Schrödinger’s Equation with Potential Energy: Introduction to Operators

Abstract

The wave equations of electromagnetism are a natural consequence of Maxwell’s equations. The time and spatial derivatives of the fields are found to be connected in a simple manner. But what physical interpretation can be given to the quantity ∇2ψ(r, t) that appears in Schrödinger’s equation? To understand the physical significance of this term, I have to introduce the concept of operatorsoperators. Operators play a very important role in quantum mechanics but can be a bit confusing; specifically a distinction must be made between operators and functions. The plan is to look at the free particle Schrödinger equation and to show that the −ħ2/2m∇2$$-\left ( \hbar ^{2}/2m\right ) abla ^{2}$$term can be represented as p̂2/2m=p̂⋅p̂/2m$$\hat {p}^{2}/2m=\mathbf {\hat {p}\cdot \hat {p}}/2m$$, where p̂$$\mathbf {\hat {p}}$$is an operator. I will then be able to generalize Schrödinger’s equation to account for situations in which an external potential is present. To simplify matters, I work with the Schrödinger equation in one dimension; in the appendix, an analogous development is given for the Schrödinger equation in three dimensions.