1. | are the eigenstates of a Hermitian operator H (H is, for example, the Hamiltonian of an arbitrary physical system). Assume that the states | > form a discrete orthonormal basis. The operator U(m, n) is defined by: U(m, n) = | Om > < n \ a. Calculate the adjoint U(m, n) of U(m, n). b. Calculate the commurator [H, U(m, n)]. c. Prove the relation: U(m, n)U*(p, q) == nq 8ng U(m, p) d. Calculate Tr { U(m, n) }, the trace of the operator U(m, n). = e. Let A be an operator, with matrix elements Amn <m A. Prove the relation: A = Σ Amn U(m, n) m.n f. Show that Apa = Tr { AU¹(p, q) }.