Chapter2_25 - + + + Thus, ) ( 2 ) ( 2 ) ( ) ( 2 2 L R x L x...

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PHY4604 R. D. Field Department of Physics Chapter2_25.doc University of Florida Delta-Function Potential – Scattered States Consider particles entering from the right traveling to the left with E > 0 encountering a potential of the form ) ( ) ( x x V αδ = . We look for solutions of the time-independent Schrödinger equation: ) ( ) ( ) ( ) ( 2 2 2 2 x E x x dx x d m ψ = + h with h / ) ( ) , ( iEt e x t x = Ψ . For E > 0 in the region x 0 we have ) ( ) ( 2 ) ( 2 2 2 2 x k x mE dx x d = = h with 2 2 h mE k = and m E 2 2 2 κ h = The most general solution is ikx L ikx L L e B e A x + + = ) ( and ikx R ikx R R e B e A x + + = ) ( , but B R = 0 since no particles are entering from the right traveling to the left. Boundary Conditions: Require ψ (x) be continuous and d ψ (x)/dx to be continuous ( except where V is infinite ). Thus, at x = 0 ) 0 ( ) 0 ( R L = which implies that A L +B L = A R . Also notice that for 0 ε we have ) 0 ( 2 ) ( ) ) ( ( 2 ) ( ) ) ( ( 2 ) ( 2 2 2 2 2 α h h h m dx x E x m dx x E x V m dx dx x d = = =
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Unformatted text preview: + + + Thus, ) ( 2 ) ( 2 ) ( ) ( 2 2 L R x L x R m m dx x d dx x d h h = = = + = which implies that R L L R A m ikB ikA ikA 2 2 h = + . Thus, ( ) 1 = ir A B L L and ( ) 1 = ir irA A L R where m k r 2 h = . Reflection and Transmission Probability: 2 2 2 1 1 | | | | r A B j j P L m k L m k L L R + = = = h h r s and 2 2 2 2 1 | | | | r r A A j j P L m k R m k L R T + = = = h h r r It is easy to show that P R + P T = 1 . Note that P R and P T do not depend on the sign of ! x E V(x) = (x) left region right region...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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