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barrier - x = L Ce qL De-qL = Fe ikL qCe qL-qDe-qL = ikFe...

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Physics 225/315 March 11 2008 One Dimensional Barrier Barrier extends from x = 0 to x = L and has height V . Incoming energy is E and k = q 2 mE/ ¯ h 2 . E < V . In barrier region q = q 2 m ( V - E ) / ¯ h 2 The wavefunctions: to left of barrier ψ ( x ) = A e ikx + B e - ikx To right of barrier ψ ( x ) = F e ikx . Within barrier ψ ( x ) = Ce qx + De - qx . The boundary conditions (wavefunction and derivatives are continuous) at x = 0 A + B = C + D ikA - ikB = qC - qD The boundary conditions (wavefunction and derivatives are continuous) at
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Unformatted text preview: x = L Ce qL + De-qL = Fe ikL qCe qL-qDe-qL = ikFe ikL These equations can be solved for B, C, D, F in terms of A , the incoming amplitude. The transmission factor is | F | 2 / | A | 2 and is given in the practice problem. For a “strong” barrier, qL >> 1, it simplifies to the decreasing exponential form given in the practice problem. T is proportional to: e-2 qL...
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