The quantum mechanical behavior of a particle between ellipses

Abstract

The quantum mechanical behavior of a particle constrained between two confocal ellipses is investigated as a model for cyclic pi systems with nondegenerate energy levels. The solutions of the Schrödinger wave equation for this system are analytic: the Mathieu functions. Effects of boundary curvature and annulus constriction on the wave functions and eigenvalues are determined and discussed. Low lying states may exhibit tunneling in regions of negative angular kinetic energy. As examples of an application to chemical systems, the model is calibrated to reproduce low lying π–π* transitions in several substituted benzenes. The limiting one dimensional problem of a particle-on-an-ellipse is discussed.

Hypervirial analysis of enclosed quantum mechanical systems. I. Dirichlet boundary conditions

“…At variance with the usual requirement that wave functions should approach zero at infinity, it is sometimes useful and necessary to consider enclosed quantum mechanical systems. This type of model has been widely applied in several fields, among which we can mention (a) the effect of the high pressure on the properties of atoms and molecules [l-lo]; (b) an electron gas in a uniform positive background [ll]; (c) the treatment of magnetic behavior [12]; (d) periodic solid state problems [13-151; (e) optical activity of polynucleotides [16]; (f) cyclic 7r systems with nondegenerate energy levels [17]; (g) exchange forces [18]; (h) catalytic properties of certain energy-transducing biological systems [ 191; (i) apparent quark confinement [20]; etc. Diagonal hypervirial relations [21,22] (4KH WIl4) = 0 (1) are valid when W4 belongs to the domain of H. In this case H is a Hermitian operator, 4 is the wave function, and W is the hypervirial operator.…”