Homework problems #5
1. Computer problem: a) Make a function that calculates numerically an integral
F3/2(y) =
1
Γ(3/2)
∞
0
x1/2
ex−y +1
dx.
Calculate F3/2(0). b) Using an appropriate numerical metod find ˜µx that fulfills
F3/2( ˜µx) = x ≡
Nλ3
T
gV
for a given x. Calcullate ˜µ80, ˜µ10 a ˜µ0.1. c) Plot a graph of function
N(ε, ˜µx) =
1
eε− ˜µx +1
for determined values ˜µ100, ˜µ1, and ˜µ0.01.
2. Evaluate the mean value of free particle hamiltonian from quantum mechanical partition function
Z =
V
λ3
T
. (1)
Calculate in momentum representation.
3. Density matrix of left-handed and right-handed polarized light in a basis of vectors of linearly polarized
light is
ˆρL =
1/2 i/2
−i/2 1/2
, ˆρR =
1/2 −i/2
i/2 1/2
. (2)
Using ˆρL and ˆρR determine the density matrix of unpolarized light ˆρn and calculate ˆρ2
L, ˆρ2
R a ˆρ2
n . Which
matrices correspond to a pure state?
4. Density matrix of a harmonic oscillator with a hamiltonian
ˆH =
1
2m
ˆp2
+
1
2
mω2
ˆx2
has in a position space form of
ρ(x,x′
,T) =
mω
π¯h
tanh
¯hω
2kT
exp −
mω
2¯hsinh ¯hω
kT
x2
+x′2
cosh
¯hω
kT
−2xx′
.
(a) Calculate the mean value of energy E = ˆH .
(b) Show that for T → ∞ the equipartition theorem holds for the mean value of energy.
5. Approximately calculate the heat capacity at the constant volume for a gas with interatomic potencial
U(r) (the unknownd integral can be denoted appropriatelly). Particles can be considered as point masses.
6. The approximate solution of the Boltzmann equation in a presence of temperature gradient T = T0 +αy
is
f = f0 +ατvy
T0
2(T0 −αy)7/2
p2
mk(T0 −αy)
−5 n0
2π¯h2
mk
3/2
e
− p2
2mk(T0−αy)
, (3)
where f0 is equilibrium velocity distribution. Calculate a mean value of momentum flux mvy|vy| ; in a
velocity distribution you can approximate T0 −αy ≈ T0. The nonzero momentum flux causes, for example,
the motion of a light mill.
The solution should be submitted not later than on May 20th and not earlier than on May 16th.
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