Homework problems #3
1. Computer problem: Verify the Maxwell-Boltzmann distibution of a velocity vector of magnitude v of
ideal gas particles using a random number generator. Calculate the result for N = 103, 106 particles. Express
the velocities relatively to 2kT/m. Hint: Let us assume that one has a function that generates
random numbers in an interval x ∈ [0,1] with uniform distribution. What transformation shall be performed
to get a random variable from interval y ∈ [−1,1]? Using a computer experiment and histogram
verify that a function erf−1
(y) gives a Gaussian distribution of random variable, where erf−1
is an inverse
function to the error function erf(y) = 2/
√
π y
0 exp(−t2)dt. Maxwell-Boltzmann distibution of a magnitude
of velocity vector can be derived by combining three random variables with Gaussian distribution
2. A system with two energy levels E0 and E1 is populated by N indistinguishable particles in distinguishable
states at temperature T. Assuming canonical distribution determine
(a) mean energy per particle,
(b) limit of mean energy for temperatures T → 0 and T → ∞,
(c) heat capacity of the system,
(d) limit of the heat capacity of the system for T → 0 and T → ∞.
3. As a result of entanglement of rotational and vibrational movement of a diatomic molecule, angular
momentum depends partially on the vibrational state. In such a case, the rotational-vibrational spectrum
can be approximated by
En,l = ¯hω n+
1
2
+
¯h2
2I
l(l +1)+αl(l +1) n+
1
2
, (1)
where first two terms correspond to the vibrational and rotational movement and the last term is a small
correction due to the entanglement of rotational and vibrational movement. The constant satisfy
¯hω ≫
¯h2
2I
≫ α. (2)
Determine energy of an ideal gas composed of diatomic molecules for temperature
¯hω ≫ kT ≫
¯h2
2I
.
The solution should be submitted not later than on April 30th.
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