Schrödinger Equation for One-Dimensional Systems

Jerry Goodisman

A.P. French, Edwin F. Taylor

This chapter examines some explicitly one-dimensional systems, including the finite square well, the harmonic oscillator, and the application of harmonic oscillator theory to molecular vibrations. It analyzes the computer solutions of the Erwin Schrodinger equation. Although there are no real one-dimensional square wells in nature, many real systems have potentials that resemble the square well. Near the center of force the different wave functions must differ from one another or else they would not satisfy the Schrodinger equation for different values of energy. In contemporary research almost all the manipulation of the Schrodinger equation is done not analytically but rather by computer using numerical methods. A major advantage of the dimensionless form of the Schrodinger equation is that a single solution for one kind of system is trivially easy to extend to all similar systems. The dimensionless form of the Schrödinger equation can be solved numerically if it is approximated by a so-called difference equation.

John D. McGervey

Emanullah Hızel, Nurettin Cenk Turgay, Bahri Guldoùgan

Schrödinger-Newton (SN) equations are modifications of the Schrödinger equation proposed by Roger Penrose involved in a gravitationally-induced wave function collapse scheme. In this paper, the three dimensional time independent Schrödinger-Newton (SN) equation is analyzed. The Lie group formalism is applied to deduce symmetries of the time indepen-dent SN equation for three dimensional case. One and two dimensional optimal systems and reduced equations are obtained.

H. Puszkarski

We propose a new simplified procedure for finding the analytical solutions of the stationary one-dimensional Schrödinger equation, with asymmetric boundary conditions imposed on the equation.The essence of the method consists in expressing the general solution explicitly in terms of the boundary parameters in a form which, by itself, satisfies one of the boundary conditions involved; then, the other boundary condition gives straightforwardly the characteristic equation.This method may turn out to be beneficial with regard to the recently growing interest in one-dimensional quantum systems.