One-dimensional Stationary Schrodinger Equation Research Articles

On One Implementation of the Numerov Method for the One-Dimensional Stationary Schrödinger Equation

We present accurate numerical results for the one-dimensional stationary Schrodinger equation in the case of three quantum problems: quantum harmonic oscillator, radial Schrodinger equation for a Hydrogen atom, and a particle penetration through the potential barrier. All of them were solved by the Numerov method with high accuracy and we plot their wave functions using the results of numerical calculations. Furthermore, we offer accurate numerical methods for solving boundary value problems, eigenvalue problems, matrix elimination.

Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials

<p style='text-indent:20px;'>In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential <inline-formula><tex-math id="M1">\begin{document}$ u(\omega t) $\end{document}</tex-math></inline-formula>. We show that if the frequency vector <inline-formula><tex-math id="M2">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula>. In contrast with previous results, the conditions of small potential <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> or large energy <inline-formula><tex-math id="M5">\begin{document}$ E $\end{document}</tex-math></inline-formula> are no longer needed.

A Schrödinger Potential Conditionally Integrable in Terms of the Hermite Functions

We introduce a biconfluent Heun potential well for the one-dimensional stationary Schrodinger equation which is composed of a confining fraction-power term and a repulsive centrifugal-barrier core. This is a conditionally integrable potential in that the strength of the centrifugal barrier is fixed to a constant. The potential supports a countable infinite number of bound states. We present the general solution of the Schrodinger equation, deduce the exact equation for the energy spectrumand derive a highly accurate approximation for energy levels. The bound state wave functions are written as irreducible linear combinations with constant coefficients of two Hermite functions of a scaled and shifted argument.

Schrödinger potentials solvable in terms of the confluent Heun functions

We show that if the potential is proportional to an energy-independent continuous parameter, then there exist 15 choices for the coordinate transformation that provide energy-independent potentials whose shape is independent of that parameter and for which the one-dimensional stationary Schrodinger equation is solvable in terms of the confluent Heun functions. All these potentials are also energy-independent and are determined by seven parameters. Because the confluent Heun equation is symmetric under transposition of its regular singularities, only nine of these potentials are independent. Five of the independent potentials are different generalizations of either a hypergeometric or a confluent hypergeometric classical potential, one potential as special cases includes potentials of two hypergeometric types (the Morse confluent hypergeometric and the Eckart hypergeometric potentials), and the remaining three potentials include five-parameter conditionally integrable confluent hypergeometric potentials. Not one of the confluent Heun potentials, generally speaking, can be transformed into any other by a parameter choice.

Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well

We consider the one-dimensional stationary Schrodinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the formulas known in the case of a mirror-symmetric potential. We consider the case of higher energy levels and the case of energies close to the potential minimums. We present an example of tunneling transport in an asymmetric double well and also consider the problem of tunnel perturbation of the discrete spectrum of the Schrodinger operator with a single-well potential. Exponentially small perturbations of the energies occur in the case of local potential deformations concentrated only in the classically forbidden region. We also calculate the leading term of the asymptotic expansion of the tunnel perturbation of the spectrum.