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Interaction between a nucleus and electrons is in gravity(not considering) and electrostatics. Due to electrostatics nucleus attracts electrons. The force that describes this process is

$$F=k\dfrac{q_1 \, q_2}{r^2}$$

What I want to say, is that, the force is "smooth" depending on distance: it doesn't looks like a sinusoidal, I mean there are no strongly-marked values.

Why, then atom has a the straight discrete energy levels?

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To simplify for you: a large number of electrons populating for example the surfaces of two spheres will exert forces upon each other just as you describe.

The picture begins to change when we focus instead on the behavior of a single electron, and it changes completely when we confine that single electron to a very small volume of space- as for example when it is orbiting close to the nucleus of an atom, under the attractive influence of the protons in that nucleus.

That is when we discover that the electron, when confined in this way, cannot possess any energy level it wants, but instead is forced to possess energies which are discontinuous and discrete- and which we can observe and measure as the so-called "line spectrum" of that atom.

Quantum mechanics was invented to furnish an accounting of why those energy levels were discrete, and a host of other things that physicists had discovered but could not explain using the tools that worked well for large objects consisting of trillions of atoms.

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  • $\begingroup$ So, the answer is "we don't know"? $\endgroup$ – user207227 Sep 20 '18 at 18:43
  • $\begingroup$ @ArturKlochko, you should specify which part of the answers you do not understand. $\endgroup$ – wcc Sep 20 '18 at 18:49
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    $\begingroup$ we know why this is so; great volumes of books have been written about quantum mechanics over the past 100 years, and it is one of the most precise models we have ever developed to make sense of our world. What is still not known is why the break point between the classical world and the quantum world happens where it does, and whether or not it is possible to construct a workable model of the world without including the principles of quantum mechanics. $\endgroup$ – niels nielsen Sep 20 '18 at 18:51
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    $\begingroup$ Which part I do not understand. All, before the third paragraph are empty words, the third paragraph sound like just a rewriting the question - "there are energy because, there are energy levels". The last paragraph - "the quantum mechanics actually describes what you are asking for". Answer, hey, where you are? $\endgroup$ – user207227 Sep 20 '18 at 18:59
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    $\begingroup$ @ArturKlochko, if you delve into the details of the Schrodinger equation, which is the equation of motion for quantum states, you find that bound states in a potential have to satisfy certain boundary conditions. That forbids distinct (more properly, orthogonal) bound states to have a continuous spectrum. It's analogous to what you see on drum surface (here, you have a smooth tension along the surface, but the normal mode spectrum is discrete). $\endgroup$ – wcc Sep 20 '18 at 19:11
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The coulomb force is not only " "smooth" depending on distance: it doesn't looks like a sinusoidal, I mean there are no strongly-marked values." In a real experiment of two macroscopic charges under an attractive potential, there will be acceleration and a continuous radiating spectrum, the two charges neutralizing each other with big sparks.

At the microscopic level commensurate with h, the planck constant, instead of the electron falling on the proton with a continuous radiative spectrum and neutralizing it, one observes discrete spectra, not predictable by Maxwell's equations.

The classical mathematical model had to be modified, first with the Bohr atom, which postulated stable orbits,still thinking classically, and then with the solutions of the schrodinger equation which developed into the theory of quantum mechanics, postulates and all.

The difference introduced with quantum mechanics is that it is all about probabilities, i.e. orbitals, and not orbits. It is a predictive theory which determines probabilities for finding a system in a specific state.These probabilities have a wave nature, manifest in the single particle at a time double slit experiment, and as far as the spectra go, the wavefunctions give the probabilities for transition from one spectral line to the other.

It is an observational fact that there are discrete energy levels, and quantum mechanics models them successfully, and predicts innumerable other possible observations correctly. Similar to the a falling apple: it is an observational fact modeled by Newtons gravitational laws which predict successfully all new possibilities of gravitational interactions in their range of validity .

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  • $\begingroup$ Hydrogen has 4 energy levels? Then electron moves away? $\endgroup$ – user207227 Sep 21 '18 at 6:16
  • $\begingroup$ Hydrogen has an infinite number of possible energy levels in which an electron can be stable for a while, before cascading down. to the ground level, in the shcrodinger equation solution hyperphysics.phy-astr.gsu.edu/hbase/hyde.html $\endgroup$ – anna v Sep 21 '18 at 6:18
  • $\begingroup$ Another user said, atom that at $E_1$ energy state can't jump instantly to $E_3$ energy: it should absorb $E_2-E_1$ and then $E_3-E_2$. Now You say there are infinity energy levels before ionization $\endgroup$ – user207227 Sep 21 '18 at 6:27
  • $\begingroup$ They are not filled. they are possible locuses an electron would be caught in, those orbitals ( not orbits) $\endgroup$ – anna v Sep 21 '18 at 8:07