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Antisymmetry principle (Read 4295 times)
Gerrit-Jan Linker
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Antisymmetry principle
01.03.08 at 11:47:42
 
Antisymmetry principle
 
The basis symmetry law in Quantum Physics:
If a physically measurable property depends on the coordinates (both spatial coordinates and spin: 4 degrees of freedom therefore) of identical particles, the emasurement result of this property must be independent of any attempt of labelling the particles. This involves that the property must be symmetrical to particle exchange.
 
Consider a system with 2 electrons.
 
In systems containing identical particles (electrons in our case) these particles must be considered indistinguishable. The probability of finding an electron somewhere must then be invariant to the exchange of the 2 electrons.
 
Let us use the vector x defined as x Ξ (r , σ) where r is the position vector and σ is the electron spin (possible values +1/2 and -1/2)
 
xi(j) indicates the electron j is at xi
ψ : state function describing the 2 electrons
 
The probability of finding electron 1 at x1 and electron 2 at x2 is:
 
| ψ( x1(1) , x2(2) ) |2 dx1dx2
 
The 2 electrons are indistinguishable so:
| ψ( x1(1) , x2(2) ) |2 dx1dx2 = | ψ( x1(2) , x2(1) ) |2 dx1dx2
 
It follows that:
| ψ( x1(1) , x2(2) ) |2 = | ψ( x1(2) , x2(1) ) |2  
 
This can be satisfied in two ways:
 
ψ( x1(1) , x2(2) ) = ψ( x1(2) , x2(1) ), namely that the wavefunction is symmetrical with regards to particle exchange
 
or
 
ψ( x1(1) , x2(2) ) = - ψ( x1(2) , x2(1) ), namely that the wavefunction is anti symmetrical with regards to particle interchange
 
There are 2 postulates with regards to the behavior of particles in relation to exchange:
1. All fundamental particles are described by means of functions belonging to one or another class.
2. THe elemental particles cannot change from one class to another.
 
From these 2 postulates the antisymmetry principle is stated:
All particles belonging to the class of half integral spin quantum numbers are described by antisymmetrical functions.  
All particles belonging to the class of integral spin quantum numbers are described by symmetrical functions.
 
Fermions:
Particles with half integral spin quantum numbers are named fermion and they obey the Fermi-Dirac statistics.
 
Bosons:
Particles with integral spin quantum numbers are named bosons and bosons obeythe Bose-Einstein statistics.
 
Fermions repel eachother more than classical particles. Bosons repel less than classical particles.
 
See also:
Probability density
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« Last Edit: 17.05.08 at 19:59:59 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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