01-27-1985}, title = {Quantum particle in the wrong box (or: the perils of finite-dimensional approximations)}, author = {Fischer, Felix and Burgarth, Daniel and Lonigro, Davide}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1985}, month = jan, year = {2026} }► References [1] D. J. Tannor. ``Introduction to Quantum Mechanics: A Time-Dependent Perspective''.

Theoretica Chimica Acta 44, 9–26 (1977). https://doi.org/10.1007/BF00548026 [25] B. Klahn and W. A. Bingel. ``The Convergence of the Rayleigh-Ritz Method in Quantum Chemistry''.

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