3) A Particle Trapped in a Shallow Defect This is a simple model for a shallow trap or defect in a semiconductor, for example, or a more realistic model for a quantum dot. We are interested in the trap states, i.e., states where the particle is localized in the trap. Hence this requires E < Vo where E is the energy of the particle. V (x) Vo Dir I II 0 L L 2 2 Since the symmetry of the Hamiltonian is even, parity is a good quantum number and the solutions must be overall even or odd. For the even solution, use the following solution with A = C: 4₁(x) = A ex (x) = n(x) = B cos kx (4(x)=Ce-xx k²= 2mE 2m K² = (Vo- h² For the odd solution, use the following solution with A' = - C': 4₁(x) = A' exx x < (x) = ₁(x) = B′ sin kx -≤x≤ (4m(x)=C'e-xx dif(x) | -E) > 0 You should also apply in each case the continuity conditions: 4₁ (x = − 1) = 4₁ (x = − 1) Pu (x = + ²) = m ( x = + 2) III = dx |x==1/12 di(x)| dx dèµ(x)| dx = + |x=+1/2 Use these conditions in the solution to find a set of two homogeneous equations of two unknowns. Solve these equations to find a relation between k and x and plot the solutions on a graph. dip(x)| dx

3) A Particle Trapped in a Shallow Defect This is a simple model for a shallow trap or defect in a semiconductor, for example, or a more realistic model for a quantum dot. We are interested in the trap states, i.e., states where the particle is localized in the trap. Hence this requires E < Vo where E is the energy of the particle. V (x) Vo Dir I II 0 L L 2 2 Since the symmetry of the Hamiltonian is even, parity is a good quantum number and the solutions must be overall even or odd. For the even solution, use the following solution with A = C: 4₁(x) = A ex (x) = n(x) = B cos kx (4(x)=Ce-xx k²= 2mE 2m K² = (Vo- h² For the odd solution, use the following solution with A' = - C': 4₁(x) = A' exx x < (x) = ₁(x) = B′ sin kx -≤x≤ (4m(x)=C'e-xx dif(x) | -E) > 0 You should also apply in each case the continuity conditions: 4₁ (x = − 1) = 4₁ (x = − 1) Pu (x = + ²) = m ( x = + 2) III = dx |x==1/12 di(x)| dx dèµ(x)| dx = + |x=+1/2 Use these conditions in the solution to find a set of two homogeneous equations of two unknowns. Solve these equations to find a relation between k and x and plot the solutions on a graph. dip(x)| dx