EE372 Assignment 4 Solutions
1. Nanoparticles
A hot research topic at the moment are nanoparticles, very small grains of various semi
conductors. An electron that is inside a nanoparticle is equivalent to an electron inside a
three dimensional square well. The potential outside the well is not infinite but instead is
roughly the work function of the material. However, the first few infinite 3D square well
energies are a good first approximation for the quantized energies of the electron inside a
nanoparticle.
a) Use the formula for the electron’s energy in the infinite 3D square well to derive a formula
for the energy difference between the first excited state and the ground state.
b) The nanoparticles can be induced to give off light by exciting the electron. The formula
derived in part (a) gives the energy of the photon emitted when the electron goes from
the first excited state to the ground state. Calculate the dimension
a
required so that the
photons emitted are red (1.8 eV), green (2.4 eV), or blue (3.5 eV). Such nanoparticles might
someday be used in colour displays.
Answer:
a) The energy of an electron in an infinite 3D square well is
E
n
1
n
2
n
3
=
¯
h
2
π
2
2
ma
2
n
2
1
+
n
2
2
+
n
2
3
The ground state energy is
E
111
and the first excited state energy is
E
211
.
So the energy
difference is
Δ
E
=
3¯
h
2
π
2
2
ma
2
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 Fall '10
 Johanson/Kasap
 Energy, ground state, ev, ﬁrst excited state

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