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Introduction to Quantum and Statistical Mechanics
Homework 5 Solution
Prob. 5.1
A quantum well laser is operated by electrons relaxing from the first excited state
E
2
to the ground state
E
1
in a finitepotential well, which is formed by a superlattice of
semiconductors with different bandgap.
Assume the electron has an effective mass
m
*
of
0.067
m
0
where
m
0
is the rest mass of electrons in vacuum.
Given the well width as
L
and well
depth as
V
o
,
(a)
To avoid transition in other levels, it is desirable to have exactly two levels in the
quantum well.
Express the relation of
V
o
with
L
and with
2
*
2
2
2
L
m
E
L
π
=
(8 pts)
E
L
≤ V
o
< 4E
L
.
(b)
Will
E
2
= 4E
1
?
Will this be a reasonable approximation?
(6 pts)
From the plot of the transcendental equation,
E
2
will not be exactly four times of
E
1
, but
however, the closer
V
o
is to
4E
L
, the better the approximation.
For example, when
V
o
= 3E
L
,
E
1
= 0.53E
L
and
E
2
= 1.96E
L
which are pretty close to 4 times.
(c)
If the emitted wavelength is at 10.6µm (middle infrared), use
E
2
and
E
1
close to the
infinite well levels to obtain an estimate of the quantum well width in nm.
(8 pts)
2
*
2
2
1
2
2
3
L
m
E
E
c
h
λ
=

=
;
L
= 6.9nm.
(d)
Graphically justify that the approximation of using
E
2
and
E
1
from the infinite potential
well is acceptable from the transcendental equation plot.
Notice that this approximation
makes the dependence on
V
o
negligible. (
8 pts
)
x=0
x=L/2
V
o
x=L/2
E
2
E
1
1
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