FIGURE 40.3 shows a particle of mass $m$ confined in a rigid, one-dimensional box of length $L$. The walls of the box are assumed to be perfectly rigid, and the particle undergoes perfectly elastic reflections from the ends. This situation, which we looked at in Chapter 38, is known as a “particle in a box.”

FIGURE 40.3 A particle in a rigid box of length $L$.

A classical particle bounces back and forth between the walls of the box. There are no restrictions on the speed or kinetic energy of a classical particle. In contrast, a wave-like particle characterized by a de Broglie wavelength sets up a standing wave as it reflects back and forth. In Chapter 38, we found that a standing de Broglie wave automatically leads to energy quantization. That is, only certain discrete energies are allowed. However, our hypothesis of a de Broglie standing wave was just a guess, with no real justification, because we had no theory about how a wave-like particle ought to behave.

We will now revisit this problem from the new perspective of quantum mechanics. The basic questions we want to answer in this, and any quantum-mechanics problem, are:

• What are the allowed energies of the particle?

• What is the wave function associated with each energy?

• In which part of the box is the particle most likely to be found?

We can use Problem-Solving Strategy 40.1 to answer these questions.