8 - ECE331 Step Potential Problem Boundary conditions V(x)...

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1 Lu ECE331 Step Potential Problem Boundary conditions V(x) = 0, x<0 V(x)=V 0 , x>0 If x<0 (reg. I), x k B x k A mE k Be Ae x Solution x mE x x x jk x jk 1 1 1 1 2 2 2 cos ' sin ' 2 , ) ( , 0 ) ( 2 ) ( 1 1 + = = + = Ψ = Ψ + Ψ h h Lu ECE331 Step Potential Problem (cont) h h ) ( 2 , ) ( , 0 ) ( ) ( 2 ) ( 0 2 2 2 0 2 2 2 2 V E m k Be Ae x Solution x V E m x x x jk x jk = + = Ψ = Ψ + Ψ If x>0 (reg. II), (1) If E>V 0 , x k D x k C x 2 2 2 cos ' sin ' ) ( + = Ψ Apply boundary conditions, At x=0, Ψ , Ψ ’ are continuous.
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2 Lu ECE331 Step Potential Problem (cont) 0 2 0 1 2 1 ) 0 ( ) 0 ( = = Ψ = Ψ Ψ = Ψ x x dx d dx d If x>0 (reg. II), OR, B’ = D’ A’k 1 = C’k 2 Lu ECE331 Step Potential Problem (cont) So, 0 , cos ' sin ' ) ( 0 , cos ' sin ' ) ( 2 2 2 1 2 1 1 1 > + = Ψ < + = Ψ x x k B x k A k k x x x k B x k A x
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3 Lu ECE331 Step Potential Problem (cont) h ) ( 2 ' , ' ) ( 0 2 ' 2 2 E V m k e D x x k = = Ψ h h ) ( 2 ' , ' ' ) ( 2 , ) ( 0 2 ' ' 0 2 2 2 2 2 2 E V m k
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This note was uploaded on 04/09/2010 for the course ECE 331 taught by Professor Rajan during the Spring '09 term at Ohio State.

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8 - ECE331 Step Potential Problem Boundary conditions V(x)...

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