Question 6: (20 marks) For a particle in one dimensional box problem, a) Find P (L/4...

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Question 6: (20 marks) For a particle in one dimensional box problem, a) Find P (L/4 <x<L/2), i.e., the probability of finding the particle between the region L/4 and L/2, for the state, n=2. b) Show that the n = 2 and n= 3 states are orthonormal. c) Is (x) = etk*, where k is a positive constant, an eigenfunction of the Hamiltonian for the particle in 1D box for the region, where V= 0? If so, is it an acceptable solution to the time independent Schrodinger's equation for the system? Justify your answer. Question T: (4 marks) A proton is accelerated to a speed of, v=(7.365±0.017)×10° m/s. With what maximum accuracy can its position be determined? (mo, 1.6726x102 kg) Question 8: (6 marks) Check whether the function (fe) = e-3ikx) is an eigenfunction of the one-dimensional (1D) Cartesian kinetic energy operator. If so, what is the eigen value? Question 9: (12 marks) d Define what is meant by a commutator operater. If  = 3x² and B =, then show that  and B donot commute with respect to the function f(x) = sin x. (a) %3D dx (b) Write the Cartesian form of the Laplacian operator, V2. Examine whether the function f (x, y, z) = x² + y² + z² is an eigen function of the Laplacian operator %3D or not. (c) Show if the functions, Y1(x) = (-) cos x, & ¥2(x) = sin x, are orthogonal or not in the interval 0< x < 2n. Question 6: (20 marks) For a particle in one dimensional box problem, a) Find P (L/4 <x<L/2), i.e., the probability of finding the particle between the region L/4 and L/2, for the state, n=2. b) Show that the n = 2 and n= 3 states are orthonormal. c) Is (x) = etk*, where k is a positive constant, an eigenfunction of the Hamiltonian for the particle in 1D box for the region, where V= 0? If so, is it an acceptable solution to the time independent Schrodinger's equation for the system? Justify your answer. Question T: (4 marks) A proton is accelerated to a speed of, v=(7.365±0.017)×10° m/s. With what maximum accuracy can its position be determined? (mo, 1.6726x102 kg) Question 8: (6 marks) Check whether the function (fe) = e-3ikx) is an eigenfunction of the one-dimensional (1D) Cartesian kinetic energy operator. If so, what is the eigen value? Question 9: (12 marks) d Define what is meant by a commutator operater. If  = 3x² and B =, then show that  and B donot commute with respect to the function f(x) = sin x. (a) %3D dx (b) Write the Cartesian form of the Laplacian operator, V2. Examine whether the function f (x, y, z) = x² + y² + z² is an eigen function of the Laplacian operator %3D or not. (c) Show if the functions, Y1(x) = (-) cos x, & ¥2(x) = sin x, are orthogonal or not in the interval 0< x < 2n.