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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: Y(x,t)=, 27x - sin 3Tx where E, 2ma а 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,r) = ¥° (x,t) Y(x,t) and sketch what this looks like for t=0, 3th , and 1 = (E,-E.) 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.) d. Calculate the average position of the particle, (x), for the times listed in part c. Does it move or stay the same as time progresses? Alternatively, use Ehrenfest theorem to show that (x) is time dependent. e. Let's say that you measure the energy of your system, and it returns a value of E,. Show that the probability of getting this energy is time independent by calculating (43|4). (Þ3| is the bra for the n = 3 state. Note that (3| is time dependent.

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: Y(x,t)=, 27x - sin 3Tx where E, 2ma а 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,r) = ¥° (x,t) Y(x,t) and sketch what this looks like for t=0, 3th , and 1 = (E,-E.) 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.) d. Calculate the average position of the particle, (x), for the times listed in part c. Does it move or stay the same as time progresses? Alternatively, use Ehrenfest theorem to show that (x) is time dependent. e. Let's say that you measure the energy of your system, and it returns a value of E,. Show that the probability of getting this energy is time independent by calculating (43|4). (Þ3| is the bra for the n = 3 state. Note that (3| is time dependent.

Chemistry
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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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 part c. Does it move or stay the same as time progresses? Alternatively, use Ehrenfest theorem to show that (x) is time dependent. e. Let's say that you measure the energy of your system, and it returns a value of E,. Show that the probability of getting this energy is time independent by calculating (µ3|4). (Þ3| is the bra for the n = 3 state. Note that (3| is time dependent.
Transcribed Image Text: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 part c. Does it move or stay the same as time progresses? Alternatively, use Ehrenfest theorem to show that (x) is time dependent. e. Let's say that you measure the energy of your system, and it returns a value of E,. Show that the probability of getting this energy is time independent by calculating (µ3|4). (Þ3| is the bra for the n = 3 state. Note that (3| is time dependent.
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