Consider a particle moving in one dimension under the influence of a potential V(x). Suppose its wave function can be written as exp [iS (x, t) / h]. Prove that S(x, t) satisfies the classical Hamilton-Jacobi equation to the extent that h can be regarded as small in some sense. Show how one may obtain the correct wave function for a plane wave by starting with the solution of the classical Hamilton-Jacobi equation with V(x) set equal to zero. Why do we get the exact wave function in this particular case?