1. The problem statement, all variables and given/known data Consider a one-dimensional time-independent Schrodinger equation for an electron in a double quantum well separated by an additional barrier. The potential is modelled by: V (x) = -γδ(x - a) -γδ(x + a) + βδ(x) Find algebraic equations which determine the energies (or k-values) of electron bound states for γ > 0 and arbitrary real β (positive or negative). Describe the symmetry of their wave functions in terms of even and odd solutions. How many bound states do you expect for this system? 2. Relevant equations You may nd it useful to work in units ħ = 1 and m = ½, and to introduce k defined as E = -k2, where E < 0 is the energy of a bound state, so that k is real. 3. The attempt at a solution I've tried to solve this for arbitrary E, since I don't understand what happens at the barrier for E<0. So I split the problem into 4 parts: x<-a, -a<x<0, 0<x<a and x>a. This results in the wavefunctions: ψ1 = AeiE½x + Be-iE½x ψ2 = CeiE½x + De-iE½x ψ3 = FeiE½x + Ge-iE½x ψ4 = HeiE½x Using the conditions of continuity and differential continuity results in the following conditions for the coefficients: Ce-iE½a + DeiE½a = Ae-iE½a + BeiE½a -Ce-iE½a + DeiE½a = -(1+iγ/E½)Ae-iE½a + (1+iγ/E½)BeiE½a F+G=C+D -F+G=(iβ/E½-1)C+(iβ/E½+1)D HeiE½a= Fe-iE½a + GeiE½a -He-iE½a = (1+iγ/E½)Fe-iE½a - (1+iγ/E½)GeiE½a And now I'm not sure what to do. I can start to solve for transmission and reflection coefficients, but as far as I am aware bound states don't have these? It feels like at this point I need to specifiy the sign for E, but for E<0 I don't know what happens at the barrier since barriers don't have bound states.