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Question 11 This question concerns a particle of mass m in a one-dimensional infinite square well, described by the potential energy function 0 for -L/2 ≤ x ≤ L/2 V(x)= elsewhere. In the region -L/2 ≤ x ≤ L/2, the normalized energy eigenfunctions take the form ᎡᏆ COS for n = 1,3,5,... L Un(2) = sin (*) for n 2, 4, 6, ... L (a) Write down the time-independent Schrödinger equation for this system in the region -L/2 < x < L/2. Verify that ₁(x) and 2(x) (as defined above) are solutions of this equation, and find the corresponding energy eigenvalues. (b) Show that the expectation value of position is equal to zero in the state described by 2(x), and calculate the expectation value of x² in this state. Hence derive the uncertainty Ax for a measurement of position in this state. (c) Using your answer to part (b), give a lower bound for the uncertainty Apa for a measurement of the momentum in the state described by u₂(x). (d) Is the ground-state energy of a particle in a finite square well (also of width L) larger than or smaller than the ground-state energy of a particle in an infinite square well? Explain your answer. You may use the standard integral 773 70 L u² sin² u du = 3 2 π = [6] [6] [3]