Relation between two variational methods to calculate the energy levels

Oliwier Urbański

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.

Luigi E. Picasso

D. Ninno, Giuseppe Iadonisi

Energy levels of two-dimensional Fr\"ohlich polarons in a magnetic field are calculated variationally by use of a generalization of the Lee-Low-Pines ansatz for the variational wave function allowing the phonon distribution function to depend on the electronic coordinates. It is shown that this choice leads to the correct interpolation between the weak- and strong-field limits. Comparing our ground-state energies with those obtained within second-order perturbation theory and the path-integral method, we argue that neither of these methods gives a satisfactory description of the dynamics of the electron-phonon interaction under the influence of an external field.

David D. Keeports

The variational method is applied to the problem of a cxn potential within an infinite one-dimensional well. Approximate ground- and excited-state wavefunctions are constructed as linear combinations of two-term even and odd polynomials. Comparisons are made between calculated approximate wavefunctions and energies, and results predicted from perturbation theory and from the behavior of related systems.

H. G. Miller, R. M. Dreizler