Moment method for eigenvalues and expectation values

We present a simple technique for performing accurate calculations of the eigenvalues of quantum systems whose potential energy is a polynomial in the coordinates. The method involves the study of recursion relations between matrix elements of powers of the coordinate operator between the exact eigenstate and a conveniently chosen basis state. The general theory is developed and applied to three examples: the quartic oscillator, the octic oscillator, and two coupled quartic oscillators. Numerical results are given.