1. The problem statement, all variables and given/known data Consider a thee-dimensional ket space. If a certain set of orthonormal kets - say, |1>, |2>, and |3> - are used as the base kets, the operators A and B are represented by A = a 0 0 0 -a 0 0 0 -a B = b 0 0 0 0 -ib 0 ib 0 with a and b both real. a) Obviously A exhibits a degenerate spectrum. Does B also exhibit a degenerate spectrum? b) Show that A and B commute. c) Find a new set of orthonormal kets which are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket? 2. Relevant equations [A,B] = AB - BA = 0 if A & B commute 3. The attempt at a solution a) B has eigenvalues b,b,-b. So yes, B is degenerate. b) I have no problem showing that A & B commute. c) I know how to find eigenkets (eigenvectors) of a matrix using the matrix eigenvalues, but I do not know how to go about finding eigenkets that are simultaneous eigenkets of both A and B? I tried finding eigenvalues of the matrix AB, which come out to ab,ab,-ab (also degenerate) but can only construct 2 eigenkets, and when I construct a third orthonormal eigenket using the cross product |1>[tex]\otimes[/tex]|2> ) it does not give A|3> = a|3> or B|3> = b|3>. Any hints or suggestions would be greatly appreciated! edit: I understand that the eigenkets should satisfy A|a'b'> = a'|a'b'> & B|a'b'> = b'|a'b'>, I just don't know how to go about finding |a'b'>.