Throughout this text, we’ve emphasized the importance of models. To understand the motion of an object, we made simplifying assumptions: that the object could be represented by a particle, that friction could be described in a simple way, that air resistance could be neglected, and so on. Models allowed us to understand the primary features of an object’s motion without getting lost in the details.

The same holds true in quantum mechanics. The exact description of a microscopic atom or a solid is extremely complicated. Our only hope for using quantum mechanics effectively is to make a number of simplifying assumptions—that is, to make a quantum-mechanical model of the situation. Much of this chapter will be about building and using quantum-mechanical models.

The test of a model’s success is its agreement with experimental measurement. Laboratory experiments cannot measure $\psi \left(x\right)$, and they rarely make direct measurements of probabilities. Thus it will be important to tie our models to measurable quantities such as wavelengths, charges, currents, times, and temperatures.

There’s one very important difference between models in classical mechanics and quantum mechanics. Classical models are described in terms of forces, and Newton’s laws are a connection between force and motion. The Schrödinger equation for the wave function is written in terms of energies. Consequently, quantum-mechanical modeling involves finding a potential-energy function $U\left(x\right)$ that describes a particle’s interactions with its environment.

FIGURE 40.2 reminds you how to interpret an energy diagram. We will use energy diagrams extensively in this and the remaining chapters to portray quantum-mechanical models. A review of 《 SECTION 10.5, where energy diagrams were introduced, is highly recommended.

FIGURE 40.2 Interpreting an energy diagram.