3. Derive the partition function for the ideal monoatomic gas. Use Boltzmann statistics, and a quantum mechanical model for the gas.

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1 Tentamen i Statistisk Fysik I den tjugosjunde februari 2009, under tiden Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 0-3 = F, 4-6 = Fx, 6-9 = E, = D, = C, = B, = A. 1. The energy of a photon gas is U = av T 4. Calculate its entropy and its free energy F. 2. In a P T -diagram for water the slope of the phase boundary between solid and liquid is negative. Explain, using the Clausius-Clapeyron relation, why this is important for life on earth. 3. Derive the partition function for the ideal monoatomic gas. Use Boltzmann statistics, and a quantum mechanical model for the gas. 4. Starting from the formula P (n) = e (ɛn µn)/kt (1) Z for an ideal quantum gas, derive the expected number of particles in the one-particle state with energy ɛ, for fermions, and for bosons. 5. Modelling a quantum gas as N free particles in a box, derive an expression for its density of states g(ɛ). 6. In the mean field approximation, with n nearest neighbours, the expectation values of the spins are determined by s = tanh n s kt. (2) Show that, close to the critical temperature T c, and determine the critical exponent β. s (T c T ) β, (3) 1

2 Tentamen i Analytisk Mekanik den tjugoandra augusti 2008, under tiden Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 0-3 = F, 4-6 = Fx, 6-9 = E, = D, = C, = B, = A. 1. Consider a particle not subject to forces, and change its velocity through the transformation δx i = v i t. Write down the Lagrangian and use Noether s theorem to deduce the correspondings constant of motion. Afterwards verify by explicit differentiation that it is indeed a (vectorial) constant of the motion, even though it depends explicitly on t. 2. Prove Kepler s third law, that the square of the period of a planet is proportional to its distance from the sun. The easy way to do this is to use the scaling properties of the Lagrangian L = m 2 (ṙ2 + r 2 θ2 + r 2 sin 2 θ φ 2 ) k r. (4) 3. In a rotating coordinate system mẍ i = F i 2mɛ ijk Ω j ẋ k + m(ω 2 δ ij Ω i Ω j )x j. (5) Set up the equations for a Foucault pendulum (making many swings a day, so some terms can be ignored), and solve them. 4. State and prove a relation between the tensor of inertia relative to the center of mass, and relative to a point translated from the center of mass by the vector a i. 5. Euler s equations for a freely spinning rigid top are I 1 Ω1 + (I 3 I 2 )Ω 2 Ω 3 = 0 I 2 Ω2 + (I 1 I 3 )Ω 3 Ω 1 = 0 I 3 Ω3 + (I 2 I 1 )Ω 1 Ω 2 = 0, where Ω i is the angular velocity around the ith principal axis. The Earth is flattened at the Poles, with ellipticity 2 (6)

3 I 1 I 3 I (7) Based on Euler s equations, what period do you expect for its (small) Chandler wobble? 6. Consider the Hamiltonian and the Poisson brackets H = 1 2m p ip i (8) {x i, x j } = 0 {x i, p j } = δ ij {p i, p j } = eɛ ijk B k (x). (9) Write down Hamilton s equations in Poisson bracket form, and verify that they are the equations for a particle in an external magnetic field B i (x). Also check that these brackets really are Poisson brackets, in the sense that they are anti-symmetric and obey the Jacobi identity. To do the last part, you may find it helpful to write the magnetic field in terms of a vector potential, ɛ ijk B k = i A j j A i. (10) 3

4 Tentamen i Analytisk Mekanik den sjätte juni 2008, under tiden Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 0-3 = F, 4-6 = Fx, 6-9 = E, = D, = C, = B, = A. 1. Draw the phase space flow for the Hamiltonian H = 1 2 p2 + x(x 1)(x 2)(x 3). (11) Your picture should show show all fixed points clearly (but you do not have to locate them to two decimal places!). Into how many regions do the separatrices divide phase space? 2. Given an action S = dt L(q, q). Suppose there exists a transformation δq = δq(q, q) such that for some function Λ = Λ(q, q). motion, and derive its form. δs = t2 t 1 dt dλ dt (12) Prove that there exists a constant of the 3. Compute the inertia tensor for a cube of constant mass density, with respect to a corner, and with respect to its center. 4. Euler s equations for a freely spinning top are I 1 Ω1 + (I 3 I 2 )Ω 2 Ω 3 = 0 I 2 Ω2 + (I 1 I 3 )Ω 3 Ω 1 = 0 I 3 Ω3 + (I 2 I 1 )Ω 1 Ω 2 = 0. (13) where Ω i is the angular velocity around the ith principal axis. Prove that the top can rotate around its principal axes, and find the conditions for these solutions to be stable. Solve the equations exactly for I 2 = I A Lagrangian for the central force two body problem is 4

5 L = m 2 (ṙ2 + r 2 φ2 ) V (r). (14) Show how to reduce the equations of motion to two integrals that are doable (in principle) as soon as the function V (r) is specified. Specialize to V (r) = kr α, and deduce under what conditions on the exponent α circular orbits are stable. 6. A Lagrangian is L = 1 2 (1 v r ) v2 + ṙ v, (15) where v and r are configuration space coordinates. What are the canonical momenta? What is the Hamiltonian? 5

6 Tentamen i Analytisk Mekanik den tjugoandra augusti 2008, under tiden Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 0-3 = F, 4-6 = Fx, 6-9 = E, = D, = C, = B, = A. 1. Starting from the energy U = U(S.V ), use Legendre transformations to derive Helmholtz free energy F, Gibbs free energy G, and the enthalphy H. If you are a physicists studying boiling water, which potential do you choose, and why? If you are a chemist mixing some ingredients in a test tube, which potential do you choose, and why? 2. Maxwell s velocity distribution D(v) for an ideal gas is proportional to (the probability that a molecule has the velocity vector v) times (the number of vectors v that correspond to the velocity v). Use this to derive the correct formula! 3. A harmonic oscillator has E = nhν. Compute its partition function and the expected number of quanta n as a function of temperature. Use this to deduce Planck s expression for the energy density of a photon gas confined to a box. 4. Consider a collection of two state atoms in interaction with black body radiation of energy density u. There are N 1 atoms in the ground state and N 2 atoms in the excited state. Let A = (probability of spontaneous decay per unit time), ub = (probability of absorption per unit time), and ub = (probability of stimulated emission per unit time). Write down an equation for the rate of change of N 2 with time. Assuming equilibrium, and assuming that the ratio N 2 /N 1 is as given by Boltzmann, what do you conclude about u? 5. Given N non-interacting electrons in a box, compute the Fermi energy ɛ F and the total energy at zero temperature. Give two qualitative arguments to suggest that conduction electrons in a metal can be treated as non-interacting. 6

7 6. Write down the one dimensional Ising model for N spins, compute the partition function, and from there compute the free energy per spin in the limit of large N. 7

8 Tentamen i Statistisk Fysik I den tjugonionde februari 2008, under tiden Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 0-2 = F, 3-5 = Fx, 6-8 = E, 9-11 = D, = C, = B, = A. 1. The Clausius-Clapeyron equation relates the slope of the phase boundary in a P -T -diagram to the latent heat of the phase transition. Derive it! 2. A thermodynamical system is described by the entropy function S = κu 3/4 V 1/4, where κ is some constant. Derive the correct units for κ, and the specific heat C V of the system. What is this? What physical constants do you expect to give κ up to a numerical factor? 3. Derive the partition function for the ideal gas. You may use a classical or a quantum mechanical model for the gas, as you please. 4. Take three non-interacting indistinguishable particles, each of which can be in four different one-particle states. What is the total number of states if the particles are fermions? If they are bosons? 5. Consider a two dimensional ideal electron gas. Derive an expression for its density of states g(ɛ). 6. Use a mean field approximation to solve the three dimensional Ising model, for three different choices of lattice. What is the critical temperature? 7. Using the solution to the previous problem, derive the critical exponent β that relates the rate at which the magnetization approaches zero to the rate at which the temperature approaches the Curie temperature. 8