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Chapter2_23

# Chapter2_23 - P T is defined by P T = 1-P R = 0 Probability...

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PHY4604 R. D. Field Department of Physics Chapter2_23.doc University of Florida Infinite Step Potential Time-Independent Schrödinger Equation: Look for solutions of the time-dependent Schrödinger equation of the form h / ) ( ) , ( iEt e x t x = Ψ ψ . Substituting Ψ (x,t) into the time dependent equation yields ) ( )) ( ( ) ( 2 2 2 2 x x V E dx x d m ψ ψ = h . Consider a potential V(x) such that V(x)=0 for x < 0 and V(x) = for x 0 . Left Region (x < 0): In this region ) ( ) ( 2 2 2 x k dx x d L L ψ ψ = with 2 2 h mE k = and the most general solution is ikx L ikx L L e B e A x + = Φ ) ( Right Region (x 0): In this region ψ R (x) = 0 . Boundary Conditions: In this case ψ L (x=0) = 0 , which implies that A L + B L = 0 and hence ) sin( 2 ) ( ) ( kx i A e e A x L ikx ikx L L = = ψ Reflection Probability: The reflection probability is 1 | | | | 2 2 = = = L m k L m k L L R A B j j P h h r s . The transmission probability,
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Unformatted text preview: P T , is defined by P T = 1-P R = 0 . Probability Density: ) ( sin | | 4 | ) , ( | ) ( 2 2 2 kx A t x x L L L = Ψ = ρ , which is the same answer as taking the limit r → ∞ in the previous problem. ) ( sin | | 4 )) 2 cos( 1 ( | | 2 ) 2 sin( 1 2 ) 2 cos( 1 ) 1 ( 2 2 | | ) ( 2 2 2 2 2 2 2 kx A kx A kx r r kx r r A x L L r L L = − → + − + − + = ∞ → x x 0 = 0 E Infinite Step Potential Forbidden Region V= ∞ Step Potential: V = infinite-8-6-4-2 2 x Probability Density Max destructive interference!...
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