3 Problems on Quantum Mechanics - Homework 6 | PH 651, Assignments of Quantum Mechanics

Download 3 Problems on Quantum Mechanics - Homework 6 | PH 651 and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 651 Fall 2007 Homework #6 (due Friday, October 12, 2007) 1. (20 pts) Consider matrices 7 0 0 1 0 3 0 1 , 0 2 0 0 1 0 5 A i B i i i i        = − =       − −    . (a) Are A and B Hermitian? Write down the matrices representing †A and †B . (b) Find eigenvalues and (normalized) eigenvectors of A. What is the relationship between Tr(A) and a sum of the eigenvalues of A? Explain. (c) Show that the eigenvectors of A form a (complete and orthonormal) basis. (d) Is Tr(AB) = Tr(BA) ? Is det(AB) = det(A)det(B) ? Is det(B+) = (det(B))*? Show. (e) Calculate the commutator [A, B]. Find Tr([A,B]). (f) Calculate the inverse of A, i.e. A-1. What are the eigenvalues of A-1 ? 2. (15 pts) Consider a system whose Hamiltonian is given by ( )1 2 2 1H α ϕ ϕ ϕ ϕ= + , where α is a real number having the dimensions of energy. (a) Is H a projection operator? What about α-2H2 ? (b) Are |ϕi> (i=1,2) eigenstates of H? (c) Calculate the commutators [H, |ϕ1><ϕ1|] and [H, |ϕ2><ϕ2|]. Is there any relationship between them? (d) Find the normalized eigenstates of H and their corresponding energy values. (e) Assuming that |ϕi> (i=1,2) form a complete and orthonormal basis, find the matrix representing H in this basis. What are the eigenvalues and eigenvectors of this matrix? Compare to the result of part (d). 3. Reading assignment: Sakurai 1.4.