If A and B are Hermitian operators, show that AB is Hermitian if and
only if A commutes with B
Solution
Given that
A and B are Hermitian operators
A^+=A^B^+=B^\hat{A}^+=\hat{A}\\\hat{B}^+=\hat{B}A^+=A^B^+=B^
So we check
[A,B]+=(AB−BA)+=BA−AB=−(AB+BA)[A, B]^+=(AB-BA)^+\\=BA-AB\\=-(AB+BA)[A,B]+=(AB−BA)+=BA−AB=−(AB+BA)
This is possible for
A commutes with B. So this is answer.
So
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