Variational method for approximating energy levels

Abstract

We present here a systematic scheme for improving the variational wave functions and corresponding energy levels for quantum systems. By expanding the wave function around a variational parameter value, a family of independent functions can be systematically generated. The eigenstates are then obtained by diagonalizing the Hamiltonian matrix within the basis and optimized with respect to the variational parameter. As a test, the ground state of the quartic anharmonic oscillator has been investigated, and it is found that this scheme converges more rapidly than the conventional Lanczos method and yields better approximations of the energy levels than other variational methods. The effectiveness of this scheme for larger systems remains to be seen.