and Atoms
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 τmax = 2π/ω for such an object about that axis is 2πI/hbar . Hence a spinning person can rotate about a vertical axis no less than once every 6×1034 seconds, a spinning virus several hundred Angstroms on a side can rotate no less than once every second, and a spinning O2 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 ΨA[x] and ΨB[x] from above therefore combine to make the anti-symmetric 2-particle wavefunction Ψ2[x1,x2] = (ΨA[x1]ΨB[x2] - ΨB[x1]ΨA[x2])/√[2]. Moreover if state A and state B are the same state, subscripts in the foregoing expression become identical and one finds that Ψ2[x1,x2] is zero everywhere! In other words, sharing states between identical fermions is not a choice, and quantum mechanics, if anything, is about choices.
{Thanks. :) /pf}