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- 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 0 dij(x) dx √x = - 1²/2 2 din(x) dx = For the odd solution, use the following solution with A' = - C': 4₁(x) = A' exx (x) = {₁(x) = B′ sin kx B' m(x)=C'e-xx III = din(x) dx You should also apply in each case the continuity conditions: 4₁ (x = -²2 ) = ₁ (x = -1) Pu (x=+) = m(x=+) dpm(x) dx |x=+12/2 VI 8 -- / x VI 212 x≤ - 1²/12 ≤x≤ 11/27 ≤ x |x=+23/23 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 K and plot the solutions on a graph.(a) For the 1s wave function, calculate the probability for the electron to be found within the nucleus, considered to be a sphere of radius R. (b) Evaluate the probability for a nucleus of lead (R = 7.1 fm). (c) Suppose instead that a muon (m = 207m) were in its 1s state. What is the probability to find it inside the lead nucleus?