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
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.
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