Biomedical Engineering Reference
In-Depth Information
12
Quantum Gases
In Chapter 11, I discussed the quantum mechanical description of a particle in a selection
of potential wells, including the three-dimensional infinite cubic box of side
L
. For a single
particle of mass
m
, I showed that the energy is quantized and that each quantum state is
described by three quantum numbers
n
,
k
and
l
as follows:
n
2
h
2
8
mL
2
+
k
2
h
2
8
mL
2
+
l
2
h
2
8
mL
2
ε
n
,
k
,
l
=
U
0
+
2
L
3/2
sin
k
π
y
L
sin
l
π
y
L
sin
n
π
x
L
(12.1)
ψ
n
,
k
,
l
=
n
,
k
,
l
=
1, 2, 3, ...
We often set the constant of integration
U
0
to zero, without any loss of generality. We will
make this simplification for the rest of the chapter.
Consider now an ideal gas comprising
N
such particles, constrained to the cubic three-
dimensional infinite potential well. Ideal gas particles do not interact with each other and
the total wavefunction is therefore a product of one-particle wavefunctions (orbitals), as
discussed in Chapter 11. Also, the total energy is the sum of the one-particle energies.
Suppose we arrange that the system is thermally isolated and then make a measurement
on the system to find how the energy is distributed amongst the particles at (say) 300 K.
We will find some particles with low quantum numbers and some with high. The kinetic
energy of particle
i
is given by
ε
i
=
n
i
l
i
h
2
8
mL
2
+
k
i
+
and the total kinetic energy is therefore
N
n
i
l
i
h
2
8
mL
2
k
i
E
kin
=
+
+
(12.2)
i
=
1
