DrClaude said:Energy eigenstate means that it is an eigenstate of the Hamiltonian, and therefore has a defined (constant) energy. It is a common expression for all sorts of quantum systems.
DrClaude said:There is no "most common use," it is a matter of context. If you're talking about electronics states of an atom, then these are energy eigenstates of the Hamiltonian describing the electrons and their interaction with the nucleus. If you are talking about the vibrations of a diatomic molecule, then they can be as simplified as the energy eigenstates of the Hamiltonian for a Morse potential, without any consideration for electrons (Born-Oppenheimer approximation).
Basically, you have a Hamiltonian ##\hat{H}## describing some quantum system, so the ##\psi_i## such that
$$
\hat{H} \psi_i = E_i \psi_i
$$
are energy eigenstates. The physics being describe basically boil down to what is in ##\hat{H}##.
If this is still not clear, you will have to give a more explicit context for your question.
Energy eigenstates are the quantized energy levels that an electron can occupy in an atom or molecule. These states are described by the Schrödinger equation and are the basis for understanding the behavior of electrons in a system.
Orbitals are the regions around an atom where an electron is most likely to be found. These regions are determined by the energy eigenstates and can be visualized as the "clouds" of electron density around an atom. Each orbital corresponds to a specific energy eigenstate.
Energy bands refer to the range of energy levels that electrons can occupy in a solid material. These bands are created by the overlapping of energy eigenstates from multiple atoms, and play a crucial role in determining the electrical and optical properties of materials.
While both energy eigenstates and orbitals describe the behavior of electrons, they differ in their mathematical formulations and physical interpretations. Energy eigenstates are solutions to the Schrödinger equation and represent the quantized energy levels of electrons, while orbitals are probability distributions that describe the likelihood of finding an electron in a specific region around an atom.
Degeneracy refers to the situation where multiple energy eigenstates have the same energy level. This can occur when there are symmetries in a system that result in different quantum states having the same energy. Degeneracy has important implications in quantum mechanics and can affect the behavior of electrons in a system.