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arXiv:2602.14654 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Boundary conditions for the Schrödinger equation in the numerical simulation of quantum systems

Marco Patriarca

quant-phcond-mat.mes-hallphysics.comp-phSubmitted Feb 16, 2026• Announced Feb 17, 2026• 7 citations
arXivHTMLPDF

Abstract

We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schrödinger equation. On one hand, we show that a closed quantum system is defined by local boundary conditions. On the other hand, we argue that, because of the uncertainty principle, no local boundary condition can be defined for open quantum systems. For this reason plane waves or wave packet trains cannot be simulated on a finite numerical lattice with the usual procedures. We suggest a method that avoids these difficulties by using only a small numerical lattice and maintains the correspondence with the physical picture, in which the incident and scattered waves may be infinitely extended.

Journal reference
Phys. Rev. E 50, 1616 (1994); Phys. Rev. E 112, 059901(E) (2025)
DOI
10.1103/Phys-RevE.50.1616
Comments
7 pages, 4 figures

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We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schrödinger equation. On one hand, we show that a closed quantum system is defined by local boundary conditions. On the other hand, we argue that, because of the uncertainty principle, no local boundary condition can be defined for open quantum systems. For this reason plane waves or wave packet trains cannot be simulated on a finite numerical lattice with the usual procedures. We suggest a method that avoids these difficulties by using only a small numerical lattice and maintains the correspondence with the physical picture, in which the incident and scattered waves may be infinitely extended.
The open-system boundary conditions for the one-dimensional Schr\"odinger equation are derived by dividing the unbounded domain into a finite system and two semi-infinite reservoirs. The resulting boundary conditions on the system are non-Markovian, as they contain a convolution over the history of the system. Thus, time-irreversibility arises in a pure-state problem. The propagator which appears in the boundary condition is derived for a simple discrete model. The correctness of the boundary conditions is verified and the usefulness of the discrete model is demonstrated by a numerical calculation of the time-evolution of a wave packet.
The quantum lattice-gas automata (QLGA) have been studied to simulate the time-dependent Schrodinger equation. It is particularly well suited to implementation on a quantum computer; the many-body quantum system can be simulated very efficiently. On the other hand, its emulation on a classical computer would also have an advantage, because its unitarity might lead better behavior than standard finite-difference methods. In this study, we apply the QLGA method to the analysis of the one-particle quantum wave propagation in nanoscale devices. To this end, we develop a new algorithm to implement the open boundary condition into the QLGA method. In the QLGA simulation, the space is discretized and qubits are assigned to each node of the lattice. The each qubit contains the probability amplitude of whether the lattice node is occupied or not by the simulated particles. We have assigned the qubit array outside the simulation region to store the information about the particles which have flown out across the boundary. Then it becomes possible to simulate the open systems with the QLGA simulation.
We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schrödinger equation that models a one-dimensional quantum lattice. Using a recently developed exact solution for the discrete system, we derive the Dirichlet-to-Neumann maps analytically via Laplace transforms. This yields a convolution-type boundary condition governed by Bessel functions. We rigorously demonstrate the consistency of this discrete formulation with its continuous counterpart in the continuum limit. Additionally, we present an efficient time-discretization scheme based on the trapezoidal rule for practical implementation. Numerical experiments using a Crank-Nicolson solver verify that our proposed TBCs eliminate spurious backscattering entirely and preserve reflectionless propagation of a Gaussian wave packet exiting the computational domain
The scattered wave formalism is developed for a quantum subsystem interacting with the external environment through open boundaries. The total wave function is divided into incident and scattered components and Markovian outgoing wave boundary conditions are applied to the scattered wave function. This formalism significantly reduces the computational effort relative to other methods which rely on Green functions and memory kernels. The method is applied to one-dimensional barrier scattering and to a three-dimensional model for the field effect transistor.
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.
We review analyses of open quantum systems. We show how non-Hermiticity arises in an open quantum system with an infinite environment, focusing on the one-body problem. One of the reasons for taking the present approach is that we can solve the problem completely, making it easier to see the structures of problems involving open quantum systems. We show that this results in the discovery of a new complete set, which is one of the main topics of the present article. Another reason for focusing on the one-body problem is that the theory permits the strong coupling between the system and the environment. In the current research landscape, it is valuable to revisit the one-body problem for open quantum systems, which can be solved accurately for arbitrary strengths of the system-environment couplings. A rigorous understanding of the problem structures in the present approach will be helpful when we tackle problems with many-body interactions. First, we consider potential scattering and directly define the resonant state as an eigenstate of the Schrödinger equation under the Siegert outgoing boundary condition. We show that the resonant eigenstate can have a complex energy eigenvalue, even though the Hamiltonian is seemingly Hermitian. Second, we introduce the Feshbach formalism, which eliminates the infinite degrees of freedom of the environment and represents its effect as a complex potential. The resulting effective Hamiltonian is explicitly non-Hermitian. By unifying these two ways of defining resonant states, we obtain a new complete set of bases for the scattering problem that contains all discrete eigenstates, including resonant states. We finally mention the non-Markovian dynamics of open quantum systems. We emphasize the time-reversal symmetry of the dynamics that continuously connects the past and the future. We can capture it using the new complete set that we develop here.