In classical wave mechanics, quantization can occur simply from a finite potential well.
In quantum mechanics, the quantization is obtained from the Schrödinger equation, which is, to my knowledge, a postulate. It does not necessitate a potential well.
When a quantum wave function is in a potential well, what causes the quantization? The finiteness of the well, or only the term with $\hbar$ in Schrödinger's equation?
Is there an analogy between these two approaches? Is the Schrödinger equation fundamentally due to a sort of boundary condition, which gives its value to the Planck constant $\hbar$?
One can obtain an analog of Schrödinger's equation if space was discrete. Is it possible to derive Schrödinger equation from such a description of space and time?
In other words, I am looking for a fundamental reason why things would be quantized in quantum mechanics. Is it analogous to the classical potential well? Is it the structure of space?
Note that an answer for a non-specialist in quantum mechanics would be appreciated, although I understand its formalism.