Approximations to the Stationary Schrödinger Equation

Benjamin A. Stickler, E. Schachinger

Pedro Pereyra Padilla

Benjamin A. Stickler, E. Schachinger

Kunlapat Sansuk, Petarpa Boonserm, Tritos Ngampitipan

Quantum mechanics is a theory that can explain phenomena of atom-sized matter or smaller, which classical mechanics cannot do. The central equation of quantum mechanics is the Schrödinger equation. Mathematically, the Schrödinger equation is a second order partial differential equation which tells us how a wave function changes with time. For most problems in quantum mechanics, the Schrödinger equation cannot present exact solutions, except for a few idealized systems in which the Schrödinger equation can be solved exactly. Therefore, approximation methods are needed to find solutions to the Schrödinger equation. In this paper, systems in stationary states are of special interest, in which the Schrödinger equation is reduced to the time-independent Schrödinger equation. One of the approximation methods that work for systems in stationary states is the WKB approximation. In this paper, transmission probabilities for various potentials are calculated using the WKB approximation. The potentials studied in this paper are composed of the asymmetric square-well potential, the hyperbolic tangent potential, the squared hyperbolic secant potential, and the double squared hyperbolic secant potential.

Hichem Eleuch, Yuri V. Rostovtsev

We have found an analytical solution of the 3D stationary Schrödinger equation for any arbitrary potential. The solution has been obtained by transforming the Schrödinger equation to the generalized Ricatti equation, and it has the form of Huygens’ principle for matter waves. We have shown that the solution is more accurate than the JWKB approximation. We discuss possible applications of the solution to molecular dynamics and dynamics of charges in nanostructures.

Shabnam Siddiqui

In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).