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Chapter7_16

# Chapter7_16 - PHY3063 R D Field Exchange Forces(1 Particle...

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PHY3063 R. D. Field Department of Physics Chapter7_16.doc University of Florida Exchange Forces (1) Particle Separation: Consider the expectation value of the distance between the two particles squared: > < > < + > >=< < 2 1 2 1 2 1 2 2 1 2 ) ( x x x x x x . Two Particles in a Box (distinguishable): If the two particles are distinguishable then ) ( ) ( ) , ( ) , ( 2 1 2 1 2 1 x x x x x x D β α αβ αβ ψ ψ ψ ψ = = α β α ψ ψ > =< >= < 2 2 2 2 1 2 1 2 1 2 1 | ) ( | | ) ( | x dx x dx x x x β β α ψ ψ > =< >= < 2 2 2 2 2 2 1 2 1 2 2 | ) ( | | ) ( | x dx x x dx x x β α β α ψ ψ > < > =< >= < x x dx x x dx x x x x 2 2 2 2 1 2 1 1 2 1 | ) ( | | ) ( | and β α β α > < > < > < + > =< > < x x x x x x D 2 ) ( 2 2 2 2 1 Two Particles in a Box (classical indistinguishable): Classically if the two particles are indistinguishable then ( ) 2 2 1 2 2 1 2 1 2 1 | ) , ( | | ) , ( | ) , ( x x x x x x classical βα αβ αβ ψ ψ ρ + = and we get the same as the distinguishable case. Namely, β α β α > < > < > < + > =< > < x x x x x x classical 2 ) ( 2 2 2 2 1 Two Bosons in a Box: ) , ( ) , ( | ) , ( | ) , ( 2 1 int 2 1 2 2 1 2 1 x x x x x x x x classical S BE αβ αβ αβ αβ ρ ρ ψ ρ + = = ( α β ) where ( ) ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( Re ) , ( 2 1 * 2 1 2 1 2 1 * 2 1 2 1 2 1 * 2 1 2 1 int x x x x x x x x x x x x x x αβ
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