Question

2m dx2 + W(x). 10.15 Square potential in ld: Swo if |x|< L/2, W(x) = (1) 0 otherwise where the height of the potential Wo can be positive or negative. The corresponding Hamilton operator is I d2 (2) We consider wave functions which are continuous together with their first derivatives. Here we look at scattering states. For these, we need the solutions to Ĥu = Eu with the boundary conditions (k > 0) elkx +R(k)e-ikx if x + --, T(k)elka if x→ too, see lectures. (a) For E < Wo, show that T(K)/2 = 3 4|qk2 -2/2mW.L 4|qk|2+(\k2 + |x|2)2 sinh” q|L where you should determine 9. (R is determined via |R2 + 1T2 = 1.) (b) For E > Wo, show that (k2 - 92) sin? QL R(K)2 = 4q2k2 cosa qL+ (k2-92) sinqL where you should again determine . (T is determined via |R12 + 1T2 = 1.) (c) Explain the physical significance of your results for both cases.

quantum mechanics question