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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 for -L/2 ≤x≤ L/2 V(x) = { ∞ elsewhere. In the region -L/2 ≤ x ≤ L/2, the normalized energy eigenfunctions take the form [√√²/ COS for n 1,3,5,... Un(2) = NTX sin for n 2, 4, 6, ... (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 x is equal to zero in the state described by u2(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 Ap 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 π [ ² si sin² u du = 3 2 -π [6] [6] [2] [3]