Consider the one-dimensional harmonic oscillator. An appropriate approximate ground state wave function is phi(x) = Ne^-beta x^2 where N is a normalization constant and beta is an different constant. a. Solve for the normalization constant, N. Give the result in terms on beta. b. Solve for the energy expectation value of phi(x). c. Use the variational method to determine the value of beta which, according to the variational principle, provides the closest value to the exact ground state energy.