This is a note for a modern physics class
on why particles with half-integral spin obey the Pauli exclusion
principle (i.e. no two can occupy the same quantum state). As a
result, atoms have electronic shells which give them very
different properties for differing atomic numbers. Hence chemistry
as we know it exists, and life as we know it is possible.
Other consequences: The electrons in metals have a *wide*
range of kinetic energies, atomic nuclei don't grow without bound,
and sufficiently light-weight white dwarf and neutron stars don't
collapse no matter how much they cool.

This means that the maximum period of rotation
τ_{max} = 2π/ω for such an object about
that axis is 2π*I*/*hbar* . Hence a spinning
person can rotate about a vertical axis no less than once
every 6×10^{34} seconds, a spinning virus several
hundred Angstroms on a side *can rotate no less than* once
every second, and a spinning O_{2} molecule can rotate
no less than once every 5×10^{-12} seconds.
As you can see from these examples, the effects of this
quantization belong more to the physics of the nano-world,
than to the microscopic or macroscopic ones.

**Part I - Fermion wavefunctions anti-symmetric under 360 degree
rotation AKA "2 turns are better than 1".**

In addition to orbital angular-momentum, elementary particles have
intrinsic (a.k.a. spin) angular momenta. Some of these (like photons)
can take on integer *hbar* values for spin, while others
(like electrons, protons and neutrons) can have only half-integral
*hbar* values. The wierd thing about the half-integral spin
particles (also known as fermions) is that when you rotate one of
them by 360 degrees, it's wavefunction changes sign. For
integral spin particles (also known as bosons), the wavefunction
is unchanged.

The mathematical origins for this property were discovered in the early
part of this century, and are often derived by solving an eigenvalue
problem with Pauli spin matrices (cf. Shiff, *Quantum Mechanics*,
McGraw-Hill 1968 p. 205). One finds that the 360 degree rotation
operator multiplies a wavefunction by
Exp[*i*×2π×*spin*], which is -1 if
*spin* is half-integral. However, reasons to suspect this
might be the case were already in the hands of Balinese candle
dancers, who for centuries have known that 360 degree rotations are incomplete when it comes to your connection to the outside world. You can convince yourself of this by trying to rotate your hand palm-side up by 360 degrees. A second 360 degree rotation in the same direction is needed to undo the arm twist that results from the first. The drawing below illustrates the effect as well. Note that three strings are needed to make it rigorous.

Half-integral spin particles thus seem to be somehow connected to the world around in such a way that their wavefunction's deBroglie phase is inverted after a 360 degree rotation, as in the diagram above. (You might want to ask a string theorist if this connection to the external world can be seen as involving one or more wrapped-up spatial dimensions.) Quantum mechanics confirms this connection by associating with these particles half-integral "intrinsic" spin angular-momenta. Fortunately, this particular wierd thing is not true for extended spinning objects, like us. Otherwise, we might have to count the number of turns during a dance, to make sure the number is even at the end of the night!

**Part II - Exchanging identical particles is the same as rotating one only by 360 degrees.**

This is a simple fact of topology. The drawing below illustrates the equivalence, which as you can see is true only if A and B are indistinguishable.

**Part III - The 2-particle wavefunction for identical fermions is
anti-symmetric under particle exchange.**

Go figure. If the two particles are independent, the 2-particle wave-function is the product of two one-particle wavefunctions. From parts I and II above, exchanging identical fermions is the same as multiplying one of the two factors by -1. Then the whole shebang might as well be multiplied by -1 instead!

**Part IV - Two-particle wavefunctions don't exist for identical fermions in the same state.**

A pair of identical single-particle wavefunctions
Ψ_{A}[**x**] and
Ψ_{B}[**x**] from above therefore combine to make
the anti-symmetric 2-particle wavefunction
Ψ_{2}[**x**_{1},**x**_{2}] =
(Ψ_{A}[**x**_{1}]Ψ_{B}[**x _{2}**] - Ψ

Note: Send comments and questions to philf@newton.umsl.edu. This page contains original material. Hence if you echo, in print or on the web, a citation would be cool.

`{Thanks. :) /pf}`