Other Examples of the Schrödinger Equation

Abstract

A number of properties of the one-dimensional, time-independent Schrödinger equation can be worked out without specifying the form of the coefficient. To this purpose one examines the two fundamental solutions, which are real because the coefficient is such. One finds that the fundamental solutions do not have multiple zeros and do not vanish at the same point; more precisely, the zeros of the first and second fundamental solution separate each other. It is also demonstrated that the character of the fundamental solutions within an interval is oscillatory or non-oscillatory depending on the sign of the equation’s coefficient in such an interval. After completing this analysis, the chapter examines an important and elegant solution method, consisting in factorizing the operator. The theory is worked out for the case of localized states, corresponding to discrete eigenvalues. The procedure by which the eigenfunctions’ normalization is embedded into the solution scheme is also shown. The chapter continues with the analysis of the solution of a Schrödinger equation whose coefficient is periodic; this issue finds important applications in the case of periodic structures like, e.g., crystals. Finally, the solution of the Schrödinger equation for a particle subjected to a central force is worked out; the equation is separated and the angular part is solved first, followed by the radial part whose potential energy is specified in the Coulomb case. The first complements deal with the operator associated with the angular momentum and to the solution of the angular and radial equations by means of the factorization method. The last complement generalizes the solution method for the one-dimensional Schrödinger equation in which the potential energy is replaced with a piecewise-constant function, leading to the concept of transmission matrix.