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

Felix Fischer, Daniel Burgarth, Davide Lonigro

TL;DR

The paper reveals that finite-dimensional Galerkin truncations of infinite quantum Hamiltonians can converge to the Friedrichs extension of the restricted operator rather than to the original $H$, potentially yielding incorrect dynamics undetectable without the analytical solution. It provides abstract convergence results and applies them to the quantum particle in a one-dimensional box, showing that a boundary-blind basis often yields Dirichlet dynamics in the limit, while modest basis modifications can produce $\alpha$-periodic dynamics. Concrete constructions using associated Legendre polynomials ($m\ge4$) illustrate the Dirichlet case, and Gram-Schmidt-based bases enable convergence to periodic cases. The work emphasizes the subtle interplay between discretization, operator extensions, and boundary conditions, with implications for numerical quantum dynamics in confined systems and beyond.

Abstract

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 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.

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

TL;DR

The paper reveals that finite-dimensional Galerkin truncations of infinite quantum Hamiltonians can converge to the Friedrichs extension of the restricted operator rather than to the original , potentially yielding incorrect dynamics undetectable without the analytical solution. It provides abstract convergence results and applies them to the quantum particle in a one-dimensional box, showing that a boundary-blind basis often yields Dirichlet dynamics in the limit, while modest basis modifications can produce -periodic dynamics. Concrete constructions using associated Legendre polynomials () illustrate the Dirichlet case, and Gram-Schmidt-based bases enable convergence to periodic cases. The work emphasizes the subtle interplay between discretization, operator extensions, and boundary conditions, with implications for numerical quantum dynamics in confined systems and beyond.

Abstract

When numerically simulating the unitary time evolution of an infinite-dimensional quantum system, one is usually led to treat the Hamiltonian 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 to the space of finite linear combinations of elements of the basis. Importantly, this is generally different from 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.

Paper Structure

This paper contains 25 sections, 33 theorems, 233 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Convergence of Galerkin approximations
  3. Galerkin approximations of the particle in a box
  4. Some bibliographic remarks
  5. Outline of the paper
  6. Galerkin approximations: general results
  7. Galerkin approximations of the particle in a box
  8. Preliminaries
  9. Main results: convergence to the Dirichlet Laplacian
  10. Main results: convergence to the alpha-periodic Laplacian
  11. Summary
  12. Proofs of Section \ref{['sec:galerkin-general']}
  13. Proof of Proposition \ref{['prop:galerkin_criterion']}
  14. Proof of Proposition \ref{['prop:friedrichs']}
  15. Proofs of Section \ref{['sec:particle_results']}
  16. ...and 10 more sections

Key Result

Proposition 2.2

Let $H$ be a self-adjoint operator on $\mathcal{H}$, $(P_n)_{n \in \mathbb{N}}$ a family of projectors as in Definition def:galerkin_approximation, additionally satisfying $P_n P_m = P_n$ for all $m \geq n$, and $U_{n}(t)=\mathrm{e}^{-\mathrm{i}\mkern1mu tH_{n}}$ the unitary propagator associated wi where $U(t) = \mathrm{e}^{-\mathrm{i}\mkern1mu tH}$ is the unitary propagator generated by $H$.

Figures (4)

  • Figure 1: Approximation error $\norm*{U_W(t)\psi_0-U_n(t)\psi_0}$ for the Dirichlet time evolution $U_{\mathrm{Dir}}$ of a Dirichlet eigenvector $\psi_0(x) = \sin(5/2 \pi (x+1))$ (left) and for the periodic time evolution $U_{\mathrm{per}}(t)$ of a periodic eigenvector $\psi_0(x)=1/\sqrt{2}$ (right) over the number of basis elements $n$ and for different times $t$. In both cases, finite-dimensional truncations are constructed by means of associated Legendre polynomials $p_l^m$ with $m=4$. The error only converges to $0$ in the Dirichlet case, in agreement with our analytical results (cf. Theorem \ref{['thm:legendre-dirichlet']}).
  • Figure 2: Pictorial representation of the Galerkin approximation. The unbounded operator $H$ on the infinite-dimensional space $\mathcal{H}$ is truncated by means of a family of finite-dimensional projectors $(P_n)_{n\in\mathbb{N}}$: this yields the bounded operator $H_n = P_n H P_n$ on $\mathcal{H}$, which only acts nontrivially on the finite-dimensional subspace $\mathcal{H}_n=P_n\mathcal{H}$, where its action is described by $\hat{H}_n$, and zero everywhere else.
  • Figure 3: Time evolution of an arbitrary initial wavefunction $\psi_0(x)$ (a) and b)) and the specific symmetric initial wavefunction $\psi_0(x) = \sin(\pi/2(x+1))$ (c) and d)) in a box with hard walls (a) and c)) and with periodic boundary conditions (b) and d)). All functions are evaluated at $t = 4/ \pi$, and the imaginary part of all wavefunctions is zero.
  • Figure 4: The normalized associated Legendre polynomials $p_l^m$ (left) and their derivatives $\diff*{p_l^m(x)}{x}$ (right) for $l \in \{4,5,8\}$ and $m = 4$.

Theorems & Definitions (85)

  • Definition 2.1: Galerkin approximation
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Definition 2.7: Galerkin projector
  • Proposition 2.8
  • ...and 75 more