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.

1. 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?
2. 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 $\hbar$?
3. One can obtain an [analog of Schrödinger's equation if space was discrete](http://en.wikipedia.org/wiki/Theoretical_and_experimental_justification_for_the_Schr%C3%B6dinger_equation#Matrix_form_and_transition_amplitudes). 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.