How is heat represented on a quantum level? Heat is just a form of kinetic energy for molecules, because as temperature rises, the heated molecules are "shake" and "vibrate" more and more. But how does that show up on a quantum scale? What element actually carries the kinetic energy: the heated molecule as a whole, its atoms, the nuclei, or the electrons' orbits? (Maybe even the quarks found in the nuclei?). Or is it that the shaking described is only an analogy for a notion of energy that is more difficult to grasp as their is no real physical movement in the heated object?
 A: Heat energy, at a microscopic level, is stored in the degrees of freedom of atoms and molecules. These degrees of freedom are translational, rotational and vibrational. They all store different amounts of energy, depending on the geometry of the atom. 
Translational degrees of freedom are the atom or molecule moving around in space, and there are always 3 for the 3 dimensions of space. The rotational and vibrational modes come from the geometry of the atom/molecule. 
From quantum mechanics, we get the idea that energy stored in rotational and vibrational (and translational, if confined) modes must come in quantised packets, with a minimum size. This size depends on the form of a certain mode. For single atoms, the moment of inertia and the energy of rotation is very small. The quantum of energy that must be added to excite the rotational modes is large, and so these do not contribute to heat storage until very high temperatures. 
Molecules have much higher moments of inertia around certain axes. For example, O2 has high moment of inertia around the two axes perpendicular to its bond axis and a low moment of inertia around its bond axis. It therefore stores heat energy in those two and they contribute to the heat capacity of O2. 
Vibrational modes store much more energy than translational or rotational modes, and are active only at higher temperatures.
This is basically what heat is at a microscopic level. Quantum mechanics gives us that the energy stored in the modes must be quantised.
A: The equipartition theorem says that all modes of excitation carry heat. There may be some modes which are too energetic to be excited at a given temperature, but the remaining modes are all excited. In overly simple terms, everything that can shake will shake. 
A: It is somehow an open subject of research. No way through just subdividing you end linking quantum world to classical one. At quantum level, different classical processes reading to a particular concept, always rely on just already determined positions and energy of a system. When one goes to systems with very low entropy we tend to lose track of measurable energies whereas high energy systems are  also source of different other particles. We then find different divisions inside physics itself, where, it is not possible to reconcile small to big or low to high energy. Always different sub-domains are to be made for different scenarios. The main question to ask some times is, could be their parameters other than just scientific limitations in understanding, which kind of block different levels of understanding to combine information in a correct way, which can simply, show clearly how exits to a particular understanding are hold up by other than just personal perceptions of things. 
A: heat is not kinetic or something,heat must be emergence and to be a question how the quantums could connect with others without emergence and out side of the consequence,then the time and anything do not growing by cause and consequence theory.then the paradox occur,i absolutely don't understand it.
A: phonons  are called quantum of lattice vibrations. Just as photons are quanta of EM radiations.In solids lattice vibrate due to heat. The quantum of lattice vibration is called a phonon. It is planck constant X frequency just as you have for photons. The heat capacity of solids for example and the thermal conductivity are explained using this phonon concept. –
