Electron in a Nanocrystal Modeled by a Quantum Particle in a Sphere

This Demonstration shows the quantum effects observed on a single electron trapped in a spherical nanoparticle (also called a "quantum dot"), modeled as a particle in a sphere. We obtain the relationships among quantum energy levels , the radius of the nanoparticle , and the distance of the electron from the center of the nanoparticle by solving the Schrödinger equation. For spherical symmetry, with and :
, with the boundary condition , where is Planck’s constant and is the mass of the electron.
The solution is the wavefunction , shown on the upper left, with the allowed energy levels for (For , the solutions are spherical Bessel functions.)
The electron's probability density curve is given by the square of the wavefunction, determining the probability of finding the electron at a given radius from the center of the nanoparticle, as shown on the upper right.
The lower-left graph shows the probability density in three dimensions.
At the lower right is an energy level diagram for the electrons, showing the relative spacings of the .

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DETAILS

Reference
[1] T. Kippeny, L. A. Swafford and S. J. Rosenthal, "Semiconductor Nanocrystals: A Powerful Visual Aid for Introducing the Particle in a Box," Journal of Chemical Education, 79(9), 2002 pp. 1094–1100. doi:10.1021/ed079p1094.
Submission from the Compute-to-Learn course at the University of Michigan.
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