Thanks a lot! That makes a lot of sense. I appreciate it!MisterX said:You could always just calculate them directly by integration.
$$<KE> = \langle \psi \mid \frac{p^2}{2m}\mid \psi \rangle$$
$$<U> = \langle \psi \mid \frac{ke^2}{r}\mid \psi \rangle$$
You can also get some expectation values using the "Hellman-Feynman Theorem"
see https://en.wikipedia.org/wiki/Hellmann–Feynman_theorem#Expectation_values
Potential energy is the energy that a system possesses based on its position or configuration. It is the stored energy that can be converted into other forms, such as kinetic energy.
In the ground state, an atom's potential energy is at its minimum, meaning the atom is in its most stable state. This allows for the atom to have a lower potential energy and a higher kinetic energy.
In the ground state, the potential energy and kinetic energy of an atom are equal. This is because the atom is in its most stable state and has no excess energy to be converted into kinetic energy.
As temperature increases, the average kinetic energy of the atoms also increases. However, the average potential energy remains the same as it is determined by the atom's position and configuration, rather than temperature.
Studying these energies can help us understand the behavior and stability of atoms in their ground state. It is also important for understanding chemical reactions and the transfer of energy in various systems.