Lecture 5 - Finite potential well Continuous spectrum...

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Finite potential well Continuous spectrum
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Finite potential well: Continuous spectrum 0 V - -a/2 a/2 V I II III We again consider three different regions: / 2 / 2 / 2 / 2 I x a II a x a III x a < - - < < E
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Scattering wave functions In the case of E > 0 wave functions do not go to zero at infinities. Thus we must impose scattering boundary conditions requiring that we have only wave propagating to the right at positive infinity or to the left at negative infinity. We cannot require that both these conditions are satisfied simultaneously. Different boundary conditions would produce different states; the appropriate condition is fixed by the experimental situation. Using analogy with classical waves we can say that if we have a wave incident from the left we will have a reflected wave propagating to the left and a single transmitted wave propagating to the right. ( 29 1 1 2 2 1 1 1 2 2 3 0 1 2 , / 2 ( ) , / 2 / 2 , / 2 2 2 ; ik x ik x ik x ik x ik x Ae B e x a x A e B e a x a A e x a m E V mE k k ψ - - + < - = + - < < + = = h h
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Boundary conditions for scattering states ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 2 2 1 2 1 1 2 2 1 2 2 1 1 2 2 / 2 / 2 / 2 / 2 1 1 1 1 2 2 2 2 /2 / 2 /2 3 1 /2 /2 /2 /2 1 1 2 2 2 2 2 2 / 2 / 2 / 2 3 2 2 /2 /2 /2 1 1 2 2 ik a ik a ik a ik a ik a ik a ik a ik a ik a i ik a ik a ik a ik a ik a ik a ik k a a ik A ik e ik B e A ik e B ik e A ik e A i Ae B e A e B e A e A e B e Ae B e A e B e k e B ik e - - - - - - - - - + = + = + + = + = - = - 1 1 2 2 1 2 2 2 2 2 1 /2 /2 / 2 /2 1 1 1 1 2 2 2 2 /2 /2 /2 2 3 /2 / 2 2 1 2 / 2 / 2 3 2 2 ; a ik a ik a ik a ik a ik a ik a ik a i ik a ik a k a k k Ae B e A e B e k k k A e
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