Chapter5_1 - PHY4604 R. D. Field Two Particles in a Box Two...

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PHY4604 R. D. Field Department of Physics Chapter5_1.doc University of Florida Two Particles in a Box One Particle in a One-Dimensional Box: For one particle in a box ( V(x) = x 0, V(x) = x L and V(x) = 0 0 < x < L ) we found h / ) ( ) , ( t iE n n n e x t x = Ψ ψ with ) / sin( 2 ) ( L x n L x n π = . The energy levels are given by 0 2 2 2 2 2 2 E n mL n E n = = h where 2 2 2 0 2 mL E h = . Two Particles in a One-Dimensional Box: For two ( non-interacting ) particles we look for a solution of the form h h / 2 1 / 2 1 2 1 ) ( ) ( ) , ( ) , , ( iEt iEt e x x e x x t x x = = Ψ with E m p m p x x = + 2 ) ( 2 ) ( 2 2 2 1 . Thus, ) , ( ) , ( 2 ) , ( 2 2 1 2 2 2 1 2 2 2 1 2 1 2 2 x x E dx x x d m dx x x d m = h h ) ( ) ( ) ( ) ( 2 ) ( ) ( 2 2 1 2 2 2 2 1 2 2 1 1 2 2 2 x x E dx x d x m dx x d x m = h h E dx x d x m dx x d x m = 2 2 2 2 2 2 2 1 1 2 1 2 ) ( ) ( 1 2 ) ( ) ( 1 2 h h Hence, E = E 1 + E 2 and ) ( ) ( 2 1 1 2 1 1 2 2 x E dx x d m = h ) ( ) ( 2 2 2 2 2 2 2 2 x E dx x d m = h The total energy is therefore given by 2 2 2 2 2 2 1 2 ) ( ) ( ) ( mL E E E β α αβ + = + = h 2 2 2 2 1 2 ) ( mL E h = 2 2 2 2 2
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