A linear operator [3' is unitary if on} = 00* = I, Where I is the identity operator. This means Uf is the inverse U'1 of U, i.e. Ui = U". (a) Show that a unitary operator preserves probabilities, is. show that (Gilda/)2) = l1l2h (1) A mmstxluenee of this is that an orthonormal set of wave functions remains orthonormal after one applies a. unitary transformation to all of them. In a nite real vector space, this is just a rotation of the basis of vectors. The discussion in this problem also applies more generally. (b) A matrix function is dened by its Taylor expansion. For example, for a matrix Kl, we have on ell : lAu I . n. n20 Show that if A is lierInitiau, then {A} = (5'75 is unitary. (C) Use [1) to Show that all eigenvalues of a unitary operator have complex norm 1. [(1) Recall that eigenvalues of Ilennitian operators also simplify in a certain manner. Using the result of (l)), show that the general properties of eigenvalues [or Hermitean and for unitary operators are compatible. (0) Consider a Hamiltonian with a timeindependent real potential. Argue that the time- evolution sweeter U (t) = t:""'"'" is a unitary operator. (f) Suppose llaf(1:, (l) : En chi/1\"\"), where the 1,0,,(I) is a (:(unpleto set of eigenfmmtions of the Hamiltonian IN)" = "11:... What do you get when you act with 0(t) = c"\"'\"' on \\P(:r:,(]]'? How does this justify that U\") = (lm/'5 is called the timeevolution operator