The quantum problem of stationary states of a particle in a box is revisited by means of the unilateral Fourier transform. Homogeneous Dirichlet boundary conditions demand a finite Fourier sine transform which is actually the Fourier sine series.
Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as O((logN)c), where N is the problem size and c a constant. We propose a quantum spectral method with polylogarithmic complexity for solving non-periodic boundary value problems with arbitrary Dirichlet boundary conditions. Our method extends the recently proposed approach by Liu et al. (2025), in which periodic problems are discretised using truncated Fourier series. In such spectral methods, the discretisation of boundary value problems with constant coefficients leads to a set of algebraic equations in the Fourier space. We implement the respective diagonal solution operator by first approximating it with a polynomial and then quantum encoding the polynomial. The mapping between the physical and Fourier spaces is accomplished using the quantum Fourier transform (QFT). To impose zero Dirichlet boundary conditions, we double the domain size and reflect all physical fields antisymmetrically. The respective reflection matrix defines the quantum sine transform (QST) by pre- and post-multiplying with the QFT. For non-zero Dirichlet boundary conditions, the solution is decomposed into a boundary-conforming and a homogeneous part. The homogenous part is determined by solving a problem with a suitably modified forcing vector. We illustrate the basic approach with a Dirichlet-Poisson problem and demonstrate its generality by applying it to a fractional stochastic PDE for modelling spatial random fields. We discuss the circuit implementation of the proposed approach and provide numerical evidence confirming its polylogarithmic complexity.
Abstract The Laplace transform is a valuable tool in physics, particularly in solving differential equations with initial or boundary conditions. A 2014 study by Tsaur and Wang (2014 Eur. J. Phys. 35 015006) introduced a Laplace-transform-based method to solve the stationary Schrödinger equation for various potentials. However, their approach contains critical methodological flaws: the authors disregard essential boundary conditions and apply the residue theorem incorrectly in the inverse transformation process. These errors ultimately cancel out, leading to correct results despite a flawed derivation. In this paper, we revisit the use of the Laplace transform for the one-dimensional Schrödinger equation, clarifying correct practices in handling boundary conditions and singularities. This analysis offers a sound and consistent framework for the application of Laplace transforms in stationary quantum mechanics, underscoring their educational utility in quantum mechanics coursework.