Two \(p\) electrons \((l=1)\) can have \(L=0,1\), or 2 and \(S=0\) or 1 in the RussellSaunders case. Are all combinations of \(L\) and \(S\) permitted if the \(n\) values of the two electrons are different? Are thcy all permitted if the \(n\) values are the same?

Russell-Saunders coupling, also known as LS coupling, is a method used in quantum mechanics to determine the energy levels of atoms. It considers both the total orbital angular momentum (L) and the total spin angular momentum (S) of electrons in a multi-electron atom. This method is particularly useful for light atoms where the spin-orbit interaction is relatively weak.

In the context of two p-electrons, Russell-Saunders coupling helps us combine their individual orbital and spin angular momenta to form the total angular momenta (L and S). The possible values of L for two p-electrons ( l=1) are 0, 1, and 2, while the possible values of S are 0 and 1.

When n values of two electrons differ, they occupy different orbitals. Hence all combinations of L and S can exist. However, when n values are the same, they are in the same shell and subshell, restricting some combinations due to the Pauli exclusion principle.

The Pauli exclusion principle is a fundamental rule in quantum mechanics stating that no two electrons in an atom can have the same set of four quantum numbers. These quantum numbers are:

• n - the principal quantum number
• l - the angular momentum quantum number
• m - the magnetic quantum number
• s - the spin quantum number

This principle has critical implications for the arrangement of electrons in atoms. For the case of same n values, we must apply the Pauli exclusion principle. Let's investigate:

If both electrons have identical quantum numbers (n, l, m, s) under certain configurations, it violates this principle. Specifically, if L=0 and S=0, both electrons would be in the same state, which isn't permissible. Hence, combinations where the total orbital momentum (L) is 0 and the total spin (S) is 0 are not allowed. All other combinations of L and S (like L=1, S=0; L=2, S=0; L=0, S=1; L=1, S=1; L=2, S=1) are permissible as they do not result in identical quantum numbers for both electrons.

Quantum numbers are values that describe the unique quantum state of an electron in an atom. There are four quantum numbers:

• Principal quantum number () - indicates the energy level and size of the orbital
• Angular momentum quantum number (l) - describes the shape of the orbital
• Magnetic quantum number ( m_l) - specifies the orientation of the orbital
• Spin quantum number ( m_s) - indicates the direction of the electron's spin

For two p-electrons with l=1, the possible values of total orbital angular momentum (L) are 0, 1, or 2. The total spin (S) can be 0 or 1. These quantum numbers help us determine the possible states and their respective energies.

When two electrons have different principal quantum numbers (), they are in different orbitals, allowing for all combinations of L and S. However, when they share the same n value, the Pauli exclusion principle restricts certain combinations like L=0 and S=0 since it would result in identical quantum numbers. This interplay between quantum numbers ensures that the electrons in an atom are uniquely defined and follow the rules of quantum mechanics.