The Variational Method and the Harmonic Oscillator

Peter Bongaarts

Gang Chen

Trial wave function was used to calculate ground state energy to obtain a relatively accurate value.After adding potential δ to one-dimensional harmonic oscillator,the odd parity stationary states of the harmonic oscillator still held and the even parity stationary states were no longer correct. Therefore,the perturbation theory and variational method were used respectively to calculate the approximate even parity stationary states,and variational method was used to solve Schrodinger equation.Results show that the calculated approximate even parity stationary states match well with the desired value,especially in excited state,the relative error is less than×10-3,and all the approximate values of energy for all levels are higher than desired value,which prove that it is simple and feasible by using perturbation theory or variational method to calculate approximate even parity stationary states with δ potential added.

Stephen Bruce Sontz

W. N. Mei

When applying the variational method, it is demonstrated that the systematic improvements in calculating the eigen‐energies of quantum mechanical systems can be achieved by careful study of the general features of the trial wave functions. Two sets of trial wave functions were used for the harmonic oscillator potential. In the first example, it is shown, even with a crude initial guess, that accurate results will be obtained by adjusting the wave function without introducing any additional variational parameters. Even though the second example requires more algebraic manipulation, it is shown that it converges to the exact result much faster. It is suggested that these calculations be incorporated in computer algebra software packages for the teaching of quantum mechanics.

Farshid Mehrdoust, Hossein Aminikhah, Mohammad Ghamgosar

The goal of this paper is to present an application of variational Monte Carlo method for solving one dimensional harmonic oscillator problem.

Jochen Pade