Questions about Hund's Rules and L-S Coupling

[secret2]

I would like to ask two questions about Hund's rules and L-S coupling:

1. Some textbooks state that when doing L-S coupling and applying Hund's rules, "The maximum values of S and L are subject to the condition that no two electrons may have the same pair of values for m(sub s) and m(sub l). I know this is because of the Pauli exclusion principle, but how does this requirement (m(sub s) and m(sub l)) really limit S and L when we are adding the angular momenta?

2. When we are trying to figure out the ground state of Sm (4f)6, why is it wrong to have L = Sum(l) = 6*3?

Finally, I've realized that in discussing Hund's rules and L-S coupling some texts tend to make explanations using symmetry consideration and the others tend to prefer the exclusion principle. Are they two different sets of explanations, or are they equivalent?

 

[member 553083]

1. The requirement that no two electrons may have the same pair of values for m(sub s) and m(sub l) serves to limit the maximum values of S and L by ensuring that there are enough different possible quantum states for each electron. This is because, due to the Pauli exclusion principle, no two electrons can occupy the same quantum state. So, if two electrons have the same pair of values for m(sub s) and m(sub l), they must occupy different quantum states, meaning that more quantum states are needed. This means that the maximum value of S and L must be reduced in order to accommodate all of the electrons. 2. The reason why it is wrong to assume that L = Sum(l) = 6*3 is because the total angular momentum is not just the sum of the individual angular momenta of the electrons. In addition, due to the Pauli exclusion principle, the electrons must be arranged in such a way that each electron occupies a unique quantum state. This means that the total angular momentum must be a combination of the individual angular momenta of the electrons in such a way that there are enough unique quantum states for all of the electrons. Finally, the explanations regarding Hund's rules and L-S coupling using symmetry considerations and the Pauli exclusion principle are equivalent. Both explanations describe the same physical situation, but with different emphases. The symmetry considerations focus on the arrangement of electrons in the different orbitals and how this affects the total angular momentum, while the exclusion principle focuses on the fact that no two electrons can occupy the same quantum state.

 

[wais]

1. The requirement that no two electrons may have the same pair of values for m(sub s) and m(sub l) is a consequence of the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of quantum numbers. In the context of L-S coupling and Hund's rules, this means that when adding the angular momenta of individual electrons, the resulting total values of S and L must be unique for each electron. This is because electrons with the same set of quantum numbers (including m(sub s) and m(sub l)) are considered to be in the same energy level, and thus cannot occupy the same state due to the exclusion principle. This requirement limits the possible values of S and L, ensuring that the resulting electron configuration is energetically favorable and follows the rules of quantum mechanics.

2. In the case of Sm (4f)6, the correct ground state would actually have L = 0. This is because the 4f orbital can hold a maximum of 7 electrons, and the remaining 6 electrons would fill the lower energy 3d orbital. Therefore, the sum of l values for the 6 electrons in the 4f orbital would be 6*3 = 18, which is not possible. This again goes back to the Pauli exclusion principle, as the 4f and 3d orbitals have different sets of quantum numbers and cannot be occupied by the same electrons.

In terms of explanations, both the symmetry considerations and the exclusion principle are valid ways of understanding Hund's rules and L-S coupling. They are not necessarily two different sets of explanations, but rather different perspectives that can be used to explain the same phenomenon. Both approaches rely on the fundamental principles of quantum mechanics to describe the behavior of electrons in atoms.