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385lec12

# 385lec12 - PHYS 385 Lecture 12 Tunneling in one dimension...

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PHYS 385 Lecture 12 - Tunneling in one dimension 12 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 12 - Tunneling in one dimension What's important : tunneling in one dimension WKB approximation alpha decay Text : Gasiorowicz, Chap. 5 Square barrier Let's now take the potential well from the last lecture and invert it to form a barrier: V ( x ) 2 a E V o We take E < V 0 and have the object approach the barrier from the left. As usual, there are the usual plane wave solutions exp( ikx ) where E < V o , and exponentially decaying solutions inside the barrier. We define the normalization coefficients as follows: For x < - a u ( x ) = exp( ikx ) + R exp(- ikx ) k 2 = 2 mE / h 2 , For - a < x < a u ( x ) = A exp(- κ x ) + B exp(+ κ x ) κ 2 = 2 m ( V o - E ) / h 2 , (real) (1) For x > a u ( x ) = T exp( ikx ) k 2 = 2 mE / h 2 . We can go through the same steps as before with the square well, but make the replacement q -> i κ . Matching u ( x ) and its derivative du / dx at the discontinuities ± a give the set of equations: exp(- ika ) + R exp( ika ) = A exp( κ a ) + B exp(- κ a ) ik exp(- ika ) + (- ik ) R exp( ika ) = - κ A exp( κ a ) + κ B exp(- κ a ) (2) A exp(- κ a ) + B exp( κ a ) = T exp( ika ) - κ A exp(- κ a ) + κ B exp( κ a ) = ikT exp( ika )

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PHYS 385 Lecture 12 - Tunneling in one dimension 12 - 2 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Four equations and four unknowns yields the previous expressions for R and T with the substitution q -> i κ : T = e ! 2 ika 2 kq 2 kq cos(2 qa ) ! i ( q 2 + k 2 ) sin(2 qa ) = e ! 2 ika 2 ik " 2 ik " cos(2 i " a ) ! i ( ! " 2 + k 2 )sin(2 i " a ) = e ! 2 ika 2 k " 2 k " cos(2 i " a ) ! ( k 2 ! " 2 )sin(2 i " a ) = e ! 2 ika 2 k " 2 k " cosh(2 " a ) ! i ( k 2 ! " 2 )sinh(2 " a ) (3) (This is the complex conjugate of Gasiorowicz result). Taking the complex square of (3) yields (same as Gasiorowicz) | T | 2 = (2 k ! ) 2 (2 k ! )
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385lec12 - PHYS 385 Lecture 12 Tunneling in one dimension...

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