Bonn 2025 – scientific programme

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QI: Fachverband Quanteninformation

QI 14: Quantum Entanglement III

QI 14.4: Talk

Tuesday, March 11, 2025, 15:00–15:15, HS IX

Quantum particle in the wron box - the perils of finite dimensional approximations — •Felix Fischer, Davide Lonigro, and Daniel Burgarth — FAU Erlangen

When numerically simulating a quantum mechanically system, one usually treats the Hamiltonian as an infinite dimensional matrix given in some basis. Then, one truncates this matrix to some finite dimension and diagonalizes the approximate, finite dimensional Hamiltonians. In general, the spectra of these truncated Hamiltonians do not converge towards the spectra of the original Hamiltonian. We show that this happens in the text book example for a quantum mechanical system - The Particle in a Box. When choosing a boundary agnostic basis, the numerics converge towards the particle in box with Dirichlet boundary conditions - independently of the boundary conditions one aims to simulate. In this talk we outline why these problems arise and show that the numerics always converge to one specific Hamiltonian - the Friedrichs extension of the restriction of the original Hamiltonian onto the finite dimensional span of the basis.

Keywords: Galerkin approximation; Boundary conditions; Convergence; Numerics; Friedrichs extension