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Unformatted text preview: PHY3063 R. D. Field Spin and Statistics
Mixed State: We see that the probability density for finding one particle at
x1 and the other at x2 is ( ∗ ) mix
ρ αβ (θ , x1 , x 2 ) = cos 2 θ | ψ αβ |2 + sin 2 θ | ψ αβ |2 + sin( 2θ ) Re ψ αβψ αβ . where θ is an arbitrary angle.
Classical Physics: Classical physics corresponds to θ = 45o:
( x1 , x 2 ) = 1 | ψ αβ ( x1 , x 2 ) | 2 + | ψ αβ ( x1 , x 2 ) | 2
2 = (|ψ 1
2 ( αβ ( x1 , x 2 ) | 2 + | ψ βα ( x1 , x 2 ) | 2 ) ) Law of Nature (consequence of CPT invariance): Nature selects the
angle θ for us!
Fermions: Fermions are particles with intrinsic spin angular momentum
equal to 1 , 3 , 5 ,L (i.e. half-integral spin) and identical fermions are
antisymmetric under 1↔2 (i.e. θ = 0 called “Fermi-Dirac statistics”) and
ρ αβ ( x1 , x 2 ) =| ψ αβ ( x1 , x 2 ) | 2 = ρ αβ
( x1 , x 2 ) − ρ αβ ( x1 , x 2 ) ,
ρ αβ ( x1 , x 2 ) = Re ψ αβ ( x1 , x2 )ψ βα ( x1 , x2 ) .
Note that two identical fermions cannot occupy the same state since
ρ αα ( x1 , x 2 ) =| ψ αα ( x1 , x 2 ) | 2 = 0 . ( ) Bosons: Bosons are particles with intrinsic spin angular momentum equal to
0, 1, 2, ... (i.e. integral spin) and identical bosons are symmetric under 1↔2
(i.e. θ = 90o called “Bose-Einstein statistics”) and
( x1 , x 2 ) + ρ αβ ( x1 , x 2 ) (α ≠ β)
ρ αβ ( x1 , x 2 ) =| ψ αβ ( x1 , x2 ) |2 = ρ αβ
( x1 , x 2 ) .
ρ αα ( x1 , x 2 ) =| ψ αα ( x1 , x2 ) |2 = ρ αα Two Particles in a Box: For the problem of two particles in a one
dimensional box (0 ↔ L) we have ( 2
sin 2 (απx1 / L) sin 2 ( βπx2 / L) + sin 2 ( βπx1 / L) sin 2 (απx2 / L)
ραβ ( x1 , x2 ) = 2 (sin(απx1 / L) sin( βπx1 / L) sin(απx2 / L) sin( βπx2 / L) )
( x1 , x2 ) = ) and
Eαβ = Department of Physics h 2π 2 (α 2 + β 2 )
2 mL2 Chapter7_14.doc University of Florida ...
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This note was uploaded on 05/29/2011 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.
- Spring '07