PHY 410 – Spring 2009 Exam #2

1
PHY 410 – Spring 2009
Exam #2
(1 Hour)
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There are 4 problems. Please answer them all showing your work clearly (for partial
credit).
USEFUL CONSTANTS AND INTEGRALS
Thermal wavelength λ th =
2πh 2
Mτ
 Mτ 
Quantum concentration n Q = 

 2πh 2 
3/ 2
=
1
λ 3th
Boltzmann constant k B = 1.38066 x10 −23 JK −1
Planck’s constant h = 1.05459 x10 −34 Js
r
Energy of a photon in mode n inside a cubic box of volume L3 : hω nr = hnπc / L
∞
x3
π4
dx
=
∫0 e x − 1 15
1
2
Problem 1 (10 points)
A hemoglobin molecule can bind 4 O2 molecules at 4 different sites. The energy of the
bound O2 molecule at a given site is ε . Assume that the energy to bind N molecules is
Nε . The hemoglobin molecule is in contact with a bath of oxygen molecules at
µ /τ
temperature τand chemical potential µ. Let λ = e
be the absolute activity of the O2
molecules. What is the probability of finding 2 O2 molecules bound to a hemoglobin
molecule? What is the probability that there is no O2 molecule bound to a hemoglobin?
Z
Let’s denote the Gibb’s sum as
. Note the multiplicity factors 4, 6, 4, and 1
associated with the number of ways 1, 2, 3, and 4 particles can occupy 4 sites. This is
very important.
Z = 1 + λ 4 e −ε / τ + λ2 6 e −2ε / τ + λ3 4 e −3ε / τ + λ41 e −4ε / τ
P ( 2) =
P (0 ) =
1 + λ 4 e −ε / τ
λ2 6 e − 2ε / τ
+ λ2 6 e − 2ε / τ + λ3 4 e − 3ε / τ + λ4 e − 4ε / τ
1
1+ λ 4 e
−ε / τ
+ λ 6e
2
− 2ε / τ
+ λ3 4 e − 3ε / τ + λ4 e − 4ε / τ
2
3
Problem 2 (10 points)
The two possible states (orbitals) of an electron in the presence of an external magnetic
field have energies ε ↑ and ε ↓ for the up and down spin orientations respectively.
Derive an expression for the average number of electrons occupying these two states as a
function of the chemical potential µ ? What is the ratio of the probabilities of finding the
electron with up and down spin orientations respectively.
Z = 1+ λ e
−ε ↑= / τ
< N (ε ↑ ) >=
< N (ε ↓ ) >=
+λe
−ε ↓ /τ
λe
1+ λ e
−ε ↑ / τ
λe
1+ λ e
−ε ↑ / τ
− (ε ↑ + ε ↓ ) / τ
+ λ2 e
+λe
−ε ↓ / τ
−ε ↑ / τ
+ λ2 e
−ε ↓ / τ
+ λ2 e
+λe
−ε ↓ / τ
− (ε ↑ + ε ↓ ) / τ
+ λ2 e
− (ε ↑ + ε ↓ ) / τ
− (ε ↑ + ε ↓ ) / τ
+ λ2 e
− (ε ↑ + ε ↓ ) / τ
< N (ε ↑ ) > λ e −ε ↑ / τ + λ2 e − (ε ↑ +ε ↓ ) / τ
=
< N (ε ↓ ) > λ e −ε ↓ / τ + λ2 e − (ε ↑ +ε ↓ ) / τ
=
e
e
−ε ↑ / τ
(1 + λ e
−ε ↓ / τ
−ε ↓ / τ
(1 + λ e
−ε ↑ / τ
)
)
Note: The state where two electrons occupy the two states with energies
ε ↑ and ε ↓ respectively we will have 1 electron contributing to < N (ε ↑ ) >
and 1 electron contributing to < N (ε ↓ ) > .
3
4
Problem 3 (15 points)
Starting from the Plank’s distribution function for the photons show that the energy
density u = U / V of the black body radiation is given by
π2
U
u= =
τ 4.
V 15c 3 h 3
(Hint: After writing down the expression for the U energy in terms of integration over
modes isolate the V and τ dependence by a suitable change of variables and then use the
integral given on the first page)(6 points)
What is the heat capacity per unit volume of the photon gas?(4 points)
Starting from the above expression for the heat capacity derive an expression for the
entropy σ as a function of τ using the thermodynamic identity. (5 points)
1
U = 2∑ hω nr
r
n
(a)
hωnr
τ
1
hnπc
e Lτ
−1
Lτ
Lτ
hnπc
Define a new var iable x =
;n=
x; dn =
dx
πch
πch
Lτ
e
−1
1∞
 hnπc 
= 2. ∫ 4πn 2 dn 

80
 L 
4
∞ 3
1
x dx
V
 Lτ 
4π
= 2 3 3τ
U = 2. .4π .
 τ∫ x
8
15
 πch  0 e − 1 π c h
3
π2
U
τ4
u= =
3 3
V 15 c h
(b)
CV du
4π 2 3
=
=
τ
V
dτ 15 c 3h 3
C
 ∂σ 
CV = τ 
 ; dσ = V dτ
τ
 ∂τ V
τ
(c)
CV (τ ' )
4π 2 3
dτ ' =
τ
3 3
'
τ
45
c
h
0
σ (τ ) − σ (0) = ∫
Since σ (0) = 0
σ (τ ) =
4π 2 3
τ
45 c 3 h 3
4
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Problem 4 (15 points)
Write down the Fermi-Dirac and Bose-Einstein distribution functions for the occupation
of a single orbital of energy ε for fermions and bosons respectively. What is the
distribution function in the classical regime? When is the classical regime applicable?
(4 points)
Using the above classical limit of the distribution function derive an expression for the
chemical potential of a system of classical particles as a function of the average number
of particles <N>, volume V, and the quantum concentration nQ .(5 points)
Using the FD distribution function calculate the chemical potential at τ = 0 ( ε F ) for an
ideal gas of fermions as a function of the number density N / V = < N > / V . If the volume
of an electron gas is compressed by a factor of 2 without changing N, what is the change
in the Fermi energy? (6 points)
(a)
1
f FD (ε ) =
(ε − µ )
τ
e
1
; f BE (ε ) =
+1
(ε − µ )
τ
e
−1
(ε − µ )
τ
Classical regime : e
>> 1; f class (ε ) = e
−
(ε − µ )
τ
(b)
1
In general < N >= ∑
r
n
τ
e
Classical lim it < N >= ∑ e
r
n
µ = τ ln
;
(ε nr − µ )
±1
−
(ε nr − µ )
τ
=e
µ /τ
∑r e
n
ε nr
−− τ
= e µ / τ VnQ = e µ / τ
V
λ3th
n
<N>
; where the density n =
nQ
V
5
6
(c)
Fermions at τ = 0
< N >= 2.∑
r
n
1
(ε − µ )
⇒ 2.
∑1 as τ → 0;
r
n , n ≤ nF
e τ +1
where the Fermi energy ε F = µ (τ = 0) is related to n F by the
h 2  πn F 
relation ε F =


2m  L 
2
1/ 3
π nF3
1 nF
3< N >
2
< N >= 2. . ∫ 4π n dn =
; nF = 

8 0
3
 π 
2
h 2  π   3N 
εF =
  

2m  L   π 
2/3
h 2  π 3 3N 


=
2m  L3 π 
2/3
1/ 3
 3N 
=

 π 
h2  2 N 
=
 3π

2m 
V
2/3
If the volume V decreases by a factor of 2 without changing N then the Fermi energy
increases by a factor of 2
2/3
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