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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?

Introductory Chemistry: A Foundation
Introductory Chemistry: A Foundation
9th Edition
ISBN:9781337399425
Author:Steven S. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Donald J. DeCoste
Chapter10: Energy
Section: Chapter Questions
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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?
Transcribed Image Text: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?
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