Exercise 1: Consider a physical system whose state space, which is three-dimensional is spanned by the orthonormal basis formed by three kets |₁) 12) and 13). I-In this basis, the Hamiltonian operator H of the system and the observable A are written as: 0 0 0 H=ho 0 2 0 A=h00 1 0 0 where w is real constant. 0 1 0 And the state of the system at t=0 is: 14(0)) = 1) +192)+13) 1- Calculate the commutator [H, A]. 2- Determine the energies of the system. 3- Determine the eigen-values of A. 4 Determine the eigen-vectors common to H and B. 5- At =0, the energy of the system is measured. What value can be found, and with what probabilities? 6- Instead of measuring Hat t=0, one measures B, what results can be found, and with what probabilities? 7- Determine (A) = (4(0)|A|4(0)) 8- Calculate (A2) = (4(0)|A|4(0)) 9- Deduce AA = ((A²) - (A)²) 10- Calculate for the system in the stately (0)), the mean value (H) and mean quadratic difference AH (root mean square deviation).