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Consider a particle with energy E incident on a rectangular barrier of width a and height VB = E. E VB x 0 a The solutions of the time-independent Schrödinger equation in the region <0 are of the form 班= 1 = Ale+ ARE and in the region z>a are of the form V = Apekz (a) Why do we only use one of the two possible solutions of the TISE for the region > a? (b) Show that the solutions of the time-independent Schrödinger equation in the region 0<

Consider a particle with energy E incident on a rectangular barrier of width a and height VB = E. E VB x 0 a The solutions of the time-independent Schrödinger equation in the region <0 are of the form 班= 1 = Ale+ ARE and in the region z>a are of the form V = Apekz (a) Why do we only use one of the two possible solutions of the TISE for the region > a? (b) Show that the solutions of the time-independent Schrödinger equation in the region 0<

University Physics Volume 3
University Physics Volume 3
17th Edition
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:William Moebs, Jeff Sanny
Chapter7: Quantum Mechanics
Section: Chapter Questions
Problem 93CP: Suppose an infinite square well extends from L/2 to +L/2 . Solve the time-independent Schrödinger's...
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Consider a particle with energy E incident on a rectangular barrier of width a and height VB = E. E VB x 0 a The solutions of the time-independent Schrödinger equation in the region <0 are of the form 班= 1 = Ale+ ARE and in the region z>a are of the form V = Apekz (a) Why do we only use one of the two possible solutions of the TISE for the region > a? (b) Show that the solutions of the time-independent Schrödinger equation in the region 0<<a are of the form (z) = B₁ + B₂x. (c) Use the boundary conditions at x = 0 and z = a to derive expressions for B₁ and B2 in terms of A₁, AR and AT. Hence, derive an equation for the tunneling probability TAT/A in terms of a and k. (d) What does your equation for the tunneling probability predict for large values of k? What qualitative argument can you give to support the conclusion that this is the correct prediction?
Transcribed Image Text:Consider a particle with energy E incident on a rectangular barrier of width a and height VB = E. E VB x 0 a The solutions of the time-independent Schrödinger equation in the region <0 are of the form 班= 1 = Ale+ ARE and in the region z>a are of the form V = Apekz (a) Why do we only use one of the two possible solutions of the TISE for the region > a? (b) Show that the solutions of the time-independent Schrödinger equation in the region 0<<a are of the form (z) = B₁ + B₂x. (c) Use the boundary conditions at x = 0 and z = a to derive expressions for B₁ and B2 in terms of A₁, AR and AT. Hence, derive an equation for the tunneling probability TAT/A in terms of a and k. (d) What does your equation for the tunneling probability predict for large values of k? What qualitative argument can you give to support the conclusion that this is the correct prediction?
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