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

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Consider a particle in a 2-D box having Lx = 10 nm and Ly = 10 nm. a) Make a surface plot of all the wave functions for the first and second energy levels. b) What is the degeneracy of the second energy level? Compare and contrast the wave functions of the second energy level. c) How does the number of nodes in the x-coordinate change as n increases? How does the number of nodes in the y-coordinate change as n, increases? d) Explain whether or not those same states would be degenerate if Lx = 10 nm and Ly = 15 nm.