Answer Happy

In the following, indicate the needed steps and your reasoning. Unless specified, class notations apply. Assume orthonormal eigenstates throughout. 1. An operator Û is said to be unitary if ÛÛ† = Î, where Û† is the Hermitian adjoint of Û and Î is the identity operator. Any such operator can be decomposed as follows: [21] Û = Û + Ût 2 Û - Ût +i- 2i = A +iB. (a) Show that the eigenvalues an of Û are of unit magnitude, i, .e, |an|² = 1. (b) Show that A and B are Hermitian. (5) Prove that Û is unitary and that (y(t)|(t)) is a constant. [5] (c) Show that [A, B] = [Â, Û] = [Â, Û] = 0. [5] (d) In the Schrödinger picture of quantum mechanics, the wavefunction evolves according to (r, t) = (t)) = e-it/hp(r, 0) = Ûo. [6] [22]