By Levi A.F.J.

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Ryzhik, Table of integrals, series, and products, Academic Press, San Diego, 1980, p. 196 (ISBN 0 12 294760 6)) ∫e ax ∫e ax e ( a cos ( bx ) + b sin ( bx ) ) cos ( bx ) dx = -------------------------------------------------------------2 2 a +b ax e ( a sin ( b x ) – b cos ( b x ) -) sin ( bx ) dx = -------------------------------------------------------------2 2 a +b ψ ( p x, t ) ax ip L ⁄ 2 2 – ip L ⁄ 2 ip L ⁄ 2 2 –i p L ⁄ 2 2 1 Lπ ( e x + e x ) π( e x – e x ) + = ------------- -----------------------------------------------+ 4L --------------------------------------------------2 2 2 2 2 2 2πh L 16 π – p x L π – px L ip L ⁄ 2 –i p L ⁄ 2 –i p L ⁄ 2 ip L ⁄ 2 i ( ω4 – ω1 )t Lπ ( e x + e x ) 4 Lπ ( e x – e x ) ------------------------------------------------------------------------------------------------- 2Re e π 2 – p 2x L 2 16 π 2 – p 2x L 2 ψ (p x, t ) 2 p xL p xL = A cos 2 -------- + Bsin 2 -------+ C sin ( p x L ) sin ( ( ω 4 – ω 1 ) t ) 2 2 32L π –8L π 2Lπ - , B = ----------------------------------------2 , C = ----------------------------------------------------------------A = ------------------------------.

Describe any assumptions you have made. Outline how you might extend your calculations to include elastic scattering from n ionized impurities in a substitutionally doped crystalline semiconductor. (c) What differences in scattering rate do you expect in (b) for the case of randomly positioned impurities and for the case of strongly correlated impurity positions? Applied quantum mechanics 3 SOLUTIONS Problem 1 (a) Consider a quantum mechanical system described by Hamiltonian H( 0 ) and for which we know the solutions to the time-independent Schrödinger equation H( 0 ) | n〉 = E n | n〉 and the time-dependent Schrödinger equation – iω nt – iωn t ∂ | n〉e = H ( 0 ) |n 〉e .

Using the result in part (a), calculate the probability of transition to each excited state of the system in the long time limit, t → ∞ . PROBLEM 2 An electron is in ground state of a one-dimensional rectangular potential well for which V ( x ) = 0 in the range 0 < x < L and V ( x ) = ∞ elsewhere. It is decided to control the state of the electron by applying a pulse of electric field E ( t ) = E 0 e 2 –t ⁄ τ 2 ⋅ xˆ at time t = 0, where τ is a constant, xˆ is the unit vector in the x-direction, and E 0 is the maximum strength of the applied electric-field.