Chapter2_26 - The most general solution is x x De Ce x + +...

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PHY4604 R. D. Field Department of Physics Chapter2_26.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 = + h with h / ) ( ) , ( iEt e x t x = Ψ . Region 1: 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 Be Ae x + + = ) ( 1 . 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 κ = = h with 2 0 ) ( 2 h E V m = .
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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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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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