Time-dependent approach to 1-dimensional barrier scattering
Numerical assignment, Fall 2005
Your assignment is to calculate the energy dependent scattering cross-section
(reflection coefficient) for a free electron scattering off a 1 dimensional square
potential barrier. You will do that in a time-dependent approach, by numerically
integrating the time-dependent Schrödinger equation for an electron wave packet
moving in a 1-dimensional potential. Atomic units are used.
1. Initial state: Construct the initial state wave function on your 1-dimensional grid
as a Gaussian wave packet that has an initial wave vector k0:
 ( x  x0 ) 2 
 exp( ik 0 x)
( x, t  0)  N exp  
2
2

x


x is the initial width of the wave packet. The choice of grid spacing is
determined by the largest energy component in your wave packet. The
wavelength corresponding to this energy must be resolved on your grid. It is
important that the wave function does not reach the edges of the grid, since it will
be reflected at the end of the grid. Normalize the wave function on the grid.
2. Time evolution: V=0. Write a program that propagates (x) in time on the grid,
when there is no potential. The time-step is also determined by the largest energy
component, which corresponds to the most rapidly changing phase. It is important
to diagnose whether your code is working properly. Propagate the wave function
for some time and print out the wave function at a couple of different times as it
propagates on the grid. What happens to the width of the wave packet? What
happens if you change the initial width? The initial kinetic energy? Calculate x
as a function of time and check that it corresponds to the analytical result for wave
packet propagation. Check that the norm of the wave function is conserved. Check
that the results are converged with respect to the grid spacing, the size of the box
and the time step.
3. Time evolution: V=V0. Construct your potential and let the wave function interact
with the potential. Start the wave packet well away from the barrier and propagate
it for a long time, so that it has finished interacting with the potential barrier. You
will get the best results if you use a potential with height between 0.2 and 2 a.u.
and a width apot between 2 and 8 a.u. Repeat the diagnostics from above.
4. Momentum distribution and reflection coefficient: Calculate the momentum
distribution of the initial and final state wave functions. What do you see in the
final state momentum distribution? From the momentum distribution of the initial
and final state wave functions, calculate the scattering cross section (reflection
coefficient) for a range of energies. Compare this with the analytical result. Where
does your calculation fail in comparison to the analytical result? Why? (Make sure
your result has numerically converged before answering this question.). What is
the origin of the resonances in the reflection coefficients?