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Transcript
Introduction to Quantum Mechanics, Fall 2014 Problem Set 2 Due Tuesday, September 30 Problem 1: Calculate the exponential etM for 0 −π 0 π 0 0 0 0 0 by two different methods: • Diagonalize the matrix M (i.e. write as P DP −1 , for D diagonal), then show that −1 etP DP = P etD P −1 and use this to compute etM . • Calculate etM using the Taylor series expansion for the exponential, as well as the series expansions for the sine and cosine. Problem 2: Consider a two-state quantum system, with Hamiltonian H = −Bx σ1 (this is the sort of thing that occurs for a spin-1/2 system subjected to a magnetic field in the x-direction). • Find the eigenvectors and eigenvalues of H. What are the possible energies that can occur in this quantum system? • If the system starts out at time t = 0 in the state 1 |ψ(0)i = 0 (i.e. spin “up”) find the state at later times. Problem 3: By using the fact that any unitary matrix can be diagonalized by conjugation by a unitary matrix, show that all unitary matrices can be written as eX , for X a skew-adjoint matrix in u(n). By contrast, show that −1 1 A= 0 −1 1 is in the group SL(2, C), but is not of the form eX for any X ∈ sl(2, C) (this Lie algebra is all 2 by 2 matrices with trace zero. Hint: For 2 by 2 matrices X, one can show (this is the Cayley-Hamilton theorem: matrices X satisfy their own characteristic equation det(λ1 − X) = 0, and for 2 by 2 matrices, this equation is λ2 − tr(X)λ + det(X) = 0) X 2 − tr(X)X + det(X)1 = 0 For X ∈ sl(2, C), tr(X) = 0, so here X 2 = −det(X)1. Use this to show that p p sin( det(X)) X p X e = cos( det(X))1 + det(X) Try to use this for eX p = A and derive a contradiction (taking the trace of the equation, what is cos( det(X))?) Problem 4: • Show that M is an orthogonal matrix iff its rows are orthonormal vectors for the standard inner product (this is also true for the columns). • Show that M is a unitary matrix iff its columns are orthonormal vectors for the standard Hermitian inner product (this is also true for the rows). 2