The *Spin-Statistics Theorem* concludes that
the determining feature as to whether a particle is a boson or
a fermion is the *spin* of the particle. Just as the
planet Earth is both spinning on its axis and orbiting the Sun, so
electrons are spinning on their axis while orbiting the nucleus
in an H-atom.
With enough energy the Earth could be made to spin slower or faster
[in fact the Earth's spin is slowing over aeons] or orbit
with more or less angular momentum. While the electron's
orbital angular momentum can change (i.e., various integer
values of *l* are allowed), it always is seen with
exactly the same spin-rate. The only possible change that
can be made to an electron's spin is to change the direction
of the spin orientation: "up" () or "down"
(). Thus
the spin angular momentum of a fundamental particle is an unchanging
characteristic of the particle. The rules
of quantum mechanics guarantee that spin angular momentum
of a particle must be [*s*(*s*+1)]^{½}
, where *s* (hereafter called the "spin
of the particle") is either a whole number (0, 1, 2,...) (
so called "integer spin") or half of a odd whole number (1/2, 3/2, 5/2, ...)
(so called "half integer spin").
Integer spin particles are always bosons; half integer spin particles
are always fermions.

Fermions | |
---|---|

Particle Name | Spin |

quarks | 1/2 |

electron | 1/2 |

proton | 1/2 |

neutron | 1/2 |

3/2 | |

He^{3} nucleus | 1/2 |

O^{17} nucleus | 5/2 |

Bosons | |

photon | 1 |

gluon | 1 |

graviton | 2 |

pion | 0 |

H^{2} nucleus | 1 |

He^{4} nucleus | 0 |

B^{10} nucleus | 3 |

On the previous page we made both symmetric and antisymmetric two electron wavefunctions. However our wavefunctions just included the spatial part of the wavefunctions. The spin part of a two electron wavefunction can be:

symmetric: , ( + ),

antisymmetric: ( - ),

(Notation: the first arrow in a pair refers to electron 1 spin and the second arrow to electron 2 spin.)

Thus in order to made an overall antisymmetric wavefunction we need to pair
one of the *three* symmetric spin wavefunctions with an antisymmetric spatial
wavefunction (hence the name: triplet), or pair the *single* antisymmetric spin
wavefunction with a symmetric spatial wavefunction (hence name: singlet)

Thus the ground state of Helium can be written:

1*s*(**r**_{1})1*s*(**r**_{2})
( - ),

One of the excited states considered on the previous page might be written:

(1*s*(**r**_{1})2*p*(**r**_{2}) -
2*p*(**r**_{1})1*s*(**r**_{2}))

Note that is impossible to create a totally antisymmetric function
of three electron spins. Thus the *Pauli Exclusion Principle*
that at most two electrons can be in the same spatial wavefunction.
(If two electrons share a spatial wavefunction, they must be in the
spin "paired", i.e., singlet, state.)

Thus two electrons can fit in the 1*s* state, six
electrons can fit in the 3 (*m*=-1,0,1) 2*p* states, and
ten electrons can fit in the 5 3*d* states.

One is tempted to connect the characteristic of 2 electrons "filling up"
an orbital, with the naive idea that "you can't put two things in the
same place". However, note that a Uranium atom, with its 92 electrons
"filling" the orbitals with *n*<5, will happily latch on to the negatively
charged muon (µ^{-}: a spin 1/2 particle similar to the electron
but about 200 times more massive) which can occupy its own 1*s* orbital.
However if three muons are attached to Uranium, the third must go in the
2*s* state. Any number of negatively charged pions (a psin 0 particle
a bit more massive than the muon) can enter the 1*s* state, even
though there are two muons and two electrons in the 1*s* state.
That is to say that the Spin-Statistics Theorem says nothing about the
exchange properties of different types of particles, only identical particles.