Bound state eigenfunctions of an anharmonic oscillator in one dimension: a Numerov method approach. (English) Zbl 1359.65130

Shah, Balkrishna P.

Int. J. Adv. Appl. Math. Mech. 2, No. 2, 102-109 (2014).

Summary: Numerical methods are useful when analytical solutions are difficult to derive. An oscillator is described by the second order differential equation. In this article the numerical method for the special case of differential equation is applied for the solution of the wave function of a harmonic oscillator quantum mechanically in classical as well as non classical region. The Numerov’s method is applied for evaluating the wave function with initial condition. The computational flow is explain schematically. The method is further applied to anharmonic oscillator. The evaluated wave function is plotted along with harmonic potential. The possible solution leads to the bound state energy eigen values. The inward and outward integrations are executed to achieve smooth matching at boundaries. The exact bound state solutions are obtained by the numerical integration corresponds to \(n=0\) to \(n=4\). The numerical approach gives the symmetric and antisymmetric wave functions. Computational issues concerned is energy convergence, which is controlled by the step size \(h\), range of potential \(x_m\) and lower bound guessed eigenvalue as well. The present results obtained are also compared with that of Bahttacharya.

MSC:

65L15

Numerical solution of eigenvalue problems involving ordinary differential equations

Keywords:

Numerov method; anharmonic oscillator; inward integration; outward integration