Chapter6_10 - PHY3063 R D Field The Infinite Square Well(4...

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PHY3063 R. D. Field Department of Physics Chapter6_10.doc University of Florida The Infinite Square Well (4) Particle in a One-Dimensional Box: Consider the solution of ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V dx x d m ψ = + h , where h / ) ( ) , ( iEt e x t x = Ψ , for the case V(x) = if x -L/2 and V(x) = if x L/2 and V(x) = 0 for -L/2 < x < L/2 (Note that now V(-x) = V(x) ). As before we have 0 ) ( = x for x -L/2 and x L/2 For -L/2 < x < L/2 we have ) ( ) ( 2 ) ( 2 2 2 2 x k x mE dx x d = = h where 2 2 h mE k = . The most general solution is ) sin( ' ) cos( ' ) ( kx B kx A Be Ae x ikx ikx + = + = where A' and B' are constants. Boundary Conditions: We require that ψ (x) be “square-integrable” and that it be continuous and “single valued”. Thus at x = L/2 (1) 0 ) 2 / sin( ' ) 2 / cos( ' ) 2 / ( = + = = kL B kL A L x . At x = -L/2 we have (2) 0 ) 2 / sin( ' ) 2 / cos( ' ) 2 / ( =
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