An anti-hermitian (or skew-hermitian) operator is equal to minus its hermitian conjugate:

(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 Q̂ can be written as a sum of a hermitian operator  and an anti-hermitian operator B̂, and give expressions for  and B̂ in terms of Q̂ and its adjoint Q̂⁺.

Transcribed Image Text:

ot = -0. (3.111)