P18.2 In this problem you will derive the commutator [Îx, Îy] = iħl₂. a. The angular momentum vector in three dimensions has the form l ilx + jly + kl₂ where the unit vectors in the x, y, and z directions are denoted by i, j, and k. Determine lx, ly, and I by expanding the 3 × 3 cross product 1 = r × p. The vectors r and p are given by r = ix + jy + kz and p = ipx + jpy + kpz. = b. Substitute the operators for position and momentum in your expressions for 1, and ly. Always write the position operator to the left of the momentum operator in a simple product of the two. c. Show that [Îx, Îy] = iħÎ₂.
P18.2 In this problem you will derive the commutator [Îx, Îy] = iħl₂. a. The angular momentum vector in three dimensions has the form l ilx + jly + kl₂ where the unit vectors in the x, y, and z directions are denoted by i, j, and k. Determine lx, ly, and I by expanding the 3 × 3 cross product 1 = r × p. The vectors r and p are given by r = ix + jy + kz and p = ipx + jpy + kpz. = b. Substitute the operators for position and momentum in your expressions for 1, and ly. Always write the position operator to the left of the momentum operator in a simple product of the two. c. Show that [Îx, Îy] = iħÎ₂.
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...
Related questions
Concept explainers
Atomic Structure
The basic structure of an atom is defined as the component-level of atomic structure of an atom. Precisely speaking an atom consists of three major subatomic particles which are protons, neutrons, and electrons. Many theories have been stated for explaining the structure of an atom.
Shape of the D Orbital
Shapes of orbitals are an approximate representation of boundaries in space for finding electrons occupied in that respective orbital. D orbitals are known to have a clover leaf shape or dumbbell inside where electrons can be found.
Question
![**P18.2** In this problem you will derive the commutator \([ \hat{L}_x, \hat{L}_y ] = i \hbar \hat{L}_z\).
**a.** The angular momentum vector in three dimensions has the form \(\mathbf{L} = i L_x + j L_y + k L_z\) where the unit vectors in the \(x, y, \) and \(z\) directions are denoted by \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\). Determine \(L_x, L_y,\) and \(L_z\) by expanding the \(3 \times 3\) cross product \(\mathbf{L} = \mathbf{r} \times \mathbf{p}\). The vectors \(\mathbf{r}\) and \(\mathbf{p}\) are given by \(\mathbf{r} = i x + j y + k z\) and \(\mathbf{p} = i p_x + j p_y + k p_z\).
**b.** Substitute the operators for position and momentum in your expressions for \(L_x\) and \(L_y\). Always write the position operator to the left of the momentum operator in a simple product of the two.
**c.** Show that \([ \hat{L}_x, \hat{L}_y ] = i \hbar \hat{L}_z\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78fbc2f9-5ce5-4aa9-99e1-1139be15f6b7%2F49fafc16-9007-4582-9228-025ce46d05da%2F3wgengs_processed.png&w=3840&q=75)
Transcribed Image Text:**P18.2** In this problem you will derive the commutator \([ \hat{L}_x, \hat{L}_y ] = i \hbar \hat{L}_z\).
**a.** The angular momentum vector in three dimensions has the form \(\mathbf{L} = i L_x + j L_y + k L_z\) where the unit vectors in the \(x, y, \) and \(z\) directions are denoted by \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\). Determine \(L_x, L_y,\) and \(L_z\) by expanding the \(3 \times 3\) cross product \(\mathbf{L} = \mathbf{r} \times \mathbf{p}\). The vectors \(\mathbf{r}\) and \(\mathbf{p}\) are given by \(\mathbf{r} = i x + j y + k z\) and \(\mathbf{p} = i p_x + j p_y + k p_z\).
**b.** Substitute the operators for position and momentum in your expressions for \(L_x\) and \(L_y\). Always write the position operator to the left of the momentum operator in a simple product of the two.
**c.** Show that \([ \hat{L}_x, \hat{L}_y ] = i \hbar \hat{L}_z\).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 7 steps with 6 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, chemistry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Chemistry
Chemistry
ISBN:
9781305957404
Author:
Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781259911156
Author:
Raymond Chang Dr., Jason Overby Professor
Publisher:
McGraw-Hill Education
Principles of Instrumental Analysis
Chemistry
ISBN:
9781305577213
Author:
Douglas A. Skoog, F. James Holler, Stanley R. Crouch
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781305957404
Author:
Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781259911156
Author:
Raymond Chang Dr., Jason Overby Professor
Publisher:
McGraw-Hill Education
Principles of Instrumental Analysis
Chemistry
ISBN:
9781305577213
Author:
Douglas A. Skoog, F. James Holler, Stanley R. Crouch
Publisher:
Cengage Learning
Organic Chemistry
Chemistry
ISBN:
9780078021558
Author:
Janice Gorzynski Smith Dr.
Publisher:
McGraw-Hill Education
Chemistry: Principles and Reactions
Chemistry
ISBN:
9781305079373
Author:
William L. Masterton, Cecile N. Hurley
Publisher:
Cengage Learning
Elementary Principles of Chemical Processes, Bind…
Chemistry
ISBN:
9781118431221
Author:
Richard M. Felder, Ronald W. Rousseau, Lisa G. Bullard
Publisher:
WILEY