Prof. Dr. Roser Valent´ı M. Altmeyer, S. Backes, K. Riedl Exercises for Advanced Solid State Theory Sommersemester 2015 Problem set 5, Due date: Wednesday, 20.05.15 Exercise 1: Landau eigenstates and eigenenergies (3 points) Find the solution to the Hamiltonian for a free electron diamagnetically coupled to an external magnetic field B, 2 2 2 2 pˆ2 ˆ = pˆx + y + pˆz − eB pˆy xˆ + e B xˆ2 , H 2m 2m 2m mc 2mc2 and show that the eigenstates and eigenenergies of H are given by, respectively, ~2 kz2 1 En,kz = ~ω0 n + + 2 2m and Ψn,ky ,kz (~r) = cHn x − x0 λ e− (x−x0 )2 2λ2 eiky y eikz z , (1) (2) (3) where r eB ~ky ~ ω0 = , x0 = , λ= , (4) mc mω0 mω0 c is a constant and Hn are the Hermite polynomials. ˆ commutes with both pˆy and pˆz and then use a proper product (Hint: Show first that H ansatz for the wavefunction.) Exercise 2: Hall resistivity for the Landau levels (7 points) (a) Show that the expectation value of the current density operator 1 e Jy = ney˙ = ne py − Ay , n = electron concentration, m c (5) over the first Landau level (x−x0 )2 1 eiky y eikz z e− 2λ2 Ψ0,kx ,kz (~r) = p √ √ Ly Lz π 1/2 λ is zero. 1 (6) (b) Add to the Hamiltonian for the Landau magnetism 2 p2z p2x m 2 py H= + + ω0 x − 2m 2m 2 mω0 (7) a term corresponding to an electrical field along the x-direction, HE = −eEx, (8) and solve the new Hamiltonian H + HE . In particular, sketch the Landau levels as a function of ky . (Hint: You can rewrite the Hamiltonian as the one of a harmonic oscillator centered around a new position x1 .) (c) Calculate now the expectation value of the current density operator over the shifted harmonic oscillator that you obtained as the ground state of H + HE , showing that you recover the classical result: hJy i = nec E. B (9) (d) Show that the classical result for the Hall resistance equals the quantum case if the electron density n corresponds to completely filled Landau levels. [Hint: (i) Consider electron gas with n = Lx LNy Lz , that creates the Hall voltage IB UH = necL and (ii) assume that N = dν, where d is the degeneracy of the Landau z levels and ν is the Landau level filling.] 2
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