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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Question
![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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d7dc701-b654-4339-995f-9e7ccdd48dfe%2F57777b70-67d8-4496-a3cf-042de784a8af%2Fmwt2h1s_processed.png&w=3840&q=75)
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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