Expansion method for stationary states of quantum billiards

Abstract

A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape ͑quantum billiard͒ is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.

A difference-equation formalism for the nodal domains of separable billiards

“…Coupled with the fact that the only integrable but non-separable billiards in two dimensions-the right-angled isosceles, the equilateral and the 30 • − 60 • − 90 • hemiequilateral triangle [43,44]-are already known to exhibit such a property [22], this completes a comprehensive description of the recurrence relations in all planar, integrable billiards and hints at a signature of integrability, hidden in the nodal domain counts alone. Furthermore, our framework also prescribes a convenient methodology to obtain analytical expressions for the total number of domains of all such billiards in two dimensions, thereby mathematically quantifying the geometric nodal patterns and establishing a much-needed foundation for further statistical investigations.…”

Exploring the distinction between experimental resonant modes and theoretical eigenmodes: From vibrating plates to laser cavities

“…If we assume that the microwave is totally reflected at the inner scatter, the Dirichlet condition is imposed on the boundary of scatter to solve the wave functions. To construct the resonant modes of this nonintegrable Sinai-like billiards, we first exploit the expansion method [6] to evaluate the eigenvalues k n and eigenmodes ϕ n (x,y) under this geometry as a quasicomplete basis. The detailed mathematics are similar as the derivation in Ref.…”

“…Eigenmodes of the bounded wave systems are generally associated with the homogeneous Helmholtz equation with boundary conditions [6][7][8]. Two-dimensional (2D) scientific systems such as microwave cavities [9,10], vibrating plates [11], oscillating water tanks [12], and laser resonators [13,14] have been extensively employed to generate wave patterns of resonant modes that are presumed to be the experimental observations for theoretical eigenmodes.…”

Non-Hermitian Hamiltonian and Lamb shift in circular dielectric microcavity

“…The so-called quantum billiard systems are conservative closed systems with Dirichlet boundary condition described by Hermitian Hamiltonian with real eigenvalues [1,2]. Once these systems become open, i.e., coupled to their environment, the situation changes.…”