Quantum carpets in leaky boxes

Abstract

Quantum waves evolving or propagating inside one-dimensional boxes generate ‘quantum carpets’: intricate patterns in spacetime. The formally equivalent patterns in classical paraxial optics are observed as the Talbot effect. Boxes that are leaky, i.e. with penetrable walls, can be modelled by representing the walls as nonhermitian boundaries. The waves are superpositions of discrete eigenmodes with complex eigenvalues. Several different boundary conditions are explored, in which the Robin parameter—ratio of the mode derivatives and their values at the walls—may or may not depend on the eigenvalue. Nonhermitian models, which ignore the waves leaking outside the box, or represent them in simplified form, are approximations. For one physical model, the evolving wave in the full space can be calculated exactly, indicating that the nonhermitian model for the wave inside is highly accurate. Leaky boxes can be useful pedagogically: as introductory examples of nonhermiticity and biorthogonality (left and right eigenvectors), and as a rich source of postgraduate or advanced undergraduate student projects, exploring many variants.