HW6 - x = a and x = b , with b > a . Assuming...

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Physics 731 Assignment #6, due Monday, Nov. 2 1. Sakurai, Problem 1, Chapter 2. 2. Sakurai, Problem 4, Chapter 2. 3. Sakurai, Problem 11, Chapter 2. 4. Sakurai, Problem 16, Chapter 2. 5. (a) Show that the WKB method reproduces the exact results for the energy eigenstates of the harmonic oscillator potential, V ( x ) = 2 x 2 2 . (b) For a particle in the n th energy eigenstate of the harmonic oscillator, determine the distance ˜ x above the turning point x n ( x n > 0 ) such that an error of 1% is obtained by linearizing the potential: V ( x n + ˜ x ) - V lin ( x n + ˜ x ) V ( x n ) = 0 . 01 . (c) Given the results of part (b), and the fact that the asymptotic form of Ai( z ) is valid to 1% as long as z 5 , determine the lower bound on the value of n for which the WKB method is justified to this level of approximation, and comment on the implications for the results of part (a). 6. Within the WKB approximation, compute the bound state energies for the potential V = α | x | . 7. Consider the general scattering problem from a potential barrier with classical turning points at
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Unformatted text preview: x = a and x = b , with b > a . Assuming that the WKB approximation holds, the wavefunction in each region can be written as x<a = A p k ( x ) e i R x a k ( x ) dx + B p k ( x ) e-i R x a k ( x ) dx (1) a<x<b = C p ( x ) e-R x a ( x ) dx + D p ( x ) e R x a ( x ) dx (2) x>b = F p k ( x ) e i R x b k ( x ) dx + G p k ( x ) e-i R x b k ( x ) dx , (3) with k ( x ) = p 2 m ( E-V ( x )) h , ( x ) = p 2 m ( V ( x )-E ) h . (4) Use the WKB method to compute the 2 2 matrix M , dened by A B = M F G , (5) in terms of the parameter = exp h R b a ( x ) dx i . For G = 0 (no incoming wave from the right), write down the transmission coefcient T in the limit of a high and broad barrier ( 1 ). 1...
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This note was uploaded on 12/14/2009 for the course QUANTUM I 731 taught by Professor Everett during the Fall '09 term at University of Wisconsin.

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