Chapter7_16 - PHY3063 R. D. Field Exchange Forces (1)...

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Unformatted text preview: PHY3063 R. D. Field Exchange Forces (1) Particle Separation: Consider the expectation value of the distance between the two particles squared: < ( x1 − x 2 ) 2 >=< x12 > + < x12 > −2 < x1 x 2 > . Two Particles in a Box (distinguishable): If the two particles are distinguishable then D ψ αβ ( x1 , x 2 ) = ψ αβ ( x1 , x 2 ) = ψ α ( x1 )ψ β ( x 2 ) < x12 >= ∫ x12 | ψ α ( x1 ) | 2 dx1 ∫ | ψ β ( x 2 ) | 2 dx 2 =< x 2 > α 2 2 < x 2 >= ∫ | ψ α ( x1 ) | 2 dx1 ∫ x 2 | ψ β ( x 2 ) | 2 dx 2 =< x 2 > β < x1 x 2 >= ∫ x1 | ψ α ( x1 ) | 2 dx1 ∫ x 2 | ψ β ( x 2 ) | 2 dx 2 =< x > α < x > β and < ( x1 − x 2 ) 2 > D =< x 2 > α + < x 2 > β −2 < x > α < x > β Two Particles in a Box (classical indistinguishable): Classically if the two particles are indistinguishable then classical ρ αβ ( x1 , x 2 ) = 1 2 (| ψ αβ ) ( x1 , x 2 ) | 2 + | ψ βα ( x1 , x 2 ) | 2 and we get the same as the distinguishable case. Namely, < ( x1 − x 2 ) 2 > classical =< x 2 > α + < x 2 > β −2 < x > α < x > β Two Bosons in a Box: BE S classical int ρ αβ ( x1 , x 2 ) =| ψ αβ ( x1 , x2 ) |2 = ρ αβ ( x1 , x 2 ) + ρ αβ ( x1 , x 2 ) (α ≠ β) where int * ρ αβ ( x1 , x 2 ) = Re ( αβ ( x1 , x 2 )ψ βα ( x1 , x2 ) ) ψ * * = 1 ψ αβ ( x1 , x 2 )ψ βα ( x1 , x2 ) + 1 ψ βα ( x1 , x 2 )ψ αβ ( x1 , x 2 ) 2 2 int < x12 >= ∫ x12 ρ αβ ( x1 , x 2 ) dx1dx 2 = 1 2 = 1 2 = 1 2 ∫ x ψ αβ ( x , x )ψ βα ( x , x )dx dx + ∫ x ψ βα ( x , x )ψ αβ ( x , x )dx dx ∫ x ψ α ( x )ψ β ( x )ψ β ( x )ψ α ( x )dx dx + ∫ x ψ β ( x )ψ α ( x )ψ α ( x )ψ β ( x )dx dx ∫ x ψ α ( x )ψ β ( x )dx ∫ψ β ( x )ψ α ( x )dx + ∫ x ψ β ( x )ψ α ( x )dx ∫ψ α ( x )ψ β ( x )dx 2 1 * 1 2 1 1 2 1 2 1 1 2 * 2 1 2 2 1 1 2 * 1 2 * 1 1 2 2 2 1 1 2 2 * 1 * 1 2 1 2 2 1 2 1 2 * 1 2 1 * 2 1 2 * 1 1 2 * 1 1 2 2 =< x 2 > α δ αβ Department of Physics Chapter7_16.doc University of Florida 2 ...
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