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answerhappygod
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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) = -{º elsewhere. In the region -L/2 ≤ x ≤ L/2, the normalized energy eigenfunctions take the form 2 $ (17²), for n = 1,3,5,... L &n(x) = 2 sin ¹(¹T²), 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 is equal to zero in the state described by u2(x), and calculate the expectation value of ² 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 πT³ π [ u² sin² u du = 3 2 COS
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) = -{º elsewhere. In the region -L/2 ≤ x ≤ L/2, the normalized energy eigenfunctions take the form 2 $ (17²), for n = 1,3,5,... L &n(x) = 2 sin ¹(¹T²), 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 is equal to zero in the state described by u2(x), and calculate the expectation value of ² 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 πT³ π [ u² sin² u du = 3 2 COS