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hw09sol

# hw09sol - Homework 9 Solution Prob 9.1 For a two-level...

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Homework 9 Solution 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) A = sin θ ; B = cos θ . (b) Find c 1 (t) , c 2 (t) , p 1 (t) and p 2 (t) following the initial condition in (a). (5 pts) ( 29 ( 29 ( 29 ( 29 - - = - = - + - = - - - = - + - + + - + - t E E t p t E E t p t E i t E i t c t E i t E i t c B B B B B B 2 sin 2 sin 1 2 sin 2 sin exp cos exp sin exp exp cos sin 2 2 2 2 2 1 2 2 1 1 θ θ θ θ θ θ (c) Find A and B in terms of θ if c 1 (0) = c 2 (0) = 2 / 1 . (5 pts) ( 29 ( 29 θ θ θ θ sin cos 2 1 ; cos sin 2 1 - = + = B A (d) When is the first time p 1 (t) = 0 following the initial condition in (c). (10 pts) ( 29 ( 29 0 2 sin 4 sin 2 1 2 1 2 2 1 1 = - + = = - + t E E t c t p B θ In general, θ 4 sin 1 2 sin 2 - = - - + t E E B . Notice that this equality can ONLY be true for a specific θ that gives sin 4 θ = -1 where 8 3 1 1 cos 2 2 12 1 π θ = - + = - - H E V . If this condition is not met then p 1 (t) will never be zero (the oscillation is not full swing). If the condition is met, then the first time p1(t) is zero at - + - = E E t π .

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hw09sol - Homework 9 Solution Prob 9.1 For a two-level...

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