The intrinsic spin of fundamental particles controls their statistics, in a way that makes matter possible. Spin-1/2 particles, fermions, obey the Pauli Exclusion Principle. No two identical fermions in the same region of space can have the same set of quantum numbers. Spin-1 particles, bosons, can all occupy precisely the same quantum state in the same region of space. To see how this works, for simplicity consider two states α and β and two identical particles, labelled 1 and 2. Remember the rule that when two situations cannot be distinguished the amplitudes must be combined. We cannot actually tell which particle is 1 and which is 2. So if we put a particle in state α and the other in state β, we have to write ψ_α(1)ψ_β(2) ± ψ_α(2)ψ_β(1), for example. Both nature and theory agree that the minus sign is always associated with fermions, and the plus sign with bosons. Thus state functions for identical fermions are always antisymmetric under exchange of particle labels, while state functions for identical bosons are always symmetric. Now consider the case where both particles are identical fermions, and both are in state α. We see at once that the overall state ψ_α(1,2) = 0.

The Slater Determinant form guarantees a fully antisymmetric multi-fermion state.

It is a characteristic of systems of many identical fermions that they tend to be well-approximated by an individual particle model. That is, no matter how many particles are in the system, each tends to behave, to a good approximation, as if it were a single particle in a potential... the potential being generated by all the other fermions. This happens, basically, because the Pauli Principle greatly limits what the interaction of any two identical particles can result in. If all states around them are filled, they cannot scatter into other states unless there is energy available to reach empty states far up in excitation. Thus, the interaction tends to leave both particles in the same state, or have them exchange states, which has no physical consequence since they are identical. The protons and neutrons in the atomic nucleus form two separate sets of identical fermions, and also are well described by an individual particle model, despite the high nuclear density and the strong force acting between particles.  As shown in the text, the number of different sets of quantum numbers available in an atomic energy level with quantum number n is 2n². [Remember that for a given n, the possible ℓ values range from 0 to n − 1.] The states of the same n form a so-called shell, while the states of a given ℓ form a so-called subshell. The shells have names, in order K, L, M, etc.  A single electron outside a closed shell has a remarkably hydrogen-like spectrum, although of course the extreme degeneracy seen in the hydrogen spectrum is broken by the more complex interaction with the core charge distribution. It is vital to remember that quantum numbers like m_ℓ and m_s always change in steps of 1. Also, since photons carry an intrinsic angular momentum of 1, when an electron in an energy level of value ℓ transitions to a lower level by emission of a photon, generally speaking, Δℓ = -1.

Warning, each circle here represents 2 electrons!

Where two or more electrons are outside a closed shell, we have to consider coupling between the electron states. We can have coupling of the two or more spins, in other words a total spin state labeled by quantum numbers resulting from S = S₁ + S₂, or coupling of the angular momenta, in other words a total angular momentum state labeled by quantum numbers resulting from L = L₁ + L₂, and also the need for including these expanded operators in a spin-orbit coupling term, S·L. The parity or symmetry of angular momentum eigenstates depends on ℓ and is given by (-1)^ℓ. The intrinsic spin follows a different rule, because for example for a pair coupling to spin 1, exchanging the two electrons does not affect the overall spin state, whereas for s = 0 it does.   If we, as is conventional, factor the overall state into a product of the space state and the spin state, the overall sign change looks like (-1)^{ℓ + s + 1}.  For atoms with larger and larger Z, the situation shifts from “LS” coupling to so-called “JJ” coupling.  See the text for a bit more detail.  The choice is between adding angular momenta and spin together separately, and then coupling the two,  or instead adding angular momenta and spin together individually as j = ℓ + s, and coupling the result. It is vital to remember that quantum numbers such as m_ℓ, m_j and m_s always change in steps of 1. A very detailed presentation of aspects of the quantum description of atoms and molecules is found in the undergraduate physics course 362K.

A tremendous theoretical advance that permits one to find solutions to the Schrödinger equation for any number of interacting identical fermions is the Hartree-Fock or Self-Consistent Field approach. This approach is used on an everyday basis in atomic and molecular physics, as well as condensed-matter physics, and even applied with great success to the atomic nucleus itself.  Realistic calculations require a very large computer and a large amount of time spent on the calculation!

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