The "particle in a ring" is another relatively simple 1-D quantum mechanics problem: The Schrödinger equation for a particle of mass m constrained to move on a circle of radius a is: where I = h² d²v 2I de² = Ev(0) 0≤0≤2π ma² is the moment of inertia and 0 is the angle that describes the position of the particle around the ring. (a) Plug the wave function (0) = Aeine into the Schrödinger equation above to show that it is a valid solution, with n = ±√2 ħ 2 (b) Argue that the appropriate boundary condition is (0) = √(0+2π) and use this condition show that E n²ħ² 21 n = 0, 1, 2,... (c) Show that the normalization constant A = (d) How might you use these results for a free-electron model of benzene?

The "particle in a ring" is another relatively simple 1-D quantum mechanics problem: The Schrödinger equation for a particle of mass m constrained to move on a circle of radius a is: where I = h² d²v 2I de² = Ev(0) 0≤0≤2π ma² is the moment of inertia and 0 is the angle that describes the position of the particle around the ring. (a) Plug the wave function (0) = Aeine into the Schrödinger equation above to show that it is a valid solution, with n = ±√2 ħ 2 (b) Argue that the appropriate boundary condition is (0) = √(0+2π) and use this condition show that E n²ħ² 21 n = 0, 1, 2,... (c) Show that the normalization constant A = (d) How might you use these results for a free-electron model of benzene?

Principles of Modern Chemistry

8th Edition

ISBN:9781305079113

Author:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Publisher:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Chapter4: Introduction To Quantum Mechanics

Section: Chapter Questions

Problem 61AP: A particle of mass m is placed in a three-dimensional rectangular box with edge lengths 2L, L, and...

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