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Lect15_ProbCurrent1 - Lecture 15 Simple problems in 1D and...

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Lecture 15: Simple problems in 1D and Probability Current I Phy851 Fall 2009
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Continuity Theorem From previous Lecture: Theorem: the wavefunction and its first derivative must be everywhere continuous. Exception: where there is a δ ( x-x 0 ) or δ ( x-x 0 ) in the potential. δ ( x-x 0 ) potential discontinuity in ψ ( x ) at x = x 0 δ ( x-x 0 ) potential discontinuity in ψ ( x ) at x = x 0
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Solution to the Step Potential Scattering Problem Assuming an incoming flux from the left only, we make the ansatz: As there is no δ or δ potential, we need to impose two boundary conditions at x = 0 : x k i x k i I e r e x 1 1 ) ( + = ψ x k i II e t x 2 ) ( = ψ ) 0 ( ) 0 ( II I ψ ψ = ) 0 ( ) 0 ( II I ψ ψ = t r = + 1 t ik r ik 2 1 ) 1 ( = ) 1 ( ) 1 ( 2 1 r ik r ik + = 2 1 2 1 k k k k r + = 2 1 1 2 k k k t + = ) ( ) ( 2 1 2 1 k k i r k k i = + (1) (2) Insert (1) into (2) Collect r terms together Solve for r Plug solutions into (1) and solve for t 2 1 2 1 1 k k k k t + + =
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Case I: Tunneling into the Barrier Consider the case where E < V 0 : ( ) ( ) γ i E V m i E V m k = = = : 2 2 0 0 2 h h x II te x γ ψ = ) ( k mE k = = : 2 1 h Q: why did we
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