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Foundations of Physics

Solutions of the Time-Dependent Schrödinger Equation for a Two-State System

Jason F. Ralph, Terrence D. Clark, Helen Prance, Robert J. Prance, A. Widom, Y. N. Srivastava

Published Aug 1, 1998• 0 citations
Publisher
DOI
10.1023/a:1018822809439

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The nonlinear optimization techniques of dynamic simulated annealing and steepest descents make it feasible to solve numerically the Schrödinger equation for systems that have many degrees of freedom. These methods involve finding the solution to a set of quasi-Newtonian equations of motion derived from a fictitious, time-dependent Schrödinger equation. The formalism is reviewed and the methods are illustrated by solving the Schrödinger equation for three simple models: a two-state system, an infinite square well, and the Mathieu equation.
Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.