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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 http://www.cs.hmc.edu/$\sim$ccecka/QuantumModel/ (Best viewed in Windows Explorer and tends to be moderately variable with respect to the speed of the machine.) So our Euler approximation becomes Agreement With Theory So we code this algorithm up to produce the applet shown here. We can test the algorithm against theory to verify that it is acting accordingly. I tested the transmission coefficient Relative to Box Width Let Number of spacial points to use in integration The wave function can then be obtained from Potential Energy Function V(x) Presets Potential Energy This linear equation can be written as Derivation of Numerical Method Using atomic units (that is, all constants are set equal to 1) the Schrödinger Equation becomes Finally, this can be separated as solved to produce the following algorithm We will be working in a discrete domain so Sweet Sweet Stuff Cris Cecka Wave function. Probability distribution of finding the particle at each spacial coordinate. 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) Potential Energy Function V(x) parameters Wave Norm Over Time Quantum, and logic, require the probability of finding the particle over all space to be 1.000. The Norm (the integral over the probability distribution) should have a value of 1.000 while the algorithm is being run. Using a classic second order approximation 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. When an Euler approximation is used Other Tests 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 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 Note that over 35000 iterations (~3min), the norm deviates by initial conditions in the large figure). Impressively, the only .06%. We would have to run the algorithm for ~3 hours to original wave function will be presented at exactly the time it see a deviation of 1%, at which point the wave function would is predicted to with a qualitatively perfect representation of be of no qualitative or quantitative use to us anyway. its initial state. Overall, a lot of analysis can come out of this program. 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)