Question

-. Let Ä be an NỮN Hermitian operator corresponding to an observable in a quantum system whose Hilbert space is an N-dimensional one. Recall that the eigenvalues and eigenvectors of Ä are given by the solutions of Âlai) = ailai), i = 1, ..., N where the eigenvalues ai are all real, an the eigenvectors form a complete orthonormal set on the N-dimensional Hilbert space, meaning that (ailaj) = dij. Suppose the state vector of the system at some point in time is ). By expanding the state vector in the basis of the eigenvectors of  as 10) = {cila) where ci = (a;), show that the expectation value of  in the state ), defined as (Â) = (V|ÂIV), can be expressed as (a) = Šailcil and give a physical interpretation of this formula in terms of individual measurements of Â.

Answer 1

que have 147 = E cilar> SE 9 = 1 and Ála:> = 2:19:17 expectation value of given by . K»:::<41 Âly)  is = [ {ail CCA 19:> E 1011? Pai 1 A 19;> but Ala) = Qila;% =>{A>£10:12 <qi19:49:7 - E 16:12 0; <Q119,7 but Tail ai >= 1 > <A>= { leihai Pie

above exprennion can be expanded <A> = op lealt al cal²t... taniwl? here a; → j th eigerwalue of A . (ci12 probability that A in in - the state 197. than expectation value of  is given by the sum of at's eigenvaluer multiplied by their corresponding probability,