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hw09 - Homework 9 Prob 9.1 For a two-level system with...

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L Homework 9 Prob. 9.1 For a two-level system with static couplingin Eq. (9.18), (a) Find A and B in terms of θ if c 1 (0) = 0 and c 2 (0) =1. (5 pts) (b) Find c 1 (t) , c 2 (t) , p 1 (t) and p 2 (t) following the initial condition in (a). (5 pts) (c) Find A and B in terms of θ if c 1 (0) = c 2 (0) = 2 / 1 . (5 pts) (d) When is the first time p 1 (t) = 0 following the initial condition in (c). (10 pts) Prob. 9.2 For a two-level system with dynamic coupling, assume that the evolution equations for the expansion coefficients c 1 (t) and c 2 (t) can be described by: ( 29 ( 29 ( 29 ( 29 = - t c t c H Ve Ve H t c t c dt d i t i t i 2 1 2 1 2 1 ϖ ϖ where V is real. (a) If ( 29 ( 29 ( 29 t H i t b t c i i ~ exp - = , find the time derivative of b i (t) . (5 pts) (b) If ( 29 ( 29 / exp 1 1 t i b t b λ = and ( 29 ( 29 ( 29 / exp 2 2 t i b t b ϖ λ + = with b 1 and b 2 as time- independent, find the matrix equation with λ being the eigenvalue of the matrix. (5 pts) (c) Find the eigenvalues λ . (10 pts) (d) If c 1 (0) = 1 and c 2 (0) =0, find the expression for c 1 (t) and c 2 (t) . (10 pts) Prob. 9.3 An infinite potential well of length L is shown in the figure.
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