UNIVERSITY OF SOUTHAMPTON

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1 UNIVERSITY OF SOUTHAMPTON PHYS2024W1 SEMESTER 2 EXAMINATION 2011/12 Quantum Physics of Matter Duration: 120 MINS VERY IMPORTANT NOTE Section A answers MUST BE in a separate blue answer book. If any blue answer booklets contain work for both Section A and B questions - the latter set of answers WILL NOT BE MARKED. Answer all questions in Section A and two and only two questions in Section B. Section A carries 1/3 of the total marks for the exam paper and you should aim to spend about 40 mins on it. Section B carries 2/3 of the total marks for the exam paper and you should aim to spend about 80 mins on it. A Sheet of Physical Constants will be provided with this examination paper. An outline marking scheme is shown in brackets to the right of each question. Only university approved calculators may be used. Copyright 2012 c University of Southampton Number of Pages 10

2 Section A 2 PHYS2024W1 A1. a) Draw qualitative plots of the Fermi-Dirac, Bose-Einstein and Boltzmann distributions as a function of β(e µ), where β = 1/(k B T ), T is the temperature, E is the energy and µ the chemical potential. Choose β(e µ) as your x-axis and preferably a logarithmic scale on the y-axis [3] b) Give a physical interpretation of the differences between the three curves. [2] A2. a) For an ideal gas of N particles and energy E the statistical weight is given by W (E, N) = ce (3N 1)/2, where c is independent of E. Use this formula and the microscopic definition of the temperature in terms of the entropy to show that 1/T = k B (3N 1)/(2E). [3] b) State the equipartition theorem and explain its connection to the result obtained in a). [2] A3. a) Derive the fundamental relation of thermodynamics for a gas T ds = du + pdv, [2] from the first law of thermodynamics du = d Q + d W assuming a reversible process. Note we have assumed the number of particles to be constant dn = 0. The variable Q, W and U denote heat, work and the total energy respectively. b) Explain why the fundamental relation of thermodynamics holds irrespective of whether the process is reversible or not. [1]

3 3 PHYS2024W1 c) Using the relation between the free energy F and the total energy F = U T S, where T is the temperature and S the entropy, derive an expression for the fundamental relation of thermodynamics in terms of df instead of du. [2] A4. a) Show that the probability of finding a fermion at an energy level E + = µ + δ is the same as not finding a fermion at the energy level E = µ δ. [3] b) Investigate the probability above and below the energy E = µ to argue that the analoguous question for bosons does not make sense. [2] Help: The partition functions of non-interacting fermions and bosons, at some given energy E, are Z fermion (E) = (1 + e β(µ E) ) Z boson (E) = 1 (1 e β(µ E) ). TURN OVER

4 Section B 4 PHYS2024W1 B1. Consider N 3-dimensional (quantum) harmonic oscillators at a frequency ω. Neglecting the zero point energies the total energy is given by E = 3N E i = 3N i=1 i=1 n i hω, where n i are non-negative integers which correspond to the energy state of the individual oscillators. a) Argue that finding the statistical weight in terms of the energy is equivalent to the problem of distributing l = Ē hω objects into 3N different containers. [3] b) Show that the number of ways this can be done is equal to: W = ((3N 1) + l)! l!(3n 1)! Hint: It is worthwhile to do a drawing of the problem in a) in order to solve b). [4]. c) Use Stirling s approximation (assuming N 1) and the microscopic definition of the entropy to obtain: [ 3N + l S(E, N) 3Nk 3N 3N + l ln 3N l 3N ln l ] 3N d) Using the microscopic definition of the temperature derive: 1 T = k ( ) 3N + l hω ln, l and solve for the energy to obtain E(T, N) =. [4] 3N hω e β hω 1, β 1 kt. [3] e) Investigate the high temperature limit (kt hω) of the energy E(T, N) and comment on the result. [3]

5 5 PHYS2024W1 f) Obtain the result for the energy E(T, N) in d) using the canonical ensemble and E = ln Z β, where Z is the partition function. [3] TURN OVER

6 6 PHYS2024W1 B2. Consider a gas of photons at temperature T enclosed in a volume V. The number of photons in the energy interval [E, E + de] is g S g(e)f BE (E) de, where with β 1/k B T. f BE (E) = 1 e βe 1, g(e) = V, 2π 2 ( hc) 3E2 a) What is the value of the spin degeneracy factor g S for photons? What is the value of the photon spin? [2] b) Why is the chemical potential µ of the photons equal to zero? [3] c) Derive the Planck energy density u(λ) as a function of the wavelength λ (Recall: E = hω and ωλ = 2πc) u(λ) = 8πhc λ 5 1 e βhc/λ 1. [5] d) Derive the Rayleigh-Jeans result by taking the long-wavelength limit of the energy density given above. [2.5] e) Draw a plot of the Planck energy density and the Rayleigh-Jeans law as a function of λ (for one specific temperature). Indicate on your plot what is meant by the ultraviolet catastrophe. [2.5] f) The photon number density, N T 3, vanishes as the temperature approaches absolute zero. Argue whether a similar result would be expected if the photons were massive. [2] g) Describe the analogies and differences between the phonon gas (Debye model) and the photon gas. Comments on the spin degeneracy g S, the dispersion relation and the range of frequencies are sufficient. [3]

7 7 PHYS2024W1 B3. The grand potential Φ G for a d-dimensional, spinless, ideal Bose gas with volume V at temperature T assumes the following form: ( d d p Φ G = k B T V h ln ( 1 ze βɛ(p)) ) + ln(1 z), d where β 1/(k B T ), z e βµ and µ is the chemical potential. a) What does the term ln(1 z) correspond to and why can it not be taken into the integral for d 2? [3] b) Consider a non-relativistic dispersion relation of the form ɛ(p) = 1 2m p2 and specialize to d = 2 to show that the grand potential is given by, ( Φ G 2D = k B T V ) λ 2 2 (z) + ln(1 z), [6] thg h with λ th = 2πmkB. You should use integration by parts and you may T assume that the boundary term vanishes (of course you can also verify this explicitly but it is not required). You might find it helpful to recall the standard formula: 0 x α 1 e x z 1 1 dx = Γ(α)g α(z), where Γ(n) = (n 1)! for integer n, g α (z) = n 1 z n n α. c) Show that the number of particles is given by: ( V N 2D = λ 2 g 1 (z) + z ). [3] th 1 z Hint: Use dφ G = SdT pdv Ndµ. d) State the condition for Bose-Einstein condensation mathematically. [2] TURN OVER

8 8 PHYS2024W1 e) Can Bose-Einstein condensation occur in the 2-dimensional case considered in this question? Explain why or why not. [3] f) How is the formula for N 2D modified to take the value of the spin degeneracy g s into account? Argue whether g s lowers or raises the critical temperature, assuming Bose-Einstein condensation to occur. [3]

9 9 PHYS2024W1 B4. The energy density functions of the Einstein and Debye models are: { 3N 3E 2 ( hω g(e) Einstein = 3Nδ(E hω E ), g(e) Debye = D ) 0 E hω 3 D 0 otherwise. N denotes the number of sites. The partition function, in terms of g(e), is given by ln Z = 0 g(e) ln 1 1 e βe de, β 1/(k BT ), where the zero point energy contribution is omitted since it does not contribute below. a) What are the physical assumptions of the Einstein and the Debye model respectively? Explain the physical ideas behind the energy density functions given above. [5] b) Using the relation C V = U T = k B β 2 2 ln Z β 2 show that C V = k B 0 e βe g(e) de. [4] (e βe 1) 2(βE)2 c) For the Debye model the heat capacity assumes the following form: C V Debye = 9Nk B x 3 D xd 0 y 4 e y (e y 1) 2dy, x D = β hω D. Argue that C V T 3 for low temperatures. You need to investigate the behaviour in x D of the expression above including the limit of integration. [2.5] d) Show that the normalization 0 g(e)de = 3N together with the expression in b) leads to the Dulong Petit law, C V = 3Nk B, at high temperature. [4] e) Explain the Dulong Petit law from the viewpoint of the equipartition theorem. [2] TURN OVER

10 10 PHYS2024W1 f) Compute the high temperature correction coefficient c 1 to the Dulong Petit law for the Debye Model, C V = 3Nk B (1 + c 1 x D + O ( xd) 2 ) [2.5] by considering the formula in part c) and Taylor expanding the integrand in the variable y. END OF PAPER

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