125. For a 2D infinite square well, what is the next highest energy level (above 50EG) that exhibits an “accidental degeneracy"? List all the possible sets of quantum numbers for this state.
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- 4. Find the points of maximum and minimum probability density for the nth state of a particle in a one- dimensional box. Check your result for the n=2 state.3. For free particles in two dimensions, what is density of states (DOS) in low speed limit (=p²/2m), and in high speed limit (=pc)?14. Verify that the parities of the one-electron atom eigenfunctions 300, 310, 320, and 322 are determined by (-1)¹.
- 17. Please answer question throughly and detailed.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.5
- 7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -125. An attractive square well potential is 55 represented by -V for r a The scattering due to this potential in low energy limit is proportional to nth power of a. Here n is (1) 2 (2) 4 (3) 5 (4) 6