Accurate energy spectrum for double-well potential: periodic basis

Abstract

We present a variational study of employing the trigonometric basis functions satisfying periodic boundary condition for the accurate calculation of eigenvalues and eigenfunctions of quartic double-well oscillators. Contrary to usual Dirichlet boundary condition, imposing periodic boundary condition on the basis functions results in the existence of an inflection point with vanishing curvature in the graph of the energy versus the domain of the variable. We show that this boundary condition results in a higher accuracy in comparison to Dirichlet boundary condition. This is due to the fact that the periodic basis functions are not necessarily forced to vanish at the boundaries and can properly fit themselves to the exact solutions.

Information entropy as a measure of tunneling and quantum confinement in a symmetric double‐well potential

“…As number of basis function is increased, eigenvalues steadily improve. With N = 100, our results excellently match with the accurate estimates of [27]. They employed trigonometric basis functions satisfying periodic boundary conditions within a variational framework.…”

“…An increase in β also causes competing effects on the particle. It leads to an increase in the [27].…”

Accurate Energy Spectrum for the Quantum Yang-Mills Mechanics with Nonlinear Color Oscillations

“…This method gives highly accurate results if both the truncated domain and the number of the basis functions are adjusted properly in one [6,7,9] or two [10] dimensions.…”

“…This equation is the Schrödinger equation for the anharmonic oscillator in one-dimension and is not exactly solvable. However, we can use the optimized trigonometric basis-set expansion method to find the highly accurate solutions [6][7][8][9]. On the other hand, the semiclassical approximation (h = 1)…”

“…Thus, we expand the wave functions in terms of the trigonometric basis functions obeying Dirichlet boundary condition. Then we use the Rayleigh-Ritz variational scheme to optimize the domain of the basis functions which leads to highly accurate solutions [6][7][8][9][10]. …”