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this question is two parts, both dealing with telling if combinations of hermitian operators are hermitian.

the first combination is PX + XP, where P stands for the momentum operator, (h bar /i)(d/x), and X is the "x operator", x. i have figured out that PX and XP are not hermitian by themselves, but i don't have any idea how to go about showing their linear combination is or isn't. so far i have tried calculating the expectation values for PX + XP and it's conjugate to see if they were the same, but i get the feeling this isn't a correct method.

the second combination is XPX.. i know if you have two operators multiplied, their product can only be hermitian if their commutator is zero.. but how do you do a commutator of a product of three operators?

any help is greatly appreciated! thanks so much.