Understanding Energy Eigenstates: Orbitals and Energy Bands Explained

[lucas_]

Does Energy Eigenstates refer to each orbital from the ground state or within one orbital in terms of the kinetic and potential energy of the electron in the orbital or from energy bands of molecular system? What is the term for each case called?

 

[Jilang]

That would depend on what the state was describing. The electron will have a state and the molecule will have a different one. I guess they would be called the state of the electron and the state of the molecule.

 

[DrClaude]

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.

 

[lucas_]

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.

Can you give an example of the most common applications for this. I'd like to know to if the state of the electron or the state of the molecule is the most common use.

 

[DrClaude]

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.

 

[lucas_]

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.

In environmental decoherence.. in the selection of preferred basis in energy eigenstates (when position basis is not chosen or in addition to).. does it usually choose the energy eigenstates of the Hamiltonian for a Morse potential or the Hamiltonian describing the electrons and their interaction with the nucleus?