Consider a particle trapped in a iD box with zero potential energy with walls at x = 0and 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?En8ml?a) What is the probability of finding the particle between x = L/4 and x = 3L/4 whenthe 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 wantto 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 fromEx 4.7 in the textbook, let's assume that the 1o electrons that make up the double bondscan exist everywhere along the carbon chains. The electrons can then be considered asparticles in a box; the ends of the molecule correspond to the boundaries of the box witha finite or zero potential energy inside. In this “molecular box", 2 electrons can occupyan energy level. What are quantum states that the electrons from this molecule canoccupy in the ground state? What's the smallest frequency of light that can excite theelectron? Briefly explain why. Note that the length of a C-C bond is about 1.54A and thelength of a C=C bond is 1.34A to allow you to estimate the length of the "molecular box".

Here we have toapply probability formula

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

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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Question

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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Consider an electron trapped in a 20 Å long box whose wavefunction is given by the following linear combination of the particle's n = 2 and n = 3 states: V10 2лх - sin 3nx - sin а ¥(x,t) =, where E, а 2ma² a. Determine if this wavefunction is properly normalized. If not, determine an appropriate value for a normalization constant. b. Show that this is not an eigenfunction to the PitB problem. What are the possible results that could be returned when the energy is measured and what are the probabilities of measuring each of these results? c. Calculate |Y(x,1) = '° (x,t)\(x,t) and sketch what this looks like for 1=0. с. 3th 2th and t =- You will likely want 2(E, – E.)' to use a graphing program such as Excel, Mathematica, or Matlab for this. What happens to the most likely position to find the particle as time progresses? Does it move? If so, with what frequency does it move? (E, – E,)' 2(E, – E,)' (E,-E,) d. Calculate the average position of the particle, (x) , for the times listed in…