Playing with the Schrödinger Equation

Abstract

In order to determine the state of a microsystem computationally, we must solve its Schrödinger equation. The first step in this process is to devise the appropriate Hamiltonian operator for the problem at hand. The kinetic energy operator is universal; it depends only on the number of particles in the system. The potential energy function is more characteristic of the system, and the difficulty of solution depends on its form. It is only possible to solve the Schrödinger equation analytically in the case of the simplest potential functions. We will see three such examples in this chapter. Our first example will be the motion of a free particle in space, the simplest of all cases. The second will be that of the harmonic oscillator. In the third case, the particle is in a central Coulomb field, which is the case in the hydrogen atom. This will be discussed in greatest detail because the conclusions from this example are far-reaching for the rest of the book. It is not our intent to present a rigorous derivation of the solutions in these examples; for that, the reader is referred to specialist books on quantum mechanics.