Energy eigenvalues of quadratic, pure quartic and quartic anharmonic oscillators with variational method

TL;DR

This paper calculates energy eigenvalues for quadratic, pure quartic, and combined quartic anharmonic oscillators using a variational method with simple harmonic oscillator wave functions, improving accuracy with new trial functions.

Contribution

It introduces a new set of trial wave functions that significantly reduce the error in variational energy calculations for anharmonic oscillators.

Findings

Maximum error with initial wave functions is 1.9977%.

New wave functions reduce maximum error to 0.561%.

Energy values for multiple states and parameters are reported.

Abstract

In this work, the energy eigenvalues are calculated for the quadratic ($\frac{g^2 x^2}{2}$), pure quartic ($\lambda x^4 $), and quartic anharmonic oscillators ($\frac{g^2 x^2}{2} + \lambda x^4 $) by applying variational method. For this, simple harmonic oscillator wave functions are considered as trial wave functions to calculate the energies for the ground state and first ten excited states with $g = 1$ and $\lambda =1/4$. For quartic anharmonic oscillators, energy values are calculated at different values of $\lambda$ with $g=1$. These energies for the ground state are compared with available numerically calculated data. Maximum value of $\%$error is found to be 1.9977. To get more accurate results, a new set of trial wave functions is suggested. With the newly proposed wave functions, maximum value of $\%$ error for the energy values reduces to 0.561. In this work, energies for the ground and first five excited states of quartic anharmonic oscillators are reported at different values of $\lambda$. Dependence of $\lambda$ on the wave functions is observed and concluded that wave functions are converging (shrinking) by increasing the $\lambda$.