This question concerns a particle of mass m in a one-dimensional infinite square well, described by the potential energy

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This Question Concerns A Particle Of Mass M In A One Dimensional Infinite Square Well Described By The Potential Energy 1 (67.04 KiB) Viewed 29 times

This question concerns a particle of mass m in a one-dimensional infinite square well, described by the potential energy function 0 V(x) = for -L/2 ≤x≤ L/2 {⁰ elsewhere. In the region -L/2 ≤ x ≤ L/2, the normalized energy eigenfunctions take the form COS (TTT). for n= 1,3,5,... = sin (TTT), 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 (x) and 2(r) (as defined above) are solutions of this equation, and find the corresponding energy eigenvalues. (b) Show that the expectation value of position r is equal to zero in the state described by 2(2), and calculate the expectation value of r² in this state. Hence derive the uncertainty Ar 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 π³ π [² u² sin² u du 2 = 3