Math Methods in Physics: HW Solutions No. 13 - Commuting Matrices & Eigenvectors, Assignments of Physics

Download Math Methods in Physics: HW Solutions No. 13 - Commuting Matrices & Eigenvectors and more Assignments Physics in PDF only on Docsity! University of Alabama in Huntsville Department of Physics Spring 2007 PH 305: Mathematical Methods in Physics Homework Solutions No. 13 Solution 48: AB = ( −10 70 70 −115 ) = BA AC = ( −34 −70 −72 65 ) 6= ( −34 −72 −70 65 ) = CA AD = ( 80 −10 −10 95 ) = DA BC = ( −89 30 38 −135 ) 6= ( −89 38 30 −135 ) = CB BD = ( 30 90 90 −105 ) = DB CD = ( −146 −128 −130 35 ) 6= ( −146 −130 −128 35 ) = DC . We therefore find that A, B, D are mutually commuting. To find the eigen- vectors, take, say, A: 0 = ∣∣∣∣∣ 6 − λ −2−2 9 − λ ∣∣∣∣∣ = (λ − 5)(λ − 10) and the eigenvectors are v1 = ( 2 1 ) v2 = ( 1 −2 ) . 1 Direct substitution shows that these two vectors are also the eigenvectors for B, D, i.e., B |vi >= λ |vi > D |vi >= λ |vi > with eigenvalues 5,−15 (for B) and 15, 10 (for D). This is an example to the result that mutually commuting matrices have the same eigenvectors, but not necessarily the same eigenvalues. Solution 49: 1. From AB−BA = 2iC and AB = −BA it follows that AB = iC. Thus, −C2 = iCiC = ABAB = A(−AB)B = −(AA)(BB) = −I I = −I so that C2 = I. [B, C] = BC − CB = B(−iAB) − (−i)ABB = −i(BA)B + iAI = −i(−AB)B + iA = iA + iA = 2iA . 2. [[[A, B], [B, C]], [A, B]] = [[2iC, 2iA], 2iC] = −4[[C, A], 2iC] = −4[2iB, 2iC] = 16[B, C] = 32iA . 2