If electrons conserve energy, why they don't stop to move? Thanks.
Because stopping electrons require energy.
And what will happen if the electrons stop? Will they fall down together with the protons?
May I suggest that next time, you frame your question as clearly and in the fullest manner as possible. You'll notice that in your original question, you mention NOTHING about any protons, or the scenario that you are referring to. None of us here are able to guess what you have in mind.
Electrons in an atom doesn't stop moving, because the idea of electron "moving" in an atomic orbital is old school. Please read the FAQ in the General Physics forum for more on this.
I know that the electrons by the modern physics are waves, but what will happen if those waves fall down?
what do you mean by a wave "falling down"? Where?
I suppose instead of electrons "falling" into the nucleus, we could always just bombard the nucleus with electrons. But I don't think that's what you are asking about.
Keep in mind these electrons don't have a localized, well defined position around the nucleus.
Anyway, I know you asked Zapper this question, so I'll let him answer it. But, my opinion is that I'm not sure anyone can answer your question because it is asking for an answer to a situation that isn't described by current physics. We can calculate an electron's energy levels in a potential well using quantum mechanics, for example; but the electron in your atom has some energy, and if its energy ends up "falling" all the way (emits an integer times plack's constant quantum of energy), it "falls" into the ground state and not any more.
Maybe you'll want to rephrase your question if you understood what I said?
Separate names with a comma.