Problem Set 1 - Introduction to Quantum Mechanics II | PHY 4605, Assignments of Physics

Download Problem Set 1 - Introduction to Quantum Mechanics II | PHY 4605 and more Assignments Physics in PDF only on Docsity! PHY4605–Introduction to Quantum Mechanics II Spring 2005 Problem Set 3 Jan. 24, 2005 Due: 31 January, 2005 Reading: PH notes Remarks: On problem set 1 too many people had continued difficulty with matrix algebra. To solidify these crucial concepts, here’s more practice. 1. Matrix algebra drill. Given the matrices A =   3 5 −4 5 4 5 3 5   ; B =   1 1 −1 −1 3 −1 −1 2 0   ; C =   −1 2 2 2 2 2 −3 −6 −6   (1) (a) Find the eigenvalues λi and normalized eigenvectors vi, i = 1, 2, 3 for A, B, and C. State the degeneracy of each eigenvalue. (b) Find the determinants of A, B, and C. (c) Recall that, under a change of basis matrices, transform as M → M ′ = Û−1MÛ and vectors as v → v′ = Û−1v. Find Û such that A is brought into diagonal form by a change of basis. How is the matrix of transforma- tion Û related to the eigenvectors of A? Find the transformed eigenvectors Û−1vi. Now answer the same questions for C. (d) Find the inverses of A and B, and state why C is not invertible. (e) Show that the inverse of the 2D matrix represented by M = a01 + ~a · ~σ is M−1 = D−10 (a01−~a ·~σ), with D0 = a20−~a ·~a. Here 1 is the identity matrix in 2D and σx, σy, and σz are the Pauli matrices. [Hint: properties of the Pauli matrices you may find useful: 1) σ2i = 1 for any i; 2) σiσj = i²ijkσk for i 6= j.] 2. Phase shift in a vector potential. (a) Show that the vector potential outside an infinite solenoid containing a flux Φ has the magnitude Φ/2πρ, where ρ is the perpendicular distance from the axis of the solenoid. (b) Show that if ψA satisfies ih̄ ∂ψA ∂t = 1 2m [p− eA(r)]2ψA, then ψA(r, t) = exp ( ie h̄ ∫ r r0 A · ds ) ψA=0(r, t) Apparently ψA depends on an arbitrary initial point r0. Comment on this ambiguity. 1