The one-dimensional Schrödinger wave equation and some of its applications

Gabriel González

A duality between an electrostatic problem in a three dimensional world and a quantum mechanical problem in a one dimensional world which allows one to obtain the ground state solution of the Schrödinger equation by using electrostatic results is generalized to three dimensions. Here, it is demonstrated that the same transformation technique is also applicable to the s-wave Schrödinger equation in three dimensions for central potentials. This approach leads to a general relationship between the electrostatic potential and the s-wave function and the electric energy density to the quantum mechanical energy.

John D. McGervey

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.

Vladimír Tichý

The goal of the dissertation is to find new method of solving two-dimensional Schrödinger equation in such cases, when the separation of the variables is not applicable. The results are applied to the two-dimensional Schrödinger equation with the potentials of the form of the quartic polynomial, of the sextic polynomial and of the quartic Morse potential. For these cases, the analytical formulae for the ground state wave functions and the corresponding energies have been found. For the specific class of the potentials of the form of the quartic polynomial, analytical formula for one of the excited states and for the corresponding energy have been found.