Quantum Dot and Acoustic Enclosure Problems in Lens-Shaped Structures

A method for solving exactly the Helmholtz equation in parabolic rotational coordinates is presented using separability of the eigenfunctions and the Frobenius power series expansion technique. Two examples of interest in wave physics are considered and analyzed quasi-analytically: (I) the wavefunctions of an electron in a quantum dot confined by two paraboloids (forming a lens-shaped structure) and the associated energy spectrum, and (II) the acoustic eigenmodes and eigenfrequencies of the pressure field bounded by rigid walls as defined by two paraboloids. The quantum dot (acoustic enclosure) problem is a Dirichlet (Neumann) boundary condition problem. In both cases, eigenfunctions and eigenmodes are calculated and the shape-dependence of the first eigenvalue for the groundstate in the quantum dot case (and the fundamental mode in the acoustic enclosure case) is examined.