1) (25 points) A particle of mass m is scattered by a symmetric square well potential given by V(I) = -Vo 12| 0 (1) | | > a/2 The incoming and outgoing plane waves of even and odd parity are It ( k; z) = e-iklx] I_(k; x) = sign(x) e-iklz] (2) O+ ( k; 2) = etiklal O_ (k; a) = -sign(x) etiklzl (3) The corresponding scattering states for (x| > a/2 are 7 + ( k ; I) = It (k; x) + S+ + (k)0+(k; x) (4) _ ( k; x) = I_ (k; x) + S_ _ (k)O_(k; x) (5) where S+ + and S_- are the diagonal elements of the S matrix in parity ba- sis. Calculate S+ + and S_- and the corresponding transition/reflections coef- ficients. Use the notation q2 = k2 + Uo, Uo = 2mVo/h . Interpret the poles and zeros of S+ + and S__ in terms of bound states. 2) 25 points Carry out a similar analysis to 1) with the potential V(x) = Vo [o(x - 1) + 8(x + 1)] (6) with Vo > 0. Interpret the poles and zeros of S+ + and S_- in the complex k plane as resonances, the the case of Vo > 1. Show that, approximately, the pole position in S+ + with the smallest real part lies at TT 2 k = + 2Uo 14U2 (7) where Uo = 2mVo/h2