Homework Set 2 - Introduction to Quantum Mechanics II | PHYS 5260, Assignments of Quantum Mechanics

Download Homework Set 2 - Introduction to Quantum Mechanics II | PHYS 5260 and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Spring 2008 PHYS 5260: Quantum Mechanics - II Homework Set 2 Issued January 28, 2008 Due February 11, 2008 Reading Assignment: Shankar, Ch.17 1. Anharmonic oscillator Consider an anharmonic oscillator with a small quartic perturbing nonlinearity, H1 = λx4. (a) Use time-independent perturbation theory to show that the lowest (nontrivial) order expression for the n-th excited state is given by En = h̄ω(n + 1/2) + 3h̄2λ 4m2ω2 [1 + 2n+ 2n2] (b) Argue that no matter how small λ is, the perturbation expansion will break down for some large enough n. What is the physical reason? Hint: You might find it useful to use the 2nd quantized notation of creation and annihilation operators (rather than working in the coordinate representation). 2. Consider a spin-1/2 particle with a gyromagnetic ratio γ in a magnetic field B = B⊥r̂⊥ + B0ẑ, characterized by a purely Zeeman Hamiltonian H = −µ · B (ignore orbital degrees of freedom). (a) By using a convenient choice of a quantization axis of S, find the spectrum by solving this problem exactly. Also find the corresponding exact spinor eigenstates in the basis with ẑ as the quantization axis. Hint: You can do the latter by either directly diagonalizingH or by an appropriate unitary rotation from the eigenstates expressed in basis with quantization axiss along B to basis with quantization axis along ẑ. (b) Treating B⊥ as a perturbation calculate the first- and second-order shifts in the spectrum and first-order shift in the corresponding eigenstates. (c) Compare your results in (b) to the Taylor expansion (to appropriate order) of the exact results for the energy and the eigenstates found in (a). Hint: To simply the algebra, it is convenient (but not necessary) to pick B⊥ along x̂, namely choose a vanishing azimuthal angle. 3. Prove the Thomas-Reiche-Kuhn sum rule ∑ n′ (En′ − En)|〈n′|x|n〉|2 = h̄2 2m , (1) where |n〉 and |n′〉 are exact eigenstates of H = p2/2m+ V (x) Hint: Eliminate the En′ − En factor in favor of the Hamiltonian operator H. This should allow you to reverse our usual insertion of a complete set of states, thereby considerably simplifying the expression. Test the sum rule on the nth state of a harmonic oscillator. 4. Band structure A fairly good model of electrons in a crystalline solid is of independent particles confined to a macroscopic box, moving in a presence of a periodic potential of positively charged ions. The corresponding single-electron Hamiltonian is H = p2/2m+Vbox(x)+Vions(x). For simple (e.g., alkali) metals to zeroth order one can even simply ignore the periodic potential, approximating electron waves by that of a particle in a box (with L → ∞, and periodic boundary conditions with normalization 1/ √ L), i.e., by familiar plane waves with a quadratic spectrum E0k = h̄ 2k2/2m. (a) Use a non-degenerate perturbation theory to compute the correction to this quadratic spectrum to second-order and the eigenfunctions to first-order in the periodic potential Vions(x). Write down your answer for a generic periodic poten- tial Vion(x), expressing it in terms of the Fourier coefficients VQn of the periodic potential; Qn = nQ1 is the nth Fourier wavevector, with n running over integers and Q1 is the elementary smallest wavevector characterizing Vion(x). You obviously need not compute the infinite sum (particularly that you do not know the Fourier coefficients), but please simplify the expression as much as possible. (b) Specialize this result to a single harmonic periodic potential, i.e., with just two values of Qn = ±Q1, e.g., just taking the periodic potential to be Vions(x) = V1 cos(Q1x). (c) By examining your expression above, note that for some values of k, the above nondegenerate perturbation theory breaks down. Find these special values k∗± for which this breakdown happens. Using the property of the nonperturbed, free particle spectrum, E0k , and the nature of the perturbing Hamiltonian, explain mathematically and physically why the breakdown takes place. Drawing pictures might be useful for the former, and thinking about interference of (electron) waves scattered by the periodic potential for the latter. (d) Sketch the resulting Ek, indicating the location of special k points.