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Chapter5_6 - PHY4604 R D Field Exchange Forces(2 Two Bosons...

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PHY4604 R. D. Field Department of Physics Chapter5_6.doc University of Florida Exchange Forces (2) Two Bosons in a Box (continued): αβ α β α α β α β β α β α α β α β β α αβ βα βα αβ αβ δ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ρ > =< + = + = + = >= < 2 2 2 * 2 2 2 1 1 * 1 2 1 2 2 * 2 2 2 1 1 * 1 2 1 2 1 2 * 1 * 2 1 2 2 2 1 2 1 2 * 1 * 2 1 2 2 2 1 2 1 2 1 * 2 1 2 2 2 1 2 1 2 1 * 2 1 2 2 2 1 2 1 2 1 int 2 2 2 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , ( ) , ( ) , ( ) , ( ) , ( x dx x x x dx x x dx x x x dx x x dx dx x x x x x dx dx x x x x x dx dx x x x x x dx dx x x x x x dx dx x x x x 2 2 1 2 1 2 2 * 2 2 1 1 * 1 1 2 1 2 2 * 2 2 1 1 * 1 1 2 1 2 1 2 * 1 * 2 1 2 1 2 1 2 1 2 * 1 * 2 1 2 1 2 1 2 1 2 1 * 2 1 2 1 2 1 2 1 2 1 * 2 1 2 1 2 1 2 1 2 1 int 2 1 2 1 | | ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , ( ) , ( ) , ( ) , ( ) , ( αβ βα αβ βα αβ αβ βα β α α β α β β α β α α β α β β α αβ βα βα αβ αβ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ρ > < = > < > =< > < > < + > < > < = + = + = + = >= < x x x x x x x dx x x x dx x x x dx x x x dx x x x dx dx x x x x x x dx dx x x x x x x dx dx x x x x x x dx dx x x x x x x dx dx x x x x x x Thus, 2 2 int 2 1 int 2 2 int 2 1 int 2 2 1 | | 2 2 2 ) ( αβ αβ α δ > < > < = > < > < + > =< > < x x x x x x x x where > < dx x x x x ) ( ) ( * β α αβ ψ ψ . For α β then we see that, 2 2 2 1 2 2 1 | | 2 ) ( ) ( αβ αβ αβ > < > =< > < x x x x x classical BE For α = β then we see that classical BE x x x x αα αα > =< > < 2 2 1 2 2 1 ) ( ) ( Two Fermions in a Box: For α β then we see that 2 2 2 1 2 2 1 | | 2
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