### Consider a rectangular barrier given by the potential\[V(x) = V_0 > 0 \quad \text{for } |x| < a\]\[V(x) = 0 \quad \text{for } |x| > a\]#### (a) Transmission ProbabilityShow that the transmission probability \(|T|^2\) for a particle of energy \(E < V_0\) to tunnel completely through the barrier is determined by the expression:\[T^{-1} = 1 + \frac{{[V_0 \sinh(2ta)]^2}}{{4E(V_0 - E)}}\]where \[\hbar t = \sqrt{2m(V_0 - E)}\]#### (b) Energy Approaching Barrier HeightWhat is \(T^{-1}\) in the limit that the energy approaches the barrier height \(E \rightarrow V_0\)?#### (c) Calculate \(|T|^2\)Calculate \(|T|^2\) for an electron of energy 1 eV incident on a barrier of height 1 eV and width 1 Å.---### Additional Information**Hyperbolic Sine Function:** \[\sinh y = \frac{1}{2} (e^y - e^{-y})\]**Taylor Series:**\[e^y = 1 + y + \frac{1}{2}y^2 + \ldots\]

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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

### Consider a rectangular barrier given by the potential

\[
V(x) = V_0 > 0 \quad \text{for } |x| < a
\]
\[
V(x) = 0 \quad \text{for } |x| > a
\]

#### (a) Transmission Probability

Show that the transmission probability \(|T|^2\) for a particle of energy \(E < V_0\) to tunnel completely through the barrier is determined by the expression:

\[
T^{-1} = 1 + \frac{{[V_0 \sinh(2ta)]^2}}{{4E(V_0 - E)}}
\]

where

\[
\hbar t = \sqrt{2m(V_0 - E)}
\]

#### (b) Energy Approaching Barrier Height

What is \(T^{-1}\) in the limit that the energy approaches the barrier height \(E \rightarrow V_0\)?

#### (c) Calculate \(|T|^2\)

Calculate \(|T|^2\) for an electron of energy 1 eV incident on a barrier of height 1 eV and width 1 Å.

---

### Additional Information

**Hyperbolic Sine Function:**

\[
\sinh y = \frac{1}{2} (e^y - e^{-y})
\]

**Taylor Series:**

\[
e^y = 1 + y + \frac{1}{2}y^2 + \ldots
\]

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Step 6: Calculate the T^-1:

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

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