40.2 Solving the Schrödinger Equation

The Schrödinger equation is a second-order differential equation, meaning that it is a differential equation for $\psi \left(x\right)$ involving its second derivative. However, this textbook does not assume that you know how to solve differential equations. As we did with Newton’s laws, we will restrict ourselves to situations for which you already have the mathematical skills from your calculus class.

The solution to an algebraic equation is simply a number. For example, $x=3$ is the solution to the equation $2x=6$. In contrast, the solution to a differential equation is a function. You saw this idea in the preceding section, where Equation 40.6 was constructed so that the function $\psi \left(x\right)={\psi }_0\text{sin}\left(2\pi x\text{/}\lambda \right)$ is a solution.

The Schrödinger equation can’t be solved until the potential-energy function $U\left(x\right)$ has been specified. Different potential-energy functions result in different wave functions, just as different forces lead to different trajectories in classical mechanics. Once $U\left(x\right)$ has been specified, the solution of the differential equation is a function $\psi \left(x\right)$. We will usually display the solution as a graph of $\psi \left(x\right)$ versus $x$.