Question

For following can I get explanation for the answers. (step by step solution would be nice).

Prove, for any Hermitian operator , and any arbitrary state If>, that the quantity <fIÑ If> is real. The expectation value of any quantity represented by a Hermitian operator is always real. This is the reason for using Hermitian operators to represent measurable quantities. For each of the following matrices, state weather it is unitary and/or Hermitian. [1] « [1] -[9] «» (e :) Show explicitly in Cartesian (x, y, z) coordinates that the V2 and , operators commute. Show that (10 points) [[,, Î +] = tħÎ

Answer 1

eigenvalue is real. ea reat value] for ie · any hermition operator its in If>= alf> (where <FIMI > - <fi(ales). = 2 = real, o & for any matrix to be hermition, its transpose of complex conjugate is some with matrix ie if [A] is hermitian, then canjugate of [8]= [.]. Transpose - [ i]. This is aqual to the original matrix. so it is hermitian. ty net A=[ i] (a)= [- & It is hermitian. ] (67) 15:) :7 A It is not hermitian.

It is not hermition. 1.12=-it (ne - you ? [042] = - it on ), (u no )] - [x] - Bin]+[5 ] 22 C C SUS n { t = ha diversi ve [L2, e Iily] = [1, le] =i[l, hy sn] inny a ith Ly & (ix-iten). 1 [ly, z altle citity time a tho (Letic] Ito L4