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Chapter7_15 - PHY3063 R D Field Bosons and Fermions in a...

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PHY3063 R. D. Field Department of Physics Chapter7_15.doc University of Florida Two Fermion Ground State Probability Density 0 0.5 1 x 2 /L x 1 = 0.5 L Fermi-Dirac Bosons and Fermions in a Box Two Particles in a Box: For the problem of two particles in a one dimensional box (0 L) we have ) 2 /( ) ( 2 2 2 2 2 mL E β α π αβ + = h and ( ) ) / ( sin ) / ( sin ) / ( sin ) / ( sin 2 ) , ( 2 2 1 2 2 2 1 2 2 2 1 L x L x L x L x L x x classical απ βπ βπ απ ρ αβ + = ( ) ) / sin( ) / sin( ) / sin( ) / sin( 4 ) , ( 2 2 1 1 2 2 1 int L x L x L x L x L x x βπ απ βπ απ ρ αβ = Note that 1 ) , ( 0 0 2 1 2 1 = ∫ ∫ L L classical dx dx x x αβ ρ and αβ αβ δ ρ = ∫ ∫ L L dx dx x x 0 0 2 1 2 1 int ) , ( . Ground State for Two Fermions: The lowest energy state for two fermions is α = 1 and β = 2 (or α = 2 and β = 1 ) and FD E mL E 0 2 2 2 12 ) 2 /( 5 = = π h and ( ) ) / ( sin ) / 2 ( sin ) / 2 ( sin ) / ( sin 2 ) , ( 2 2 1 2 2 2 1 2 2 2 1 12 L x L x L x L x L x x classical π π π π ρ + = ( ) ) / 2 sin( ) / sin( ) / 2 sin( ) / sin( 4 ) , ( 2 2 1 1 2 2 1 int 12 L x L x L x L x L x x π π π π ρ = ) , ( ) , ( ) , ( 2 1 int 12 2 1 12 2 1 12 x x x x x x classical FD ρ ρ ρ = . Put one particle at x 1 = 0.5L and plot the probability of
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