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Lecture16Notes - Chem 120A Spring 2006 READING Square...

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Chem 120A Square Potentials 02/24/06 Spring 2006 Lecture 16 READING: Engel: Chapters 4 and 5 See ’McQuarrie square potentials’ supplement in Lecture Notes folder on blackboard.berkeley.edu Last time we began to look at the behavior of a quantum particle incident upon a square potential barrier (see Figure 1). The wavefunctions in the three regions of the problem are given by: Region I: Ψ I ( x ) = Ae ikx + Be - ikx Region III: Ψ III ( x ) = Qe ikx + Fe - ikx where k 2 = 2 m ¯ h 2 ( E ) Region II E < V : Ψ II ( x ) = Ce - κ x + De κ x where κ 2 = 2 m ¯ h 2 ( V - E ) E > V : Ψ II ( x ) = C e ik 0 x + D e - ik 0 x where k 2 0 = 2 m ¯ h 2 ( E - V ) We can solve for the coefficients (A, B, C, D, Q) by matching Ψ s and their first derivatives at the bound- aries. Note that terms with e + i α x (where α = k , k 0 , i κ ) represent wavepackets moving to the right in each region. Likewise terms with e - i α x represent wavepackets moving to the left. Classically if a particle hits an energy barrier with E < V it will not go through the barrier, but will be reflected (T = 0 and R = 1 where T is the transmission coefficient and R is the reflection coefficient). When E > V , the classical particle will be transmitted through the barrier (T = 1 and R = 0). In the quantum mechanical case, very different behavior is observed. The particle can still penetrate the barrier, even for E < V . In addition a quantum particle with E > V is not perfectly transmitted, except at certain resonance conditions. To investigate this behavior, we will solve for the transmision, T, and reflection, R, coefficients. These were defined last time as: Chem 120A, Spring 2006, Lecture 16 1
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T = Q A 2 R = B A 2 (1) The particle is either transmitted or reflected, so their sum is unity R + T = 1 For E > V the transmission coefficient is 1 T = 1 + 1 4 V 2 E ( E - V ) sin 2 ( k 0 a ) (2) For E < V the transmission coefficent is T = 1 1 + 1 4 V 2 E ( V - E ) sinh 2 ( κ a ) (3) (you can follow the derivation of this equation in the ’McQuarrie square potentials’ supplement in Lecture
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