Transcript A Real Time Numerical Integrator for the 1D Time Dependent

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)