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

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 :

E

n

R

r

l=0

m=0

-ψ''(r)+ψ(r)=Eψ(r)

2

h

8m

2

π

2

r

ψ(R)=0

h

m

The solution is the wavefunction (r)=sin, shown on the upper left, with the allowed energy levels = for (For , the solutions are spherical Bessel functions.)

ψ

n

2

r

R

nπr

R

E

n

2

h

2

n

8m

2

R

n=1,2,3,…

l>0

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.

r

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