A Real-Time Numerical A Real-Time Numerical Integrator for the One-Integrator for the One-

Dimensional Time-Dependent Dimensional Time-Dependent Schrödinger EquationSchrödinger Equation

Cris CeckaCris Cecka

April 29April 29thth 2004 2004

Harvey Mudd CollegeHarvey Mudd College

PurposePurpose

To derive a Numerical Integration method To derive a Numerical Integration method for the One-Dimensional Time-Dependent for the One-Dimensional Time-Dependent Schrödinger Equation.Schrödinger Equation.

To determine validity and accuracy of To determine validity and accuracy of method.method.

It’s all Greek…It’s all Greek…

A whole lotta GreekA whole lotta Greek

Almost there…Almost there…

SweetSweet

Check it OutCheck it Out

http://http://www.cs.hmc.edu/~ccecka/QuantumModelwww.cs.hmc.edu/~ccecka/QuantumModel

Accuracy BabyAccuracy Baby

Other TestsOther Tests

Other Other TestsOther Other Tests

The eigenfunction expansion of the wave The eigenfunction expansion of the wave form can be shown to be conserved over form can be shown to be conserved over long periods!!long periods!!

AstoundingAstounding

Future PlansFuture Plans

User defined potentialUser defined potentialTime-Dependent potentialTime-Dependent potential

Dirac SmashingDirac SmashingMathematical implication of complex-Mathematical implication of complex-

valued potentialsvalued potentialsMomentum spaceMomentum spaceDerivation of eigenfunction expansion Derivation of eigenfunction expansion

using interference patternsusing interference patterns

ReferencesReferences A. Askar and A.S. Cakmak, Explicit Integration Method for the Time-A. Askar and A.S. Cakmak, Explicit Integration Method for the Time-

Dependent Schrodinger. Equation for Collision Problems, J. Chem. Phys. Dependent Schrodinger. Equation for Collision Problems, J. Chem. Phys. (1978).(1978).

Visscher, P. B. A fast explicit algorithm for the time-dependent Schrodinger Visscher, P. B. A fast explicit algorithm for the time-dependent Schrodinger equation.equation.

Robert Eisberg and Robert Resnick, Quantum Physics (John Wiley \& Sons, Robert Eisberg and Robert Resnick, Quantum Physics (John Wiley \& Sons, Inc., New York, 1974)Inc., New York, 1974)

L. G. de Pillis, private communcation, 2004L. G. de Pillis, private communcation, 2004