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Atom, as a quantum system can have only certain energies. Also, energy, specifically kinetic energy depends on velocity, which should mean, that an atom should move only on some strict velocities, and also move only continuously, because atom jump from one energy level to another immediately.

In the observers frame of reference we have an not-moving atom, i.e. quarks and electrons, that exist around these quarks. If I understand correctly, even in this frame of reference electrons have only certain possible energies. Now it (electrons and quarks, i.e. atom) start to move. What will happen to electrons?

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Discrete energy values are the energies of the atom in the reference frame where it rests. Motion of center of mass of the atom does provide additional, much finer, control over total energy of the system. But as in free space motion of center of mass is fully separable from internal motion of the atom — i.e. from relative motions of electrons and the nucleus, it's rarely spoken of.

In real gases you can actually experimentally see the influence of motion of the center of mass of an atom on its spectrum. The effect is called Doppler broadening, and it makes the discrete energy levels fuzzy.

This is, in particular, used in Doppler cooling to lower down the temperature of multiatomic systems.

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  • $\begingroup$ Also, discrete energy levels are only present for certain potentials and energies. $\endgroup$ Commented Aug 26, 2019 at 20:49
  • $\begingroup$ @AaronStevens we are talking about an atom, there discrete levels are always present. $\endgroup$
    – Ruslan
    Commented Aug 26, 2019 at 20:50
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    $\begingroup$ I suppose this comment is better suited for the OP. The OP seemed to follow the reasoning that quantum systems must have discrete energy levels $\endgroup$ Commented Aug 26, 2019 at 20:52
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Energy levels are internal states of an atom and are not affected by the motion of the atom. The photon given off by an excited state decaying to a lower energy state can be affected by the motion of the atom. The photon ejected can be red-shifted or blue-shifted based on the direction of the atom relative to the photon detector used to measure the photon. This is the Doppler effect.

The quarks in an atom are coupled to create the protons and neutrons of the nucleus. When someone talks about atoms they are usually referring to the case with a nuclei composed of protons and maybe neutrons (Hydrogen atom has no neutrons). So the problem is much different if your including free quarks.

An excited atom does not decay immediately, they have short by measurable lifetimes.

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