Question

Formula for total energy of ideal non relativistic fermi gas in three dimension

Answer 1

Three-dimensional box. Ideal Fermi
and Bose gases
This “Tillegg” starts with the three-dimensional box (8.1), which relevant e.g.
when we consider an ideal gas of fermions (8.2) and the ideal Bose gas (8.3).
8.1 Three-dimensional box
[Hemmer 5.2, Griffiths p 193, B&J p 331]
8.1.a Energy levels
The figure shows a three-dimensional box with volume V = LxLyLz, where the potential
is zero inside the box and infinite outside. To find the energy eigenfunctions and the energy
levels, we use the well-known results for the one-dimensional box:
ψnx
(x) = q
2/Lx sin knxx; knxLx = nxπ, nx = 1, 2, · · · .
Hcx ψnx
(x) = −
h¯
2
2m
∂
2
∂x2
ψnx
(x) = Enxψnx
(x), Enx =
h¯
2
k
2
nx
2m
=
π
2h¯
2
2mL2
x
n
2
x
.
In the three-dimensional case, the Hamiltonian is Hc = Hcx + Hcy + Hcz, where all four
operators commute. As energy eigenfunctions we may then use the product states
ψnxnynz
(x, y, z) = s
8
LxLyLz
sin
nxπx
Lx
· sin
nyπy
Ly
· sin
nzπz
Lz
, (T8.1)
which are simultaneous egenfunctions of H, c Hcx, Hcy and Hcz (and which are equal to
zero on the walls of the box). The energy eigenvalues are
Enxnynz =
π
2h¯
2
2m

n
2
x
L2
x
+
n
2
y
L2
y
+
n
2
z
L2
z

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