What is degeneracy as opposed to a degenerate state? How can we know what orbitals are higher in energy?

Degeneracy and a degenerate state are related concepts in quantum mechanics, particularly in the context of atomic and molecular physics.

Degeneracy:

Degeneracy refers to the situation where two or more quantum states have the same energy. In other words, if multiple quantum states have the same energy level, they are said to be degenerate.

For example, in the context of electrons in an atom, when multiple electrons occupy the same energy level (e.g., the same shell and subshell), they are in degenerate states because they have the same energy. This concept arises from the quantization of energy levels in quantum systems.

Degenerate State:

A degenerate state is one of the individual quantum states that is part of a degenerate set. In other words, it's one of the distinct quantum states within a group of states with the same energy.

In the context of electron orbitals, a degenerate state could refer to the different electron spin states within a given orbital. For example, in a p orbital, there are three degenerate states corresponding to the three different spatial orientations of the orbital (px, py, pz). These states have the same energy but differ in their spatial distribution.

Regarding your question about knowing which orbitals are higher in energy:

In multi-electron systems like atoms and molecules, it's essential to understand the relative energies of different electron orbitals. Here's how you can determine which orbitals are higher in energy:

Principal Quantum Number (n): In general, orbitals with higher principal quantum numbers (n) are higher in energy. For example, 2p orbitals are higher in energy than 1s orbitals.

Azimuthal Quantum Number (l): For a given principal quantum number (n), the azimuthal quantum number (l) determines the subshell and, to some extent, the energy within that shell. Orbitals with higher values of l within the same shell have higher energy. For example, in the n=2 shell, 2p orbitals are higher in energy than 2s orbitals because l=1 for p orbitals and l=0 for s orbitals.

Effective Nuclear Charge (Zeff): The effective nuclear charge experienced by an electron in an orbital also influences its energy. Orbitals that experience a higher effective nuclear charge are lower in energy. This is due to the increased attraction between the electron and the nucleus, stabilizing the orbital.

Electron-Electron Repulsion: Electrons in the same shell experience electron-electron repulsion, which can cause deviations from a purely "increasing energy with n and l" pattern. For example, 2s and 2p orbitals do not have the expected energy order due to electron-electron repulsion.

In summary, the energy of orbitals in multi-electron systems is influenced by several factors, including the principal quantum number (n), azimuthal quantum number (l), effective nuclear charge (Zeff), and electron-electron repulsions. These factors collectively determine the relative energies of the orbitals within an atom or molecule.