Chapter7_14 - PHY3063 R. D. Field Spin and Statistics Mixed...

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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 A S A S 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: classical A S ρ αβ ( 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 222 antisymmetric under 1↔2 (i.e. θ = 0 called “Fermi-Dirac statistics”) and int FD A classical ρ αβ ( x1 , x 2 ) =| ψ αβ ( x1 , x 2 ) | 2 = ρ αβ ( x1 , x 2 ) − ρ αβ ( x1 , x 2 ) , Pauli Exclusion Principle! where int * 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 FD A 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 int BE S classical ( x1 , x 2 ) + ρ αβ ( x1 , x 2 ) (α ≠ β) ρ αβ ( x1 , x 2 ) =| ψ αβ ( x1 , x2 ) |2 = ρ αβ BE S classical ( 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) 2 L 4 int ραβ ( x1 , x2 ) = 2 (sin(απx1 / L) sin( βπx1 / L) sin(απx2 / L) sin( βπx2 / L) ) L classical ραβ ( 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.

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