mathematical physics - Reason for the discreteness arising in quantum mechanics?

What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the boundary conditions, and the quanta in harmonic oscillator arise due to the commutation relations of the ladder operators, which give energy eigenvalues differing by a multiple of $\hbar$. But what actually is the reason for the discreteness in quantum theory? Which postulate is responsible for that. I tried going backwards, but for me it somehow seems to come magically out of the mathematics.

Answer

If I'm only allowed to use one single word to give an oversimplified intuitive reason for the discreteness in quantum mechanics, I would choose the word 'compactness'. Examples:

1. 
   

   The finite number of states in a compact region of phase space. See e.g. this & this Phys.SE posts.

   
   


2. 
   
   

   The discrete spectrum for Lie algebra generators of a compact Lie group, e.g. angular momentum operators. See also this Phys.SE post.

   
   


3. 
   

   On the other hand, the position space $\mathbb{R}^3$ in elementary non-relativistic quantum mechanics is not compact, in agreement that we in principle can find the point particle in any continuous position $\vec{r}\in\mathbb{R}^3$. See also this Phys.SE post.