Simple systematics in the energy eigenvalues of quantum anharmonic oscillators

Abstract

In an earlier paper we proposed (Bhattacharya R, Roy D and Bhowmick S 1998 Phys. Lett. A 244 9) a remarkably simple scheme for evaluating the ground state energy of λx2m quantum anharmonic oscillators. In the present paper we extend the scheme to evaluate the excited state energies as well, and provide a motivation for the scheme. The only information needed in the calculation is the first term of the standard strong coupling expansion for each state of the oscillators. We have calculated the first few terms of the standard strong coupling expansion for some excited states for the quartic, sextic and octic oscillators and, for a given m, we provide simple expressions for the strong coupling constants as functions of n, which seem to reproduce any excited state energy with a reasonable accuracy. Moreover, for the ground state, we propose a simple expression for the first term of the strong coupling expansion which is globally true, i.e., it gives the value of the constant for any m. We then calculate a large number of excited state energies for the quartic, sextic and octic oscillators over a wide range of values for the coupling parameter. For any of the states the problem reduces to one of solving a polynomial equation and the predicted values closely agree with those obtained by other methods.