A New Approximation Method for the Schrödinger Equation

Seogjoo Jang

И. Д. Феранчук, L I Komarov

Michael Jennings

The paper outlines the development of a new, spectral method of simulating the Schrödinger equation in the momentum domain. The kinetic energy operator is approximated in the momentum domain by exploiting the derivative property of the Fourier transform. These results are compared to a position-domain simulation generated by a fourth-order, centered-space, finite-difference formula. The time derivative is approximated by a four-step predictor-corrector in both domains. Free-particle and square-well simulations of the time-dependent Schrödinger equation are run in both domains to demonstrate agreement between the new, spectral methods and established, finite-difference methods. The spectral methods are shown to be accurate and precise.

R Grimm, R. G. Storer

Chi-Ming Chang

Various methods providing approximate solutions to the time dependent Schrödinger equation proposed in a previous paper are compared using two examples for which the exact analytic solutions are known. Results show that the new method proposed in the previous paper is better than the traditional method for time dependent parameters.

Bhashkar Singh, A. S. Rawat, Seema Rawat

Solving the Schrödinger equation in quaternionic quantum mechanics (QQM) has been a nut to crack. In complex quantum mechanics (CQM), approximation methods have been used to find the solutions of the Schrödinger equation when exact solutions are difficult to obtain. This study challenges the earlier Eikonal (or WKB) approximation formalism [S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995)] developed for an imaginary quaternion potential. In this paper, the correct form of coupled equations, obeying the noncommutative structure and approximated solution for quaternionic time-independent Schrödinger equation (QTISE), has been obtained.