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Particle in a Box: Applications (Worksheet)

Particle in a Box: Applications (Worksheet)

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The particle-in-a-box is a model system but there are physical manifestations of this model system. One demonstration of the particle-in-a-box was published in the paper "Confinement of Electrons to Quantum Corrals on a Metal Surface" by M. F. Crommie, C. P. Lutz and D. M. Eigler, Science, 1993, 262, 218-220. This group activity is based on material presented in the paper.

The authors use scanning tunneling microscopy (STM) to both position Fe atoms on the Cu surface to make their "quantum corral" and to measure the result of that structure. What is scanning tunneling spectroscopy?How does it work? (you could draw a picture of the instrument)

In the paper, Figure 2A shows the image below with the caption: "Spatial image of the eigenstates of a quantum corral. (A) 48-Fe ring constructed on the Cu (111) surface (V=0.01 Volt, I=1.0 nA). Average diameter of ring (atom center to atom center) is 142.6Å. The ring encloses a defect-free region of the surface."

What does this image depict? What are the dots in the ring shown? Draw a line on the image showing the diameter as given in the caption. What are the circles inside the ring? Why did the authors make a circle instead of a square?

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Figure 2B shows a slice through the ring in Figure 2A. The caption reads: "(B) Solid line: cross section of he above data. Dashed line: fit to cross section using a linear combination of \(|5,0 \rangle\), \(|4,2 \rangle\), and \(|2,7 \rangle\) eigenstate densities.

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What is a fit to data? How did the authors make the fit to the data that they show with the dashed line? Why doesn't the fit follow the peaks at about -75Å and 70Å?

What is a linear combination of eigenstates?