# The time-dependent Schrodinger equation with open boundaries

by Cemal Yalabik, Bilkent University, Physics Department. April 1998

This applet was generated from our research programs (written in fortran - of course) which we use to study electron transport properties in small geometries. It has been vastly simplified, but it still runs very slow on a not-so-fast machine.

This applet shows you a computer simulation of the time development of the wavefunction of a particle moving in an arbitrary potential. When the applet starts, you should be seeing the potential (in red) corresponding to two barriers at the bottom of the page. At the top, you will be seeing a Gaussian wave-packet (real part in green, imaginary in yellow, and amplitude in white) that is moving in this potential. You can modify the potential by moving the cursor to the area where the potential has been plotted at the bottom, and "drawing" in your modifications with the left mouse button down. (The reset button will recover the original potential.)

You also have the choice of changing the type of the wave to one that is being injected (with constant wave-number) from the left, by clicking on the second button. On "slow" machines, wait until you see the effect of a mouse event (such as clicking a button or plotting in a modification for the potential) before you attempt a second mouse event.

Things to watch:

• The wave-packet:
• Note the large oscillations between the barriers as the wave passes through. The wave is near resonance, so quite a bit of it passes through.
• The part of the wave that does pass through is relatively "monochromatic", i.e. it has a constant wavelength, due to the filtering effect of the resonant structure. The reflected part has a complicated distribution of wavelengths.
• The part that passes through moves to the right, and encounters little interference. The reflected wave however, does have considerable interference.
• At larger times, the wave tends to "hang", with a very slow variation. That is because the shorter wavelength (high energy) components have left the picture, and only the slower components remain.
• The injected wave:
• Note again the large oscillations between the barriers as the wave passes through.
• Once the wave has "stabilised", note the interference on the left due to the incoming and the reflected waves. The wave on the right in contrast experiences no such interference.
• Modify the potential. Plot in your favorite potential. (Do not modify the potential too close to the boundaries at the left and the right, as that will interfere with the absorbing boundary conditions.) Some things you can try are :
• Put a barrier in front of the wave. Notice how the wavelength decreases whenever the wave "slows down" at larger values of the potential.
• Put in quantum wells and try to construct resonant structures. (You should adjust the length of the features to "fit" a multiple of the wavelength of the wave.)

Technical details in:

"A numerical implementation of absorbing and injecting boundary conditions for the time-dependent Schrodinger equation"
M. C. Yalabik, M. Ihsan Ecemis, Phys. Rev. B51, 2082 (1995).