Why is energy always quantized ?
You ask a very interesting question!
The fact that energy is quantized probably feels weird to you because you are looking at it from the real domain. We must expand our horizon to the Complex plane.
Energy is a continuous analytic function
When you move from one energy level to another, you are basically move from one Riemann surface to another and it is continuous !!!
Imagine it like those those swirling car parking places that go round and round. You get from one floor to the another rather smoothly right ?
BUT when you look at it from the outside, it feels as if you have jumped from one floor to the other, but in reality you just moved from one Reimann surface to another.
The same thing applies to energy as well.
I understand that’s a lot math. But I hope it helps :D