Question

 If A, B and C are Hermitian operators, determine if the following combinations are Hermitian: (a) A + B (b) 1 2i [A, B] (c) (ABC − CBA) (d) A2 + B2 + C 2 (e) (A + iB)

Answer 1

Answer

Given that

A, B ,C are Hermitian operators

So

$\hat{A}^+=\hat{A}\\\hat{B}^+=\hat{B}\\ \hat{C}^+=\hat{C}$

Therefore checking hermitian

$(a) (A + B ) ^+= A^+ + B^+\\= A + B$

$(b) (2i[A , B ] )^+=(-2i) [B,A]\\=2i [A , B ]$

d) $(A^2 + B^2+C^2 ) ^+\\= A^2 + B ^2+C^2$

So options a b d are Hermitian and other are not.