Candle Dances [2pi-rotated electron] 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.
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Other resources of possible interest:
*The newsgroup post by Joshua Burton that inspired this page can be found
here.
*For more on spins & Pauli exclusion, see John Baez' notes at
http://math.ucr.edu.
*Try focussing a high-res electron microscope image on-line!
*Consider the view a la deBroglie of a microscope's electrons.
*Pick a map-frame in relativity to get the most from Newton & Galileo.
*Animate electron and photon wave packets. What's different?
*Figure traveler time on a "1-gee" trip to Andromeda galaxy.
*Browser-interactively solve your own constant-acceleration problems.
*Does making a hotdog require 50 nanoseconds of life's power stream?
*Is statistical physics dead, or is there a paradigm change afoot?
*At UM-StLouis see also: cme, i-fzx, phys&astr, programs, stei-lab, &
wuzzlers.
*Some current and previous courses: p111, p112, p231, p341, p400.
*Cite/Link: http://newton.umsl.edu/~philf/candles.html
*This release dated 12 Nov 1996 (Copyright by Phil Fraundorf 1988-1996)
*Since 5 Dec 1996, you are visitor number [Image].
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Preface
One must begin with the understanding that rotation (more specifically
extended-object or orbital angular momentum) components are quantized in
units of hbar (Planck's constant / 2 Pi = 1.05x10^-34 joule seconds). Modern
physics students might convince themselves of this by calculating the energy
eigenvalues for a bead on a loop (i.e. a particle in a periodic box, cf.
Garrod, 20th Century Physics, Faculty Publishing 1984 p.138). As a result,
the minimum projected angular velocity Omega of something with moment of
inertia I obeys hbar = I Omega.
This means that the maximum period of rotation Tmax = 2Pi / Omega for such
an object about that axis is 2Pi 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 O2 molecule can rotate no less than once
every 5x10^-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.
The Four-Part Argument for Pauli Exclusion
Adapted from USENET postings and Feynman Quotes/pf961108
Part I - Fermion wavefunctions are anti-symmetric under 360 degree rotation.
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*2Pi*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.
[360 degrees inverts, 720 degrees doesn't]
Half-integral spin particles 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. Quantum mechanics then
confirms this connection by associating with them 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.
[Exchanging two is the same a rotating one by 360 degrees]
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.
Two single-particle wavefunctions PsiA[x] and PsiB[x] combine to make the
anti-symmetric 2-particle wavefunction Psi2[x1,x2] = (1/Sqrt[2])
(PsiA[x1]*PsiB[x2] - PsiB[x1]*PsiA[x2]). If state A and state B are the
same, this means that Psi2[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.
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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}