PHYS3113, 3d year Statistical Mechanics Tutorial problems. Tutorial 1, Microcanonical, Canonical and Grand Canonical Distributions

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1 1 PHYS3113, 3d year Statistical Mechanics Tutorial problems Tutorial 1, Microcanonical, Canonical and Grand Canonical Distributions Problem 1 The macrostate probability in an ensemble of N spins 1/2 is P n = 1 2 N N! n!(n n)! Here n is the number of spins up. Represent n as n = N 2 + δ Using Stirlings formula for x! at x 1, show that the probability is peaked at δ = 0. Show that P (δ) has a Gaussian form and find the width of the distribution. Problem 2 Consider 1D quantum oscillator H = p2 + mω2 x 2. 2 Energy levels are ɛ n = h (n + 1/2), n = 0, 1, 2... The oscillator is in a contact with heat bath with temperature T. Calculate (i) Partition function of the oscillator. (ii) Free energy of the oscillator. (iii) Average energy of the oscillator. (iv) Probability to find the oscillator in state n. Problem 3 Show that the density of quantum states in the one dimensional phase space p, x is dγ = dpdx 2π h. Hint: it is convenient to consider a particle in a box of length L where L is very large. Problem 4 Consider 1D classical (non quantum) oscillator H = p2 + mω2 x 2 2 Using the result of Question 3 calculate (i) Partition function, Z = e H(p,x)/kT dγ, of the oscillator. (ii) Free energy of the oscillator. (iii) Average energy of the oscillator. (iv) Probability to find particle at position x. (v) Probability to find particle with momentum p..

2 2 (vi) Compare the classical case with quantum one in Question 2. Problem 5 Consider Helium gas in thermal equilibrium at 25000K. Calculate the ratio of the number of atoms in the first excited level to the number of atoms in the ground state, N ex /N gr. The ground state has internal angular momentum J = 0. The first excited level has angular momentum J = 1. The energy of the excited level is 19.82eV above the ground state. The degeneracy of an atomic energy level is 2J + 1. Problem 6 A system consists of N weakly interacting subsystems, the temperature is T. Each subsystem possesses only two energy levels, a lower one ɛ 1 which is two-fold degenerate and the upper one ɛ 2 which is nondegenerate. Derive expressions for (i) the total energy, (ii) the heat capacity. Sketch dependence of these quantities on T. What is the high temperature limit of the energy. What is entropy of the system at T=0? Is the entropy consistent with 3d law of thermodynamics? Problem 7 Calculate the heat capacity of classical 1D anharmonic oscillator with Hamiltonian H = p2 + αx4. Compare your result with that for a harmonic oscillator and explain the reason for difference.

3 3 Tutorial 2, Diatomic molecules, Maxwell velocity distribution, Plank s black-body radiation Problem 8 Consider gas of H 2 molecules at T=2000K. For H 2 molecule the characteristic rotational temperature is θ rot = h2 2Ik B = 85K and the characteristic vibrational temperature is θ vib = hωe k B = 6200K (see lecture notes). The molecule has the following degrees of freedom: (i) translations, (ii) vibrations, (iii) rotations. Here we neglect the molecule dissociation since the dissociation energy, E 4.7eV 55000K is sufficiently high. At the given temperature, T = 2000K, (i) Which degrees of freedom are in the classical regime and which are in the quantum regime? Explain your answer. (ii) Calculate the contribution to the specific heat (per molecule) from translational degrees of freedom. (iii) Calculate the contribution to the specific heat (per molecule) from vibrational degrees of freedom. (iv) Calculate the contribution to the specific heat (per molecule) from rotational degrees of freedom. (v) Hence, calculate the total specific heat of the gas (per molecule). Problem 9 Consider carbon dioxide molecule, CO 2. CO 2 is a linear molecule, therefore it has two rotational degrees of freedom and several vibrational degrees of freedom. The bending vibrations have relatively low energy, you can find value of the energy on Internet. At room temperature only rotations and bending vibrations are important. All other vibrations are of sufficiently high energy and hence they are practically frozen. Calculate the specific heat of internal degrees of freedom of CO 2 molecule at T=300K and compare your result with experimental data. Problem 10 Consider oxygen gas, O 2, at room temperature, T=300K. Calculate fraction of molecules having speed (i) higher than 400m/sec, (ii) smaller than 100m/sec. Problem 11 Consider Helium gas at room temperature, T=300K. Calculate fraction of atoms having speed (i) higher than 400m/sec, (ii) smaller than 100m/sec. Problem 12 Cosmic microwave background radiation is described by Plank s law. The temperature of the radiation is T=2.7K. (i) Calculate the energy density (Joules per cubic meter) of the background radiation. (ii) Compare this with the average energy density of the Universe. Problem 13 Temperature of the sun surface is T 5800K, radius of sun is R km. The sun is losing its mass due to light radiation. Calculate amount of mass radiated per second.

4 4 Problem 14 A photon has dispersion ɛ = cp and this has been used in problems 12,13, c is speed of light. Suppose photons had a dispersion ɛ = p2 where m = 1eV/c2. (Of course this dispersion violates Lorenz invariance, but we forget about this issue) All other things besides the dispersion being equal to that in problem 13, surface temperature of the sun and the sun radius. What would be the rate of the sun radiation mass loss in this case?

5 5 Tutorial 3, Bose-Einstein condensation, Degenerate Fermi gas Problem 15 Consider gas of sodium atoms with number density n = cm 3. A sodium atom has one electron on the external shell. Therefore the electron spin is s = 1/2 and spin of Na nucleus is I = 3/2. Altogether spin of the atom F = s + I is integer, so it is a boson. Assume that all atoms are in the same spin state F = 1, F z = 1. This implies that the spin degeneracy is g=1. (i) Derive formula for temperature of Bose condensation T c and calculate value of T c in Kelvins. (ii) Derive formula for fraction of condensate atoms at T < T c and calculate the fraction at T = T c /2. Problem 16 Consider two dimensional ideal gas of bosons with quadratic dispersion, ɛ = p2. The number of particles is conserved, the number density per unit area is n. (i) Calculate chemical potential as function of temperature, consider specially the cases of high T and low T. (ii) Calculate the temperature of Bose-Einstein condensation. (iii) Sketch the plot of n p at a low temperature. Problem 17 Consider two dimensional ideal gas of bosons with quadratic dispersion, ɛ = p2. There is also an attractive potential which can bound a particle. The energy of the bound state is ɛ 0 = ɛ 0. The number of particles is conserved, the number density per unit area is n. The bound state is shallow, ɛ 0 h2 n m (i) Calculate temperature of Bose-Einstein condensation. (ii) Derive formula for fraction of condensate atoms at T < T c. Problem 18 The number density of protons in nuclear matter is n = cm 3. (i) Calculate the Fermi energy of protons. Express your answer in MeV. (ii) Compare the Fermi energy with the rest energy of proton. Problem 19 Calculate specific heat of two dimensional degenerate ideal Fermi gas. Problem 20 Two dimensional ideal Femi gas of particles with quadratic dispersion, ɛ = p2, is placed in the external potential U(x) = { 0, x < 0 U 0, x > 0 The temperature is zero. Due to the potential the number density of particles depends on x n(x) = { 2n0, x < 0 n 0, x > 0, where n 0 is known. Find the value of U 0. Express your answer in terms of n 0, m, and h.

6 Problem 21 Consider a White Dwarf composed of a gas of non-relativistic electrons ɛ = p2. The White Dwarf exists in a two dimensional universal, yet the gravitational force still obeys inverse square law. In this problem the energy scales are such that k B T ɛ F, and hence the approximation T 0 is fine. Calculate the characteristic radius of the White Dwarf; i.e. the radius that balances (minimises the energy between) gravitational attraction and radiative pressure due to the electron gas. Problem 22 Consider again the White Dwarf problem. Again T 0, but we return to three dimensional space, and the electron gas now has dispersion ɛ = ap σ, a =const. Find the condition on σ such that the White Dwarf can form a stable state; i.e. has a finite, non-zero radius. Problem 23 Consider a dense White Dwarf composed of a gas of ultra-relativistic electrons ɛ cp. At such high energies it becomes possible to create positrons (with the same dispersion). Assume the White Dwarf is composed of 1 2 N Helium nuclei, such that N = N e N p is the difference between the number of electrons and positrons. This guarantees charge neutrality. The electron and positron gases generate pressure, while the heavy Helium nuclei dominate the gravitational attraction. Find the ratio of N p /N e that minimises the energy of the system. 6

First Problem Set for Physics 847 (Statistical Physics II)

First Problem Set for Physics 847 (Statistical Physics II) Important dates: Feb 0 0:30am-:8pm midterm exam, Mar 6 9:30am-:8am final exam Due date: Tuesday, Jan 3. Review 0 points Let us start by reviewing