Quantum Behavior of Time Dependent Systems; Ionization Problems on the Half Line
Student: Matt Kohut
Dr. Ovidiu Costin and Dr. Rodica Costin (Rutgers Math Department)
For more information on my project, please see Michael
Project Description: The physical situation we are examining is a particle in a quantum mechanical system (such as an atom) which is essentially free to move anywhere on the positive real axis. The region to the left of zero is taken to represent some type of physical barrier (such as a wall) which is impossible to cross. The particle is subject to a periodic external force given by W(x)sin(wt) and a purely spatial dependent potential given by the so-called Delta function. The Delta function is defined to be zero on the entire real line except for one point a at which it is effectively infinite (so that the integral over the real line of the Delta function is 1). Our objective is to determine what the long term behavior of such a system is like; specifically, we'd like to determine if ionization of the system occurs. and how this depends on the parameters of the system. For those who don't know, ionization refers to the phenomenon where a particle eventually leaves the system, such as the ejection of elelectrons in atoms.
Methodology: Our method will be to analyze solutions to an instance of Schrodinger's Equation, which is the governing equation for quantum mechanical systems. Schrodinger's Equation gives the wave function of the particle, which is essentially a probability density function for the system at a time t (so integrating the wave function over a subset of the real line for a fixed t gives the probability of finding the particle in that set at time t). Given initial conditions, we'll show that our particular wave function goes to zero as time goes to infinity (which is equivalent to saying that ionization has occurred). In order to study the long term time dependence of the wave function (that is, to study the growth order), we will take the Laplace Transform of Schrodinger's Equation and instead study the analyticity of the Laplace Transformed Wave Function. This method is justified if one notes that Laplace Transforms turn multiplication into differentiation (in a sense).