Joseph Lesnefsky with Rolfe Petschek
Optimization of the Static First Hyperpolarizibility of One or Many Non-Interacting Fermions in One Dimension
Started Fall 2011, Finished Summer 2012
Large non-linear electronic polarizabilities would be advantageous for a variety of devices and have been intensely studied for around three decades. This project will examine theoretical limits on the static first non-linear electronic susceptibility of a material, β, known as the hyperpolarizability. Kuzyk et. al [1] suggests that there is a theoretical maximum to the hyperpolarizability of N fermions. Numerical optimizations for a single fermion in a one-dimensional potential [1,2] strongly suggest that the actual value converges to 0.708951 times the theoretical limit. We will continue the study of a single electron problem [2] by modeling the potential starting from the simple quantum harmonic oscillator (SHO) and modifying the Hamiltonian with linear combinations of Hermite polynomials to maximize the hyperpolarizability. We will then examine the Hessian curvature near the maximum to deduce what parts of the wave function and the potential are integral to attaining this maximized hyperpolarizability. The hyperpolarizability for multiple non-interacting fermions will then be investigated using similar techniques. This may suggest significantly stronger limits on the hyperpolarizability of many electron systems.