This chapter summarizes some qualitative properties of the solutions of the one-dimensional Schrödinger equation. The one-dimensional equation represents not only the idealized situations but is of interest also for solving the Schrödinger equation for three-dimensional central potentials as in that case the problem reduces to solving one-dimensional equations. The study of the qualitative properties enables one to extract useful physics information without solving the equation which more often than not is a formidable task. The question of identification of exactly solvable one-dimensional potentials and their explicit solutions is addressed in subsequent chapters.

Asymptotic Behaviour

Consider a particle of mass m constrained to move in one dimension in a time-independent potential V(x). Its wave function of definite energy E solves the Schrödinger equation

where U(x) and are the scaled potential and energy defined by

We study first the asymptotic properties of the solutions of (9.1) under the condition as , where V± are constants. The solution of (9.1) as is then given by

where. It is real for. In that case, as we will see in Section 10.1, the first term in (9.3) represents the particle moving freely in the direction, whereas the second one represents the one moving freely in the −x direction.

However, if E < V+m, then k+ is imaginary and (9.3) reads