Abstract
The numerical solutions of the time independent Schrödinger equation of different one-dimensional potentials forms are sometime achieved by the asymptotic iteration method. Its importance appears, for example, on its efficiency to describe vibrational system in quantum mechanics. In this paper, the Airy function approach and the Numerov method have been used and presented to study the oscillator anharmonic potential V(x) = Ax2α + Bx2, (A>0, B<0), with (α = 2) for quadratic, (α =3) for sextic and (α =4) for octic anharmonic oscillators. The Airy function approach is based on the replacement of the real potential V(x) by a piecewise-linear potential v(x), while, the Numerov method is based on the discretization of the wave function on the x-axis. The first energies levels have been calculated and the wave functions for the sextic system have been evaluated. These specific values are unlimited by the magnitude of A, B and α. It’s found that the obtained results are in good agreement with the previous results obtained by the asymptotic iteration method for α =3.
Highlights
The description of vibratory system in quantum mechanics is very important for young researchers and scientists and to have a deep understanding for its future applications
The Airy function approach is based on the replacement of the real potential V(x) by a piecewiselinear potential v(x), while, the Numerov method is based on the discretization of the wave function on the x-axis
In this paper, using both the Airy Function Approach (AFA) and the Numerov Method (NM), we study the anharmonic oscillator potentials V(x) = Ax2α + Bx2
Summary
The description of vibratory system in quantum mechanics is very important for young researchers and scientists and to have a deep understanding for its future applications. Alhendi et al.[15] used a power-series expansion and Fernández[16] applied the Riccati–Padé method, to calculate their accurate eigenvalues. Barakat[25] used the asymptotic iteration method (AIM) to calculate the eigen-energies for the anharmonic oscillator potentials V(x) = Ax2α + Bx2, he introduced an adjustable parameter β to improve the AIM rate of convergence. AlFaify[26] have applied the airy function approach to study the anharmonic oscillator potentials V(x) = x8 ± x2. Anharmonic oscillator potentials V(x) = Ax2α + Bx2 for α=2, 3 and 4, we sketch the variations of the normalized waves functions associated to the first six states of energies as a function of the position, and we compare our results with the previous published ones
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