1) (25 points) A particle of mass m is scattered by a symmetric square well potential given by V(x) = -Vo |z| 0 (1) |z| > a/2 The incoming and outgoing plane waves of even and odd parity are It (k; a) = e iklal I_ (k; a) = sign(x) e-iklal (2) O+ ( k; 2) = etiklal O_(k; x) = -sign(x) etiklal (3) The corresponding scattering states for (x| > a/2 are 2/ + ( k; I) = It (k; 2) + S+ +(k)0+(k; 2) (4) 2 _ (k; z) = I_ (k; z) + s_ _(k)O_(k; z) (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 4U2 (7) where Uo = 2mVo/h2