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Consider a particle trapped in a iD box with zero potential energy with walls at x = 0 and x = L. The general wavefunction solutions for this problem with quantum number, n, are: Esin Wn(x) = The corresponding energy (level) for each wavefunction solution is: n?h? En 8ml? a) What is the probability of finding the particle between x = L/4 and x = 3L/4 when the particle is in quantum state n = 1, 2 and 3. %3D

Consider a particle trapped in a iD box with zero potential energy with walls at x = 0 and x = L. The general wavefunction solutions for this problem with quantum number, n, are: Esin Wn(x) = The corresponding energy (level) for each wavefunction solution is: n?h? En 8ml? a) What is the probability of finding the particle between x = L/4 and x = 3L/4 when the particle is in quantum state n = 1, 2 and 3. %3D

Principles of Modern Chemistry
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
Chapter5: Quantum Mechanics And Atomic Structure
Section: Chapter Questions
Problem 6P: Using Table 5.2, write down the mathematical expression for the 2px wave function for an...
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Consider a particle trapped in a iD box with zero potential energy with walls at x = 0 and x = L. The general wavefunction solutions for this problem with quantum number, n, are: Vn(x) = The corresponding energy (level) for each wavefunction solution is: n?h? En 8ml? a) What is the probability of finding the particle between x = L/4 and x = 3L/4 when the particle is in quantum staten = 1, 2 and 3. %3! You can use calculator or a numerical program to do the integral. For people who want to try doing the integral by hand, the following identity will be helpful: sin²(x) = (1 – cos (2х))/2. b) Is the particle always uniformly distributed throughout the box? Explain your answer. c) Consider the molecules: CH2=CH-CH=CH-CH=CH-CH=CH-CH=CH2. Follow from Ex 4.7 in the textbook, let's assume that the 1o electrons that make up the double bonds can exist everywhere along the carbon chains. The electrons can then be considered as particles in a box; the ends of the molecule correspond to the boundaries of the box with a finite or zero potential energy inside. In this “molecular box", 2 electrons can occupy an energy level. What are quantum states that the electrons from this molecule can occupy in the ground state? What's the smallest frequency of light that can excite the electron? Briefly explain why. Note that the length of a C-C bond is about 1.54A and the length of a C=C bond is 1.34A to allow you to estimate the length of the "molecular box".
Transcribed Image Text:Consider a particle trapped in a iD box with zero potential energy with walls at x = 0 and x = L. The general wavefunction solutions for this problem with quantum number, n, are: Vn(x) = The corresponding energy (level) for each wavefunction solution is: n?h? En 8ml? a) What is the probability of finding the particle between x = L/4 and x = 3L/4 when the particle is in quantum staten = 1, 2 and 3. %3! You can use calculator or a numerical program to do the integral. For people who want to try doing the integral by hand, the following identity will be helpful: sin²(x) = (1 – cos (2х))/2. b) Is the particle always uniformly distributed throughout the box? Explain your answer. c) Consider the molecules: CH2=CH-CH=CH-CH=CH-CH=CH-CH=CH2. Follow from Ex 4.7 in the textbook, let's assume that the 1o electrons that make up the double bonds can exist everywhere along the carbon chains. The electrons can then be considered as particles in a box; the ends of the molecule correspond to the boundaries of the box with a finite or zero potential energy inside. In this “molecular box", 2 electrons can occupy an energy level. What are quantum states that the electrons from this molecule can occupy in the ground state? What's the smallest frequency of light that can excite the electron? Briefly explain why. Note that the length of a C-C bond is about 1.54A and the length of a C=C bond is 1.34A to allow you to estimate the length of the "molecular box".
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