# Does electron-proton interaction and electron-electron interaction in an atom gives rise to a microscopic potential energy?

When studying thermodynamics we come across a property of a system called internal energy, which is the sum of all energies possessed by the system at the microscopic level. Internal energy has two components. A kinetic energy component and a potential energy component.

I read that, potential energy at the microscopic level arises because of the interaction

• Between the molecules of a system
• Between the atoms of a molecule
• Between the nucleons - protons and nuetron

I wonder if some potential energy arises because of the interaction

• between electrons in an atom
• between electrons and protons in an atom

Is some potential energy associated with electron-electron and proton-electron interaction?

It’s a little weird that your textbook discusses intermolecular and internucleon potential energies, but leaves out electron-nucleus interactions. At ordinary temperatures, the nuclear degrees of freedom are completely frozen out; as temperatures increase, the electronic excitations become accessible long before the nuclear degrees of freedom.

Perhaps your text is treating internal energy in the classical limit, where a “jiggling atom” may have any distance from its neighbors and therefore any continuous value for the internal energy. Electronic excitations in atoms, by contrast, can only store energy in discrete lumps; internal energy is added to or removed from the electron-nucleus field by absorbing or emitting photons.

The resolution to this inconsistent treatment of “electrons” versus “jiggling” is that “jiggling” energies are also quantized, and their energy also comes in lumps. However, the energies of the allowed oscillator states for an atom in a solid at room temperature are very close together, so the continuum approximation is okay. For a discrete state with energies $$E$$ above the “ground state,” the probability of finding a system in that state goes like the “Boltzmann factor” $$e^{-E/kT}$$. These probabilities mean effectively that all of the system’s states up to the “thermal energy” $$kT$$ are sort of equally likely to be found, while higher-energy states rapidly become forbiddingly improbable. The quantum nature of atomic vibrations in solids and liquids is responsible for some surprising things they do at low temperature. A famous example is zero-entropy superfluidity in helium (though there’s lots of other details to that story.)

You might know that the energies for an electron in a hydrogen atom are $$E_n =-13.6\,\mathrm{eV}/ n^2$$. At room temperature $$T≈300\,\mathrm K≈25\,\text{milli-eV}/k$$, the Boltzmann factor for hydrogen’s first excited state is roughly $$e^{\rm -10\,{eV}/25\,meV}≈10^{-10^3}=0$$. At room temperature, you can completely ignore electronic excitations in most atoms when you are computing things like heat capacities, which depend on internal energy.

However, in a hot enough system, electronic excitations can play an important role in thermodynamics. In the outer layers of the Sun, there is a region hot enough that the $$n=3$$ and higher hydrogen states develop a significant population. In this region, you get spontaneous transitions $$n=3\to 2$$, which emits a photon with a characteristic pink color called “hydrogen-alpha.” You can see this pink part of the Sun with your naked eye during a total solar eclipse, if you’re lucky. A camera filter which transmits only hydrogen-alpha light can take an image of a relatively thin region of the Sun’s surface; image searches are helpful.

At room temperature, the electronic internal energy is there, but you can’t change it, so it’s not really important for thermodynamics. The same is true for the interaction energies between protons and neutrons, whose quantized excitations start at about a mega-eV.

It depends on your point of view. We say that a spring stores potential energy. In microscopic models, part of that is electrostatic, but part is due to "Pauli force" between electrons. However, in quantum field theory, Pauli force isn't a force and doesn't have a potential. Instead, it's associated with increased electron kinetic energy.

I'm a bit of a positivist here: things that register on force gauges should be understood as forces, so the energy stored in the spring should be understood as potential energy. I say yes to your question.