FIGURE 40.4 is the energy diagram of a particle in the rigid box. You can see that $U=0$ and $E=K$ inside the box. The upward arrows labeled $\infty$ indicate that the potential energy becomes infinitely large at the walls of the box ($x=0$ and $x=L$).

FIGURE 40.4 The energy diagram of a particle in a rigid box of length $L$.

Next, we need to establish the boundary conditions that the solution must satisfy. Because it is physically impossible for the particle to be outside the box, we require

That is, there is zero probability $(\text{i.e., }\left|\psi \left(x\right)\right|{}^2=0)$ of finding the particle outside the box.

Furthermore, the wave function must be a continuous function. That is, there can be no break in the wave function at any point. Because the solution is zero everywhere outside the box, continuity requires that the wave function inside the box obey the two conditions

In other words, as FIGURE 40.5 shows, the oscillating wave function inside the box must go to zero at the boundaries to be continuous with the wave function outside the box. This requirement of the wave function is equivalent to saying that a standing wave on a string must have a node at the ends.

FIGURE 40.5 Applying boundary conditions to the wave function of a particle in a box.