Quantum Mechanics I

PREFLIGHT

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The following three questions refer to the material you were to read in preparation for the lesson. Questions one and three require you to write a three or four sentence response. Number two is a multiple choice question. Click in the appropriate circle.

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1. Quantum Corrals

Scientists discovered a new method for confining electrons to artificial structures at the nanometer lengthscale. Surface state electrons on Cu(111) were confined to closed structures (corrals) defined by barriers built from Fe adatoms. The barriers were assembled by individually positioning Fe adatoms using the tip of a low temperature scanning tunneling microscope (STM). A circular corral of radius 71.3 Angstrom was constructed in this way out of 48 Fe adatoms.
(This web page was downloaded from IBM, sorry for the jargon. Cu(111) is a copper crystall that is "cleaved" along one plane of the crystal producing a very smooth surface of Cu, which is the "floor" of this picture. "Adatoms" are "adsorbed atoms" that just stick to the Cu crystal, without being incorporated. The STM is explained in your textbook.)
From this information, estimate the energy of the electrons "in the corral". Do they satisfy the Heisenberg uncertainty relation?

2.

The best explanation for the "pond ripples" in the above photo is:

Electrons have wave properties, so this is just what an electron looks like.

The Fe adatoms form a cage for the electron, this is just the standing waveinside the cage.

Many electrons are in the cage, and this is the interference pattern produced by the circle of Fe adatoms acting as "slits" .

An STM can only see atoms, so the ripples are really vibrations of Cu atoms in the "floor".

3. Quantum Mechanical Scattering...

This applet integrates the Schrödinger wave equation. Under what circumstances does the Gaussian wavepacket "tunnel"? Under what circumstances does the wavepacket "reflect"? What is the physical meaning of the blue and green curves?

The wave function Psi(x,t) is initially a guassian wave packet moving to the right.
The probability density p(x,t) = |Psi(x,t)|^2 is shown in black.
The real and imaginary parts of Psi(x,t) are shown in blue and green.
The potential energy function v(x) is shown in red.
The boundary conditions are periodic, so that waves which exit to the right will return on the left.
The algorithm for integrating the Schrödinger wave equation is from: Richardson, John L., Visualizing quantum scattering on the CM-2 supercomputer, Computer Physics Communications 63 (1991) pp 84-94

honors extra

Scanning tunneling microscopes (STM) can readily write and read atomic scale surface features. This density is roughly equivalent to a million gigabits per square inch. A famous example of atomic scale writing is the IBM logo spelled out with individual xenon atoms using a low temperature STM.
What is the QM limit of "spatial" data storage? Is a million gigabits (=10¹⁵) bits/in² close to the limit?

Below is a space for your thoughts, including general comments about today's assignment (what seemed impossible, what reading didn't make sense, what we should spend class time on, what was "cool", etc.):

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