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Chapter6_36

Chapter6_36 - The most general solution is x x De Ce x −...

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PHY3063 R. D. Field Department of Physics Chapter6_36.doc University of Florida Quantum Mechanical Tunneling (1) Consider a barrier potential of the form + = 0 0 ) ( 0 V x V 0 0 < > x L x L x Now consider particles with E < V 0 entering from the left (Region 1) and traveling to the right. Classically all particles would be reflected back at x = 0 and none would ever reach Region 3! Time Independent Schrödinger Equation: We look for solutions of the time-independent Schrödinger equation ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V dx x d m ψ ψ ψ = + h with h / ) ( ) , ( iEt e x t x = Ψ ψ . Region 1: In this region V(x) = 0 and hence ) ( ) ( 2 ) ( 2 2 2 2 x k x mE dx x d ψ ψ ψ = = h with 2 2 h mE k = and m k E 2 2 2 h = The most general solution is ikx ikx Be Ae x + + = ) ( 1 ψ . Region 2: In this region V(x) = +V 0 and hence ) ( ) ( ) ( 2 ) ( 2 0 2 2 2 x x E V m dx x d ψ κ ψ ψ =
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Unformatted text preview: The most general solution is x x De Ce x − + + = ) ( 2 . Note that the wave function is not zero in the classically forbidden region. Region 3: In this region V(x) = 0 and hence ) ( ) ( 2 ) ( 2 2 2 2 x k x mE dx x d − = − = h with 2 2 h mE k = and m k E 2 2 2 h = The most general solution is ikx ikx Ge Fe x − + + = ) ( 3 , but in this region G = 0 since there are no particles entering from the right traveling to the left in region 3. x x = 0 E Barrier Potential Classically Forbidden V “Region 1” “Region 2” “Region 3” x = L...
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