4. Consider a rectangular barrier given by the potentialV(x) = Vo > 0 for x

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Consider a rectangular barrier given by the potential for x 0 V(x) = 0 for xa (a) Show that the transmission probability |T|² for a particle of energy E < Vo to tunnel completely through the barrier is determined by the expression T-¹ = 1 + [Vo sinh (2 t a)] 4E (Vo - E) where ht = [2 m (Vo - E)]¹/2 (b) What is T-¹ in the limit that the energy approaches the barrier height E Vo? (c) Calculate |T|² for an electron of energy 1 eV incident on a barrier of height 1 eV and width 1 Å

Consider a rectangular barrier given by the potential for x 0 V(x) = 0 for xa (a) Show that the transmission probability |T|² for a particle of energy E < Vo to tunnel completely through the barrier is determined by the expression T-¹ = 1 + [Vo sinh (2 t a)] 4E (Vo - E) where ht = [2 m (Vo - E)]¹/2 (b) What is T-¹ in the limit that the energy approaches the barrier height E Vo? (c) Calculate |T|² for an electron of energy 1 eV incident on a barrier of height 1 eV and width 1 Å

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

4. Consider a rectangular barrier given by the potential
V(x) = Vo > 0 for x <a
V(x) = 0
for xa
(a) Show that the transmission probability |T|² for a particle of energy E < Vo to tunnel completely
through the barrier is determined by the expression
T-¹ = 1 +
[Vo sinh (2 t a)]
4 E (Vo - E)
where ht= [2 m (Vo - E)] ¹1/2
(b) What is T -¹ in the limit that the energy approaches the barrier height E
-1
Vo?
(c) Calculate |T|² for an electron of energy 1 eV incident on a barrier of height 1 eV and width 1 Å
Hyperbolic Sine Function: sinh y = ½ (e-e).
Taylor series: ey = 1 + y + ¹ y² +

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Step 1: a) Calculate the T^-1:

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Step 2: Calculate the T^-1:

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Step 3: Calculate the T^-1:

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Step 4: Calculate the T^-1:

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Step 5: Calculate the T^-1:

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Step 6: Calculate the T^-1:

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Step 7: Calculate the T^-1:

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Step 8: b) Use of E tends to V(0) limit:

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Step 9: c) Calculate the value of T^2:

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