
Section 11 Exercise: The Schrödinger Equation


CVS

[about 1 minute]

To extract the exercise files for this weeks exercise, do
cd ~/ComputerPhysics
cvs update d
In case of problems, see the CVS
update help page.
To avoid problems with "undefined procedures or functions"
make sure IDL
can find the necessary procedures.
The xpacket
experimentarium

[about 75 minutes]

Today's Xwidget interface should help refresh your understanding of the
basic
properties of solutions to the Schrödinger equation.
Starting Up
Flat Potential: Wave Packets
The initial condition corresponds to a wave packet bouncing back and
forth

Observe how the bouncing of the wave packet creates interference at
the walls

Look at the width of the wave packet.
than the speed of the wave envelope (the group speed)?


Measure how far the package travels, for a number of steps of size
1,
twice as many steps of size 0.5, and four times as many steps of
size 0.25
(choose numbers such that you can easily double/half the values).

In order for the complex wave to be reasonably resolved, we should
also limit
its wave length to be longer than about 6 grid zones.
(Check the code, the theory, or the plot.)


You may want to make the plot points more visible, and "zoom
in", using something similar to
IDL> !p.psym=1
IDL> !x.range=[50,100]
To reset back to what it was, do
IDL> !p.psym=0
IDL> !x.range=0
Gaussian Barrier
The "Gauss" button creates a potential barrier in the form of Gaussian.

Study the reflection at, and tunneling through the Gaussian barrier.
Vary
the size (width) of the wave packet, within reasonable limits (it
should
still be a "packet" of waves).
Harmonic Oscillator
A parabolic potential corresponds to a harmonic oscillator.

Place the potential in the middle

Choose a potential height at the walls well over the packet energy

Place the initial wave packet near the center

Let the solution run for a long time
Step Function
The step function is not infinitely steep; its shape may be controlled
by
the width parameter.

Set the height of the step to about 150% of the packet energy.
Double Well

Adjust the control parameters until you have a double well, with a
thin barrier
in the middle. (The "potential position" should be at about 65  70%
of the
length of the box, and the width large enough to make the barrier
"thin";
i.e. a few zones. The height of the barrier should be about the same
as the
packet energy, or slightly smaller.)

Put the wave packet entirely within one of the wells.

Study the leakage from one side of the double well to the other.
Experiment
with the height of the barrier.
Additional
experiments

[about 75 minutes]

The following experiments should add to your understanding of the
Schrödinger
equation. First we are going to investigate the FFT of the signal, and
then change
integration algorithm to prove that the straightforward brute force does
not work
for this case.

Add a common block for storing the time evolution, in your own copy
of
setup.pro, ala
common TimeEvol, ff ; common block
...
ff=complexarr(n,nt) ; array for (pos,time)
...
...
ff(*,t) = phi ; save solution

Choose a simple potential (narrow parabola or well), with the lowest
possible
energy, and high walls.

Choose enough timesteps for the for wave to bounce a number of times
(choose deltat = max).
QUIT and look at the Fourier transform of the solution:
IDL> common TimeEvol, ff
IDL> n=n_elements(ff(0,*)) ; number of t's
IDL> ft=ff ; allocate space
IDL> for i=0,n1 do ft(*,i)=fft(ff(*,i),1) ; FFT
IDL> shade_surf,ft*conj(ft) ; Shaded surface plot
IDL> tvscl,ft*conj(ft) ; Flat image of the array
Now, lets change to the time integration algorithm. The previous part of
the exercise
showed that the implicit approach worked  preserved the total norm of
the probability
in time. We also hinted that this is not the case using an explicit
method. Now is the
time for proving that:

Replace the time evolution by a first order (Euler) method (Eq 116
in the Notes)
dphidt = ...
phi = phi + dt*dphidt

Try different values of dt.

Replace the time evolution with a 4th order RungeKutta method (copy
from Exercise 10).
Include the new dfdt roiutine in the setup.pro
FUNCTION dfdt,t,f,v
; set end points to complex(0,0.)
...
END
...
rk4,....
...

Try different values of dt
Home work

[about 2 hours]

The second Project is introduced one Wednesday. You should read (have
read
:) the sections on Partial Differential Equations (in Numerical Recipes
 cf. the Chapter 11 lecture notes for details).
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