## Identical particles

Postulate: Every elementary particle is either a **fermion or a boson**. A state of many identical particles is totally
**antisymmetric** with respect to the
interchange of any two particles if they are fermions, and it is totally
**symmetric** if they are bosons. No two
identical fermions can have exactly the same set of quantum numbers. This
is called the Pauli exclusion principle.

### Atomic Spectra

**LS coupling**

We assume that the non-central part of the electrostatic interaction is much
bigger than the spin-orbit interaction. (This is usually true for light
multi-electron atoms.) The electrostatic interaction leads to a splitting of
the level corresponding to a given electron configuration into a number of
sublevels characterized by different values of the total orbital angular
momentum of the electrons, L, and their total
spin, S. The operator for the electrostatic
interaction commutes with L = l_{1}+l_{2}+l_{2}+... and S
= s_{1}+s_{2}+s_{3}+... . Some of the values of L
and S obtained from the general rules for addition of angular momenta can
correspond to states forbidden by the Pauli principle. Filled shells do
not contribute to the total orbital angular momentum L and the total spin S. To each
allowed term LS belong
(2L+1)(2S+1) states, differing by the values of M_{L} and M_{S}.
The spin-orbit interaction leads to a splitting of the term LS into a number of
components corresponding to different values of the total angular momentum J.
But it does not completely remove the degeneracy. Each J component is
degenerate with a multiplicity of 2J+1.

∑_{J}(2J+1) = (2L+1)(2S+1).

In the LS coupling scheme, a term is designated by ^{2S+1}L_{J}.
2S+1 is called the **multiplicity** of the term.

Which terms corresponding to a given configuration have the lowest
energy?

Hund's Rule (established empirically to find the ground state term)

- The level with the largest multiplicity has the lowest energy.
- For a given multiplicity, the level with the largest value of L has the
lowest energy.
- For less than half-filled shells:
- The component with the smallest value of J has the lowest energy (normal
order).

Examples of less than half-filled shells: np^{2}, nd^{2}

- For more than half-filled shells:
- The component with the largest value of J has the lowest energy
(inverted order).

Examples of more than half-filled shells: np^{4}, nd^{8}

- When the number of electrons is 2l + 1, i.e. when the shell is half filled,
there is no multiplet splitting.