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

.