A Real-Time Numerical Integrator for the Spring 2004 Scientific Computing – Professor L. G. de Pillis A Real-Time Numerical Integrator for the One-Dimensional Time-Dependent Schrödinger Equation Abstract In this paper, I investigate a numerical method of integrating the One-Dimensional Time-Dependent Schrödinger Equation. A numerical method is derived using a method that is reminiscent of Runge Kutta, but implicit in the algorithm. This method is written into an integrator and tested for validity of results with respect to quantum constructs as well as accuracy with a quantum tunneling benchmark. A Java applet is used to show results of particle wave propagation in real time for various potential energy functions and initial conditions. The applet can be found online at (Best viewed in Windows Explorer and tends to be moderately variable with respect to the speed of the machine.) Derivation of Numerical Method Using atomic units (that is, all constants are set equal to 1) the Schrödinger Equation becomes We will be working in a discrete domain so Using a classic second order approximation When an Euler approximation is used This does not give a good approximation however since the system is “stiff” (the eigenvalues of the Jacobian matrix differ greatly, resulting in divergent results). Runge Kutta is difficult to use since we do not have a closed form differential function with respect to time. Note that Sweet Sweet Stuff So we code this algorithm up to produce the applet shown here. Wave Norm Over Time Quantum, and logic, require the probability of finding the particle over all space to be The Norm (the integral over the probability distribution) should have a value of while the algorithm is being run. Note that over iterations (~3min), the norm deviates by only.06%. We would have to run the algorithm for ~3 hours to see a deviation of 1%, at which point the wave function would be of no qualitative or quantitative use to us anyway. Agreement With Theory We can test the algorithm against theory to verify that it is acting accordingly. I tested the transmission coefficient The maximum deviation from the theoretical value was 1.25%, while the average deviation was 0.7%. These results are incredibly accurate and definitely surprised and pleased me. Other Tests One interesting test is the interference pattern of a particle in a box with no initial momentum. As expected, each eigenfunction will present itself in time due to the evolution of the interference of phasors for each eigenfunction. (The initial conditions in the large figure). Impressively, the original wave function will be presented at exactly the time it is predicted to with a qualitatively perfect representation of its initial state. Overall, a lot of analysis can come out of this program. So our Euler approximation becomes Let The wave function can then be obtained from This linear equation can be written as Finally, this can be separated as solved to produce the following algorithm Cris Cecka Number of spacial points to use in integration Potential Energy Function V(x) Presets Relative to Box Width Potential Energy Our method implicitly forces the ends of the wave function to be zero. This corresponds to requiring infinite potential walls on both sides of any potential energy function V(x) Wave function. Probability distribution of finding the particle at each spacial coordinate. Potential Energy Function V(x) parameters Acknowledgments Professor L. G. de Pillis A. Askar and A.S. Cakmak, Explicit Integration Method for the Time-Dependent Schrödinger Equation for Collision Problems, J. Chem. Phys. (1978). Visscher, P. B. A fast explicit algorithm for the time-dependent Schrödinger equation. Robert Eisberg and Robert Resnick, Quantum Physics (John Wiley & Sons, Inc., New York, 1974)