how can we directly demonstrate the existence of energy level within an atom??
Look at the spectrum of ionized gas, such as from a discharge tube, using, for example, the diffraction grating spectrometer.
so atoms can emit spectra is the proof ? y so, how does that proof?
Each of the spectra, at a specific angle, corresponds to the energy that it was emitted from. From there, you get correspondence with the atomic energy level.
You may want to look this up and see why this is so, because the theory requires algebra and geometrical description.
Or you can send a beam of electrons with a kinetic energy of a few tens of eV into a gas, and study the way they scatter via collisions. It becomes apparent that there are inelastic collisions in which atoms absorb certain fixed amounts of kinetic energy, corresponding to the differences between energy levels. Do a Google search on "Franck-Hertz experiment" for more details.
what you are essentially asking is why energy must be quantized...
..the answer is that this avoids the "ultraviolet catastrophe" where classical physics predicted that if there is an non-quantized energy continuum then blackbody radiation would have infinite energy. This was one of the first historical clues that something was wrong with describing atomic phenomena classically, and led to quantum theory - which of course is experimentally validated from the spectra that others have mentioned above.
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