Question

Exercisel: Consider a physical system whose state space, which is three-dimensional is spanned by the orthonormal basis formed by three kets lu, lu2) and lu). 1- In this basis, the Hamltonian operator H of the system and the observable A are written as H-h 1 0 0A where w is real constant. And the state of the system at tu0 is: 19(0)--lu:) + luz) + lus) 1- Calculate the commutator [H, A]. 2- Determine (H)s(Y(0)[H1Ψ(0) 3- Calculate ΔH,[H-hy-VIP-R2 = ((H2)-(HPF 0 0 -1 11- Consider the kets luPo) and IY,) defined by: 1- Determine the representatives matrix of lupi). Ipa) . (Ψ11 and (Ψ21 2- Calculate the norm of the kets |Ψ:) and ΙΨ2) 3- Calculate the matrix representing P,-14, 4 Calculate the matrix representing Pa Iw,l Exercise2: Consider an orthonormal basis (lu,), lu2)), in this basis, a linear operator A is defined by: A (c 0 Determine the eigenvalues and the eigenvectors of the operator A. where e is a real constant. Good Lack

Answer 1

Го Lo o O 672.O o

F H 2 to) 2. 2 2.

(IT) 3 0 on 0

0 0 for eigen valus A : O

and

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