 quiz5slns.pdf-Name: Physics 403 Quiz #5 ...
quiz5slns.pdf-Name: Physics 403 Quiz #5 1)
Showing 1-2 out of 2
quiz5slns.pdf-Name: Physics 403 Qui...
quiz5slns.pdf-Name: Physics 403 Quiz #5 1)
##### Page 1
Name:
Physics 403 Quiz #5
1) The Fermi-Dirac distribution is
1
e
β
(
E
-
ζ
)
+1
, the Bose-Einstein distribution is
1
e
β
(
E
-
ζ
)
-
1
, and
the Maxwell-Boltzmann distribution is
e
-
β
(
E
-
ζ
)
. In three dimensions, the density of states
for a spinless nonrelativistic particle is
g
(
E
)=
1
4
π
2
(
2
m
~
2
)
3
2
E
1
2
.
All problems below are in
three dimensions.
a) What is the mean number of electrons at zero temperature?
The electron is a spin 1/2 particle and has spin degeneracy 2
S
+ 1 = 2. It is a fermion
as it has half-integral spin; thus
<N>
=
V
Z
0
2
g
(
E
)
dE
e
β
(
E
-
ζ
)
+1
As
T
= 0,
ζ
=
E
F
and
1
e
β
(
E
-
E
F
)
+1
= 1 if
E<E
F
and
1
e
β
(
E
-
E
F
)
+1
= 0 if
E>E
F
. Thus
<N>
=
V
Z
E
F
0
2
g
(
E
)
dE
=
2
3
V
1
2
π
2
2
m
~
2
3
2
E
3
2
F
b) What is mean energy of a spin 1 particle at temperature
T
and zero chemical potential?
The particle has spin degeneracy 2
S
+ 1 = 3. It is a boson as it has integral spin; thus
<U>
=
V
Z
0
3
Eg
(
E
)
dE
e
βE
-
1
Evaluating
<U>
=
V
3
4
π
2
2
m
~
2
3
2
1
β
5
2
Z
0
x
3
2
dx
e
x
-
1
=
V
3
4
π
2
2
m
~
2
3
2
1
β
5
2
Γ(
5
2
)
ζ
(
5
2
)
The mean energy per particle is given by
<U>
<N>
where
<N>
=
V
Z
0
3
g
(
E
)
dE
e
βE
-
1
evaluated similarly;
<N>
=
V
3
4
π
2
2
m
~
2
3
2
1
β
3
2
Γ(
3
2
)
ζ
(
3
2
)
c) What is the chemical potential of low density gas of helium at room temperature
T
r
consisting of
N
atoms in a container of volume
V
?
Helium is a spin zero boson; at low density at room temperature, its distribution is the
Maxwell-Boltzmann distribution.
1

##### Page 2
<n>
=
Z
0
g
(
E
)
e
-
β
(
E
-
ζ
)
dE
Evaluating
<n>
=
1
4
π
2
2
m
~
2
β
3
2
e
βζ
π
2
Thus
ζ
=
1
β
(ln
<n>
-
ln(
1
4
π
2
2
m
~
2
β
3
2
π
2
))
Alternately,
ζ
=
∂F
∂N
where
F
=
-
1
β
ln
Z
and
Z
=
Z
N
1
N
where
Z
1
=
V
Z
0
g
(
E
)
e
-
βE
dE
yields, after calculation, the same answer.
d) What is the mean energy of a spin 3/2 particle at zero temperature?
A spin 3/2 particle has spin degeneracy 2
S
+ 1 = 4. It is a fermion as it has half-integral
spin; thus
<U>
=
V
Z
0
4
Eg
(
E
)
dE
e
β
(
E
-
ζ
)
+1
As
T
= 0,
ζ
=
E
F
and
1
e
β
(
E
-
E
F
)
+1
= 1 if
E<E
F
and
1
e
β
(
E
-
E
F
)
+1
= 0 if
E>E
F
. Thus
<U>
=
V
Z
E
F
0
4
Eg
(
E
)
dE
=
2
5
V
1
π
2
2
m
~
2
3
2
E
5
2
F
The mean energy per particle is given by
<U>
<N>
where
<N>
=
V
Z
E
F
0
4
g
(
E
)
dE
evaluated similarly;
<N>
=
2
3
V
1
π
2
2
m
~
2
3
2
E
3
2
F
thus
<U>
<N>
=
3
5
E
F
.
2

Browse thousands of Study Materials & Solutions from your Favorite Schools UBC Vancouver
UBC_Vancouver
School:
Statistical_Mechanics
Course:
Great resource for chem class. Had all the past labs and assignments  Leland P.
Santa Clara University
Introducing Study Plan
Using AI Tools to Help you understand and remember your course concepts better and faster than any other resource. Find the best videos to learn every concept in that course from Youtube and Tiktok without searching. Save All Relavent Videos & Materials and access anytime and anywhere Prepare Smart and Guarantee better grades