Imperial College London BSc/MSci EXAMINATION May 2014

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5 Imperial College London BSc/MSci EXAMINATION May 2014 This paper is also taken for the relevant Examination for the Associateship MOCK-EXAM QUESTIONS: QUANTUM THEORY OF MATTER For 4th-Year Physics Students Mock-Exam Questions A selection of questions Marks shown on this paper are indicative of those the Examiners anticipate assigning. General Instructions Complete the front cover of each of the THREE answer books provided. If an electronic calculator is used, write its serial number at the top of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the box on the front cover of its corresponding answer book. Hand in THREE answer books even if they have not all been used. You are reminded that Examiners attach great importance to legibility, accuracy and clarity of expression. c Imperial College London /PT4.5 1 Go to the next page for questions

6 1. Many properties of simple metals such as Na, K, Mg, Cu,... can in good approximation be described in terms of the Sommerfeld model comprising the following assumptions: 1. Ideal Fermi-gas in a volume V = L Periodic boundary conditions on V. 3. Constant lattice potential V(r) = const.. (i) Write down the eigenenergies and eigenfunctions. [12 marks] (ii) Calculate the Fermi-energy and Fermi-wavevector as a function of the electron density n=n/v. [6 marks] (iii) How is the average energy per electron related to the Fermi-energy? (iv) Calculate the electronic density of states ρ(e). [6 marks] [6 marks] [Total 30 marks] 2014/PT4.5 2 Please go to the next page

7 2. Consider a d-dimensional crystal (N lattice sites) with one atom (of mass M) per unit cell; R n shall denote the lattice vector and u n is the elongation of the atom in the n-th unit cell. In harmonic approximation the phonon Hamiltonian may read: H = hω j (q) q, j ( b q j b q j+ 1 2 ). (1) (i) Show that for the j th component of the elongation vector we have: u n, j = 1 N e iqr h ( ) n b q j + b q j. (2) 2Mω j (q) q Hint: Use a cartesian coordinate system such that the dynamic matrix is diagonal. [8 marks] (ii) Calculate the average thermal elongation u. (iii) Show that u 2 n = 1 M [5 marks] dω n(ω) 1 coth(β hω), (3) ω where n(ω)= 1 N q j δ ( ω j (q) ) is the density of phononic states. [12 marks] (iv) Explain why u 2 n diverges for d = 1,2 but not for d = 3. Consequences? [5 marks] [Total 30 marks] 2014/PT4.5 3 Please go to the next page

8 3. Consider free electrons described by by the dispersion relation E (k)= h2 k 2 2m. (i) Derive a relation between the Fermi-wavenumber k F and the Fermi-energy E F with the particle density n e = N e /V, where N e is the total number of electrons, V is the volume. [10 marks] (ii) Calculate the density of states ρ(e). [10 marks] (iii) In the low-temperature limit (k B T E F ) and using the Sommerfeld expansion for the number of electrons per unit cell Z e = µ 0 deρ(e)+ π2 6 (k BT) 2 ρ 2 (µ)+o ( T 4), (1) with ρ = dρ/de, show that for the temperature dependence of the chemical potential µ(e) the following expression holds: µ(t)=e F π2 ρ (E F ) 6 ρ(e F ) (k BT) 2. (2) [10 marks] [Total 30 marks] 2014/PT4.5 4 Please go to the next page

9 4. In Hartree approximation, the potential for a single electron at position r of an atom comprising a positively charged nucleus (charge +Ne) and a surrounding electron cloud can be written as Φ(r)= d 3 r e 2 r r ν( r ) Ne2 r, (1) where ν(r) is the electron density. Assume further that the potential Φ(r) varies very slowly with r (local, homogenous electron gas). (i) Show that a consequence of these assumptions is: ν(r) (2m(ε F Φ(r))) 3/2 3π 2, (2) where ε F denotes the energy of the highest occupied energy level. (ii) What value does ε F = 0 hold for a neutral atom? [12 marks] [6 marks] (iii) Show that the Poisson-equation is a consequence of (1). Hence, using spherical polar coordinates and assuming spherical symmetry for r > 0, derive the Thomas- Fermi equations 3π r 2 r ( r 2 Φ(r) ) = 4e 2 [ 2mΦ(r)] 3/2. (3) r [12 marks] [Total 30 marks] 2014/PT4.5 5 Please go to the next page

10 5. (i) In the grand canonical ensemble use n ν = 1 {nα } e β α n α (ε α µ) n ν e β α n α (ε α µ) {n α } (1) for a diagonal single-particle Hamiltonian in occupation representation to show n ν = where µ is the chemical potential and β = 1 temperature T. 1 e β(ε ν µ) + 1, (2) k B T with Boltzman-constant k B and [10 marks] (ii) Then explain that for interaction-less Fermions that are described by the (diagonal) Hamiltonian H = α ε α a α a α. (3) the following relation applies for (grand canonical) expectation values: a α 1 a α 2 a α3 a α4 = a α 1 a α4 a α 2 a α3 δ α1 α 4 δ α2 α 3 (4) a α 1 a α3 a α 2 a α4 δ α1 α 3 δ α2 α 4. (iii) Show then that as a consequence of (ii), [10 marks] a n 1 a n 2 a n3 a n4 = a n 1 a n4 a n 2 a n3 a n 1 a n3 a n 2 a n4 (5) holds for interaction-less Fermions with respect to any other single-particle basis n. [10 marks] [Total 30 marks] 2014/PT4.5 6 End of examination paper

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