Two Applications of the Variational Method to Quantum Mechanics

By using the unperturbed wave function with a variable scale factor as trial-function for a perturbed Schroedinger equation, a lower bound for the second-order perturbation energy of the ground state in terms of the unperturbed and first-order perturbation energy is obtained. A minimum property of the sum of the $n$ lowest proper values is utilized in a new method for obtaining higher proper values and functions.