An approach to the solution of the Schroedinger equation

Boris V. Alekseev, A. I. Аbakumov

S. D. Bajpai, Debahuti Mishra

One of the fundamental problems in quantum mechanics is to find a solution of Schroedinger equation for different forms of potentials. The object of this paper is to obtain a series solution of a particular one-dimensional, time-dependent Schroedinger equation involving Hermite polynomials. The authors also show a relationship of their particular one-dimensional, time-dependent Schroedinger equation with an equation of heat conduction.

M. Roncadelli

A. O. G. Oakes

In this work, we intend to investigate the application of the Schroedinger equation in a minimum length scenario for a potential already known in nature.In this perspective, we will apply to the potential barrier, whose ordinary case is already widely known in the literature.We intend to show the solutions of the Schroedinger equation in this scenario, the stationary solutions and, finally, discuss the transmission coefficient of the wave.

Sergey V. Ershkov

A new type of solution for the full 3+1 dimensional space-time Schroedinger equation is presented here. We consider elegant presentation of the exact solution in a spherical coordinate system, along with the assuming of separation of the two angular co-ordinates from the radial and time variables. The separation of variables follows from an assumed product form of the full potential function, which should allow us to reduce the Schroedinger Eq. to Riccati ODE in stationary case. If the angular dependence is constant, then the azimuthal dependence is linear and the polar dependence is logarithmic. With this reduction the remaining partial differential equation links the radial and time dependence which is, in effect, the standard time-dependent spherical radial Schroedinger equation. Besides, we obtain that the time-depended 2-angles solutions exist if the axis of preferential direction of wave propagation is similar to the time arrow in mechanical processes. The other possibility is the appropriate phase-shifting of the time-depended solutions. A paraxial type approximation is obtained for the solution in the limit in which the polar angle approaches zero.

Galina A. Nemova