Apart from quantum mechanical systems, how you will accept that energies are quantized?

Question

Why energies are quantized at quantum scale? Apart from quantum mechanical calculations, how you will accept that energies are quantized?

Answer 1

Why is always hard to answer. Here’s Feynman charmingly explaining the same. Because to answer why, one must agree upon a framework (the axioms) from which to build the argument to answer.

The beauty of science is however, we always rely on nature. After all nature is what we are trying to describe. So no matter what our theories describe, at the end of the day what we must accept is what nature tells us.

And what nature tells us is that energy is quantised. One of the earliest triumphs of the quantum theories was Neils Bohr’s model of the hydrogen atom correctly describing the discrete lines of the hydrogen spectrum (Balmer series, Lyman series and so on) which had been an open problem for over two decades at that point.

Here’s the observed hydrogen spectrum. What this describes is the energy emitted/absorbed by hydrogen atoms. It’s clearly discrete. And that’s all (among other confirming evidence) it takes us to convince that the energy levels are quantised.