Question

Sur I Nano 2019/20) Qunntum physics exercices 1.1 Linear algebra and formalism Exercice 1.1.1 Basic calculations We consideran llibert prace Ey of dimension 2 and the two following vectors of En = (17.) and le >= () acting on vectors of Eh 21- We consider also the linear operator = 1+5 1 Calculate the square norms of the two vectors < Hermitian scalar products <ul> and < > 2. Calculate the eigensalues and cigarvectors of A. >, < > and their Exercice 1.1.2 Formalism with bra and ket We consider an arbitrary Hilbert space and a linear operator of En. We recall that the adjoint operator of A noted At is defined by the relation where 1 and 2 > are arbitrary elements of En. We reall also that < 1/2 >= 21 > 1. We note lp = AI > Show that i =< 1 A. (indication using the the V properties rocalled in the text, use two expressions of the Hermition product <elp with an arbitrary element > of En. 2. Show that (BA) = A B (indication consider a matrix element < 1(BA) 2 > V with two arbitrary elements of Eu 1 > and 2 > 150 the previous questions and the properties recalled in the text). Exercice l. 1.3 We consider two states la) and b) of an Hilbert space c. We deline an operator 6 of enly 00) = le) (610). for all state (1) E 1. Justify that is a linear operator of 2.1.in Hermitian operator for arbitrary states a) and (b) Find a condition so This Ternition 3. We assume that the Hamiltonian of the systen noted H is time independent. Calculated for an arbitrary state I) (0) 1. We usume now that ex is a finite Hilbert stave of dimension 2. One note (11).12) an orthonormal basis of en. Wensume that in this busis, we have la) = C ) and b) =( Explicit the components of O and Of in the basis (11).12). 5. Calculate Tro) and De(Ó).

Answer 1

As per HOMEWORKLIB RULES I have done First question.

Please hit the like button if you want to appreciate me.