Chapter2_3 - PHY4604 R. D. Field The Infinite Square Well...

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PHY4604 R. D. Field Department of Physics Chapter2_3.doc University of Florida The Infinite Square Well (1) 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 0 and V(x) = if x L and V(x) = 0 for 0 < x < L . For x 0 and x L we have 0 ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 2 = → = + x x x V E x dx x d x mV V h . For 0 < x < L 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 ikx ikx Be Ae x + = ) ( where A and B are constants. Bounday Conditions: We require that ψ (x) be “square-integrable” and that it be continuous and “single valued”. Thus at x = 0 0 ) 0 ( = + = = B A x and hence ) sin( 2 ) ( ) ( kx iA e e A x ikx ikx = = . At x = L we have 0 ) sin( 2 ) ( = = = kL iA L x which implies that kL = n π with n = 1, 2, 3,. .. Energy Levels: We see that only certain values of k are allowed which means that only the following energies are allowed: 0 2 2
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