1. Prove the following: (a) If two observables are compatible, their corresponding operators share a com- mon set of eigenvectors. Note: While this theorem holds for non-degenerate and degenerate cases, you may only consider the case where the operators are non- degenerate. (b) The kinetic energy operator in H = L²(R) is defined by the action K h² d² 2m dx²¹ Let D(K) be the domain of K consisting of continuous and infinitely differentiable functions that vanish at infinity. For any two functions p(x) and (x) in D(K), show that (pH) = (Hol). In this sense, K is said to be symmetric in its domain.

1. Prove the following: (a) If two observables are compatible, their corresponding operators share a com- mon set of eigenvectors. Note: While this theorem holds for non-degenerate and degenerate cases, you may only consider the case where the operators are non- degenerate. (b) The kinetic energy operator in H = L²(R) is defined by the action K h² d² 2m dx²¹ Let D(K) be the domain of K consisting of continuous and infinitely differentiable functions that vanish at infinity. For any two functions p(x) and (x) in D(K), show that (pH) = (Hol). In this sense, K is said to be symmetric in its domain.