LHomework 9Prob. 9.1 For a two-level system with static couplingin Eq. (9.18), (a)Find Aand Bin terms of θif c1(0)= 0 and c2(0)=1. (5 pts)(b)Find c1(t), c2(t), p1(t)and p2(t)following the initial condition in (a). (5 pts)(c)Find Aand Bin terms of θif c1(0)= c2(0)= 2/1. (5 pts)(d)When is the first time p1(t) = 0following the initial condition in (c). (10 pts)Prob. 9.2For a two-level system with dynamic coupling, assume that the evolution equations for the expansion coefficients c1(t)and c2(t)can be described by:(29(29(29(29=-tctcHVeVeHtctcdtdititi212121ϖϖwhere Vis real.(a)If (29(29(29tHitbtcii~exp-=, find the time derivative of bi(t). (5 pts)(b)If (29(29/exp11tibtbλ=and (29(29(29/exp22tibtbϖλ+=with b1and b2as time-independent, find the matrix equation with λbeing the eigenvalue of the matrix.(5 pts)(c)Find the eigenvalues λ. (10 pts)(d)If c1(0)= 1 and c2(0)=0, find the expression for c1(t)and c2(t). (10 pts)Prob. 9.3An infinite potential well of length Lis shown in the figure.
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