Question

Exercise 1: Consider a physical system whose state space, which is three-dimensional is spanned by the orthonormal basis formed by three kets |ф11ф2) and IP2). I- In this basis, the Hamiltonian operator H of the system and the observable A are written as: H- ho 0 2 0 A h0 01 where o is real constant And the state ofthe system att-os: ΙΨ(0))siip)+1P2》怡1%) 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 1-0, the energy of the system is measured. What value can be found, and with what probabilities? 6- Instead of measuring H at 1 0, one measures B, what results can be found, and with what probabilities? 7- Determine (A)()IAIV(0)) & Calculate (A2)s (p(0)Alp(0)) 9- Deduce ΔΑ=((A2)-(A) 10- Calculate for the system in the stately (0), the mean value (H) and mean quadratic difference AH (root mean square deviation).

Answer 1

IoO 0 2 aan, 1/1O o 0 2 0 2구 --户 。 4. Lara, bind dort.eeigenvechms«F 뷰 H k A cemmut, LCH,A]:o).tay vill avcomm eiyen veehrs eiym ve.chr aresponding eigen va ehrs