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
, with the boundary condition
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
, 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