Homework # 6  Solution
Problem # 1:
Show that the kinetic energy of a threedimensional electron gas of
N
electrons at
zero temperature is
U
=3/5
NE
F
.
For free electrons the density of states is given by
3/ 2
1/ 2
2
2
2
(
)
2
V
m
D E
E
π
⎛
⎞
=
⎜
⎟
⎝
⎠
=
.
Using the formula for the Fermi energy
2/3
2
2
3
2
F
N
E
m
V
π
⎛
⎞
=
⎜
⎟
⎝
⎠
=
,
we can express it as follows
1/2
3/ 2
3
(
)
2
F
N
D E
E
E
=
.
The kinetic energy of
N
free electrons at zero temperature is given by
3/ 2
3/2
0
0
3
3
(
)
2
5
F
F
E
E
F
F
N
U
ED E dE
E
dE
NE
E
=
=
=
∫
∫
.
Problem # 2:
Show that the density of states of a freeelectron gas in two dimensions is
independent of energy.
In two dimensions there is one allowed wavevector per area
(
)
2
2
/
L
π
in
k
space. Since the area
of
2
2
F
k
π
is occupied by
N
electrons, the total number of states is given by
2
2
2
2
2
(2
/
)
2
F
F
k
L k
N
L
π
π
π
=
=
.
where a factor of 2 comes from the spin degeneracy.
The Fermi energy is therefore given by
2
2
2
2
2
F
F
k
N
E
m
mL
π
=
=
=
=
.
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 Spring '08
 WORMER
 Electron, Energy, Kinetic Energy, Work, Fundamental physics concepts, Condensed matter physics, Fermi gas, Fermi

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