40.1 The Schrödinger Equation

In the winter of 1925, just before Christmas, the Austrian physicist Erwin Schrödinger gathered together a few books and headed off to a villa in the Swiss Alps. He had recently learned of de Broglie’s 1924 suggestion that matter has wave-like properties, and he wanted some time free from distractions to think about it. Before the trip was over, Schrödinger had discovered the law of quantum mechanics.

Schrödinger’s goal was to predict the outcome of atomic experiments, a goal that had eluded classical physics. The mathematical equation that he developed is now called the Schrödinger equation. It is the law of quantum mechanics in the same way that Newton’s laws are the laws of classical mechanics. It would make sense to call it Schrödinger’s law, but by tradition it is called simply the Schrödinger equation.

You learned in Chapter 39 that a matter particle is characterized in quantum physics by its wave function $\psi \left(x\right)$. If you know a particle’s wave function, you can predict the probability of detecting it in some region of space. That’s all well and good, but Chapter 39 didn’t provide any method for determining wave functions. The Schrödinger equation is the missing piece of the puzzle. It is an equation for finding a particle’s wave function $\psi \left(x\right)$.

Consider a particle with mass $m$ and mechanical energy $E$ whose interactions with the environment can be characterized by a one-dimensional potential-energy function $U\left(x\right)$. The Schrödinger equation for the particle’s wave function is

This is a differential equation whose solution is the wave function $\psi \left(x\right)$ that we seek. Our first goal is to learn what this equation means and how it is used.