Equation 40.16 defined $\beta$. The boundary condition of Equation 40.19 then placed restrictions on the possible values of $\beta :$

where the value of $\beta$ and the energy associated with the integer $n$ have been labeled ${\beta }_n$ and $E_n$. We can solve for $E_n$ by squaring both sides:

where, in the last step, we used the definition $\text{ℏ}=h\text{/}2\pi$. For a particle in a box, these energies are the only values of $\mathit{E}$ for which there are physically meaningful solutions to the Schrödinger equation. That is, the particle’s energy is quantized! It is worth emphasizing that quantization is not inherent in the wave function itself but arises because the boundary conditions—the physics of the situation—are satisfied by only a small subset of the mathematical solutions to the Schrödinger equation.

It is useful to write the energies of the stationary states as

where $E_n$ is the energy of the stationary state with quantum number n. The smallest possible energy $E_1=h^2\text{/}8m{L}^2$ is the energy of the $n=1$ ground state. These allowed energies are shown in the energy-level diagram of FIGURE 40.6. Recall, from Chapter 38, that an energy-level diagram is not a graph (the horizontal axis doesn’t represent anything) but a “ladder” of allowed energies.

FIGURE 40.6 The energy-level diagram for a particle in a box.

Equation 40.22 is identical to the energies we found in Chapter 38 by requiring the de Broglie wave of a particle in a box to form a standing wave. Only now we have a theory that tells not only the energies but also the wave functions.