Optimal series expansion method for bound states of quantum systems

Abstract

An optimal series expansion method is developed to improve variational wave functions for the bound state calculation of quantum systems. With one variational parameter residing in the exponential decay factor, the finite power series expansion of the remaining factor in the trial wave function is determined with the criterion that the Schrödinger equation be satisfied as well as possible. In this way, more and more accurate trial wave functions can be obtained without undesirably increasing the number of variational parameters. Thus, instead of the standard variational principle of minimizing the energy expectation value, we utilize an alternative variational principle with which one minimizes the deviation of the trial wave function from being a true eigenstate of the Hamiltonian. The variational principle applies to both the ground state and the excited states. We then employ the optimal series expansion method to calculate the bound states of several quantum systems, including the one-dimensional double well potential, the Morse oscillator, the two-dimensional double well potential, and the vibrational states in the lowest electronic surface of methyl iodide. Computational results indicate that this method can yield excellent results for both energy eigenvalues and eigenfunctions. Therefore, the optimal series expansion method provides an alternative approach to the bound state calculation of quantum systems.