An anti-hermitian (or skew-hermitian) operator is equal to *minus* its hermitian conjugate:
Q^†=−Q^.(3.111)
(a) Show that the expectation value of an anti-hermitian operator is imaginary.
(b) Show that the eigenvalues of an anti-hermitian operator are imaginary.
(c) Show that the eigenvectors of an anti-hermitian operator belonging to distinct eigenvalues are orthogonal.
(d) Show that the commutator of two hermitian operators is anti-hermitian. How about the commutator of two *anti*-hermitian operators?
(e) Show that any operator Q^ can be written as a sum of a hermitian operator A^ and an anti-hermitian operator B^, and give expressions for A^ and B^ in terms of Q^ and its adjoint Q^†.