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WOLFRAM|DEMONSTRATIONS PROJECT

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

​
R
0.75
n
1
2
3
4
5
6
7
8
9
10
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
E
n
, the radius of the nanoparticle
R
, and the distance of the electron from the center of the nanoparticle
r
by solving the Schrödinger equation. For spherical symmetry, with
l=0
and
m=0
:
-
2
h
8
2
π
m
ψ''(r)+
2
r
ψ(r)=Eψ(r)
, with the boundary condition
ψ(R)=0
, where
h
is Planck’s constant and
m
is the mass of the electron.
The solution is the wavefunction
ψ
n
(r)=
2
r
R
sin
nπr
R
, shown on the upper left, with the allowed energy levels
E
n
=
2
h
2
n
8m
2
R
for
n=1,2,3,…
(For
l>0
, 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
r
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
E
n
.
​
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