Final answer:
To determine whether each operator is Hermitian, we check if they satisfy A=A*. A² and AB+BA are Hermitian. None of the operators have pure imaginary expectation values or are purely numerical.
Step-by-step explanation:
To determine which of the given operators are Hermitian, we need to check if they satisfy the condition: A=A*
(i) AB: In general, the product of two Hermitian operators is not Hermitian. Therefore, AB is not necessarily Hermitian.
(ii) A²: A Hermitian operator squared is always Hermitian. Therefore, A² is Hermitian.
(iii) AB−BA: The difference of two Hermitian operators is not necessarily Hermitian. Therefore, AB−BA is not necessarily Hermitian.
(iv) AB+BA: The sum of two Hermitian operators is always Hermitian. Therefore, AB+BA is Hermitian.
(v) ABA: The product of a Hermitian operator with another operator is not necessarily Hermitian. Therefore, ABA is not necessarily Hermitian.
(a) None of the operators (i) to (v) are necessarily Hermitian.
(b) To calculate the expectation value, we need to consider the operator sandwiched between the bra and the ket vectors. Since the expectation value involves the complex conjugate of the operator, the real non-negative expectation values can be obtained from Hermitian operators. Therefore, A² and AB+BA have real non-negative expectation values.
(c) The pure imaginary expectation values can be obtained from anti-Hermitian operators. None of the given operators are anti-Hermitian.
(d) Purely numerical operators have expectation values that are real constants. Therefore, none of the given operators are purely numerical.