- Diffusion Current
It is often possible to change the physics being displayed on the screen
by a slight modification of a key procedure. One such procedure is the
CalculateGridPointDensity
function. Since the value returned by
this function is displayed in the [Graph 1]|[I(x,t)]
data
analysis graph, changing this function will allow the user to display
other dynamical properties that can be calculated at a grid point.
- a.
- Use the finite-difference approximation to write an expression
for the current density,
, associated with the diffusion
equation
in terms of the current values of the wavefunction which are represented
on the grid by 
.
- b.
- Rewrite the PASCAL
code for the
CalculateGridPointDensity
function in the WAV-GBL.PAS
unit
so that it returns the probability current when EQUATIONTYPE=DIFFUSION.
Do this by adding another match to the CASE
statement. The
current will now be shown whenever you select the
[Graph 1]|[I(x,t)]
data analysis option.
Are you able to predict the
observed distributions when the initial condition is a mode? A
Gaussian?
- Energy on a String
The classical wave equation, eq:CWE, models the transmission of
waves on a string as well as E&M waves. For small displacements we can
write the total energy density for this system as
where 
is the mass per unit length and T is the tension.
- a.
- One difference between these two systems is that the energy
density exhibits frequency-dependence in addition to the usual
amplitude-squared dependence. Find an analytical expression for the
instantaneous and average energy per unit length for a harmonic wave,
, traveling on a string.
- b.
- Find an expression for the energy density at the n-th grid
point. Use the forward finite difference approximation for the first
derivatives of
. Assume that T=1 and that 
.
- c.
- Rewrite the PASCAL
code for the
CalculateGridPointDensity
function in the WAV-GBL.PAS
unit using
this new expression. The [I(x,t)]
option under [Graph]
will now plot this density. Are you able to predict the observed
distributions when the initial condition is a mode? A Gaussian?
- Quantum Probability Current
Although not as widely used as probability density, the current density
is an equally useful quantity for many types of problems.
- a.
- Find an analytical expression for the probability current
density for right and left traveling de Broglie waves,
.
- b.
- Find an expression for the probability current at the
grid point. Use the forward finite difference approximation for the
first derivative of 
.
- c.
- Rewrite the PASCAL
code for the
CalculateGridPointDensity function in the WAV-GBL.PAS
unit so
that it returns the probability current when EQUATIONTYPE=SCHRODINGER.
Examine the current distribution for various initial conditions using
the [Graph 1]|[I(x,t)]
option. Are you able to predict the
observed distributions when the initial condition is a mode? A
Gaussian?
- d.
- Set up the Schrödinger equation with absorbing boundaries.
Build two barriers and fill the region between these barriers with a
standing wave using the parser. Is the probability current what you
expect?
- Advection Equation
It order to understand the numerical solution of eq:CWE it is often
helpful to begin by considering a simpler but related equation, the
advection equation.
This equation has many properties in common with the classical wave
equation but is has the very interesting property that it supports
propagation of a function 
in only a single
direction.[6]
- a.
- Show that the solution of the classical wave equation is given
by a linear combination of
and 
while the solution of
the advection equation is only given by 
.
- b.
- What change must be made to eq:AdE to admit
solutions of the form
?
- c.
- Use finite-difference methods to solve this equation. Find an
expression for the
in terms of previous grid points. Rewrite
the propagation code for the procedure CWStep using your new algorithm.
Is your algorithm stable?
