40.3.1 Model: Identify a Potential-Energy Function

By a rigid box we mean a box whose walls are so sturdy that they can confine a particle no matter how fast the particle moves. Furthermore, the walls are so stiff that they do not flex or give as the particle bounces. No real container has these attributes, so the rigid box is a model of a situation in which a particle is extremely well confined. Our first task is to characterize the rigid box in terms of a potential-energy function.

Let’s establish a coordinate axis with the boundaries of the box at $x = 0$ and $x = L$ . The rigid box has three important characteristics:

The particle can move freely between 0 and $L$ at constant speed and thus with constant kinetic energy.
No matter how much kinetic energy the particle has, its turning points are at $x = 0$ and $x = L$ .
The regions $x < 0$ and $x > L$ are forbidden. The particle cannot leave the box.

A potential-energy function that describes the particle in this situation is

U_{rigid box} (x) = {\begin{array}{l} 0 & 0 \leq x \leq L \\ \infty & x < 0 or x > L \end{array}

(40.9)

Inside the box, the particle has only kinetic energy. The infinitely high potential-energy barriers prevent the particle from ever having $x < 0$ or $x > L$ no matter how much kinetic energy it may have. It is this potential energy for which we want to solve the Schrödinger equation.