40.2.2 Quantization

We’ve asserted that the Schrödinger equation is the law of quantum mechanics, but thus far we’ve not said anything about quantization. Although the particle’s total energy $E$ appears in the Schrödinger equation, it is treated in the equation as an unspecified constant. However, it will turn out that there are no acceptable solutions for most values of $E$. That is, there are no functions $\psi \left(x\right)$ that satisfy both the Schrödinger equation and the boundary conditions. Acceptable solutions exist only for discrete values of $E$. The energies for which solutions exist are the quantized energies of the system. Thus, as you’ll see, the Schrödinger equation has quantization built in.