University:
Massachusetts Institute of Technology
Course:
6.728 | Applied Quantum and Statistical Physics
Academic year:

2021
Views:

127

Pages:

2
Author:

Aliza Leon

WKB Approximation Applied to Tunneling ΨI = eikx + rI e−ikx tII (x)e−θ(x) + rII (x)eθ(x) � 2m k = E ¯h2 � x� 2m θ(x) = (V (x� ) − E) dx� ¯h2 0 ΨII = ΨIII = AIII eikx Note, we use t(x) and r(x) in ΨII , but they can only take the WKB AWAY from the classical turning point. Now we assume rII = 0. This is safe as long as the barrier remains relatively thick (so the reﬂected wave has small amplitude). Now consider the coundary condition at x = 0: ΨI (0) = tII (0)e−θ(0) 1 ⇒ ΨII (x) = � e−θ(x) ΨI (0) � , ζ(x) where ζ(x) = 2m (V (x) − E). h̄ Keep in mind that ΨII (x) is the approximate eigenstate only (e.g at x = 0, ΨII → ∞ which is clearly unphysical). Now consider the boundary condition for continuity at W : − ΨII (W ) ΨI (0)e � = 4 � W � 2m h ¯2 0 2m (V h ¯2 (V (x� )−W ) dx� = ΨIII (W ). (W ) − E) Then T = = � � � ΨIII (W ) �2 � � � ΨI (0) � � W � 2m −2 (V (x� )−E) dx� h ¯2 0 e � 2m (V (W ) − E) h ¯2 This solution has the form T = Ae−2θ(x) . In general tunneling barriers will have a dependence of √ T ∼ = e− φ0 W , which falls out of the derivation above if V (x) = φ0 , 0 < x < W (so the barrier height is a constant). More complex barriers may require careful corrections to WKB to achieve quantitative agreement with experiments. 2

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