How Do Basis Vectors Represent Orbitals in Quantum Chemistry?

[Niles]

Homework Statement

Hi all.

This post is about quantum chemistry, but my question arises when looking at the problem from a physical point of view.

The Schrödinger equation gives us the stationary states of a system, and let's say that we are looking at a system with two stationary states (Dirac notation - but the LaTeX does not work, so bear with me) |1> and |2> with an associated eigenenergy. These two orthonormal states span the Hilbert space we are working in.Now here's my question: I am looking at a figure of a molecule with six orbitals, and now each orbital is represented by an orthonormal basis |1>, |2>, |3>, |4>, ..., |6>. An eigenstate is then a linear combination of these basis-vectors (orbitals) with an associated energy.

Question: How am I do interpret these basis-vectors |1>, |2>, |3>, |4>, ..., |6>? They surely cannot represent stationary states (i.e. solutions to the time-independent Schrödinger equation), because then a linear combination of them would not have an eigenenergy.

Thanks in advance. Any help will be greatly appreciated, since I cannot get help from anywhere else at the moment.Niles.

 

[Niles]

Can I get a moderator to move this thread to the "Advanced Physics Homework Help"? I think it belongs there more than in this section.

Thanks in advance.

 

[Blueshift5]

I would like to clarify a few things about quantum chemistry and physics. The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system. It gives us the stationary states of a system, which are the solutions to the time-independent Schrödinger equation. These states are represented by orthonormal vectors in a Hilbert space, as mentioned in the post.

In the context of quantum chemistry, these stationary states represent the energy levels of electrons in an atom or molecule. Each state has an associated energy, and a linear combination of these states can also have a specific energy value, known as an eigenenergy. This is a key concept in quantum mechanics, as it allows us to describe the behavior of a system in terms of the probabilities of different energy states.

Now, to address the question about the basis vectors |1>, |2>, |3>, |4>, ..., |6>, these are not the same as the stationary states mentioned earlier. These basis vectors represent the different orbitals of the molecule, which are regions of space where the electrons are likely to be found. These orbitals are also orthonormal, meaning they are perpendicular to each other and can be used to describe any possible state of the system.

In summary, the basis vectors in quantum chemistry do not represent stationary states, but rather the orbitals of a molecule. They are used to describe the behavior of electrons in a molecule and are essential in understanding the electronic structure and properties of molecules. I hope this helps clarify any confusion and provides a better understanding of quantum chemistry and physics.