# Theory of Angular Momentum | Week 3

**Course Name: Theory of Angular Momentum**

**Course Link: Theory of Angular Momentum**

#### These are answers of Theory of Angular Momentum Coursera Week 3 Quiz

**Question 1Rotation of spin 1/2 systemUse the following information for Questions 1-6:Using the eigenstates of S**

_{^z}as the basis, construct the two-dimensional vector, (a b) for the eigenstate of S

_{^z}with eigenvalue -ℏ/2.What is the first element a?Express your answer in terms of imaginary unit i and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.

Answer: 0

**Question 2Rotation of spin 1/2 systemWhat is the second element b?Express your answer in terms of imaginary unit i and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.**

Answer: 1

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 3Rotation of spin 1/2 systemWe now use the Euler rotation matrix for spin 1/2 system to rotate the state given in Questions 1-2 such that the original z-axis coincides with the y-axis after rotation.Calculate the resultant vector (c d) after the rotation.What is the first element c?Express your answer in terms of imaginary unit i, constant pi, and reduced Planck’s constant hbar. Your expression does not have to include all these parameters. In order to avoid confusion with the autograder, express your answer in the form of Aexp(iθ) where A and θ are both real numbers.**

Answer:

**Question 4Rotation of spin 1/2 systemWhat is the second element d?Express your answer in terms of imaginary unit i, constant pi, and reduced Planck’s constant hbar. Your expression does not have to include all these parameters. Once again, in order to avoid confusion with the autograder, express your answer in the form of Aexp(iθ) where A and θ are both real numbers.**

Answer:

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 5Rotation of spin 1/2 systemThis time, consider the matrix representation of S**

_{^y}using the Pauli matrices. Find the eigenvector (c′ d′) of the matrix S

_{y} with eigenvalue of −ℏ/2.What is the first element c′?Express your answer in terms of imaginary unit i and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.Note that you can always multiply a phase factor e

^{iθ}to a vector and it would still represent the same quantum state. This is the inherent ambiguity arising from the normalization condition. In this problem, to avoid confusion with the autograder, choose a phase factor that makes c′ a real number.

Answer: 1/sqrt(2)

**Question 6Rotation of spin 1/2 systemThis time, consider the matrix representation of S**

_{^y}using the Pauli matrices. Find the eigenvector (c′ d′) of the matrix S

_{y}with eigenvalue of -ℏ/2.What is the second element d′?Express your answer in terms of imaginary unit i and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.Note that you can always multiply a phase factor e

^{iθ}to a vector and it would still represent the same quantum state. This is the inherent ambiguity arising from the normalization condition. In this problem, to avoid confusion with the autograder, choose a phase factor that makes c′ a real number.

Answer: -i/sqrt(2)

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 7Rotation of spin 1 systemUse the following information for Questions 7-15:Consider a particle with angular momentum j=1. Using the eigenstates of J**

_{^z}as the basis, construct the three-dimensional vector, (a b c) for the eigenstate of J

_{^z}with eigenvalue ℏ.What is the first element a?Express your answer in terms of imaginary unit i, constant pi, and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.

Answer: 1

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 8Rotation of spin 1 systemWhat is the second element b?Express your answer in terms of imaginary unit i, constant pi, and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.**

Answer: 1

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 9Rotation of spin 1 systemWhat is the third element c?Express your answer in terms of imaginary unit i, constant pi, and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.**

Answer: 0

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 10Rotation of spin 1 systemWe now use the Euler rotation matrix for j=1 system to rotate the state given in Questions 7-10 such that the original z-axis coincides with the y-axis after rotation.Calculate the resultant vector (d e f) after the rotation.What is the first element d?Express your answer in terms of imaginary unit i, constant pi, and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.**

Answer:

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 11Rotation of spin 1 systemWhat is the second element e?Express your answer in terms of imaginary unit i, constant pi, and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.**

Answer: 1/sqrt(2)

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 12Rotation of spin 1 systemWhat is the third element f?Express your answer in terms of imaginary unit i, constant pi, and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.**

Answer:

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 13Rotation of spin 1 systemThis time, consider the matrix representation of J**

_{^y}for j=1 system using ∣j,m⟩ basis. Here ∣j,m⟩ are the eigenkets of J

_{^2}and J

_{^z}operators.Find the eigenvector (d′ e′ f′) of the matrix J

_{y}with eigenvalue of ℏ.What is the first element d′?Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.Note that you can always multiply a phase factor e

^{iθ}to a vector and it would still represent the same quantum state. This is the inherent ambiguity arising from the normalization condition. For Q13-15, to avoid confusion with the autograder, choose a phase factor that makes d′ a real number.

Answer: 1/2

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 14Rotation of spin 1 systemWhat is the first element e′?Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.Note that you can always multiply a phase factor e**

^{iθ}to a vector and it would still represent the same quantum state. This is the inherent ambiguity arising from the normalization condition. For Q13-15, to avoid confusion with the autograder, choose a phase factor that makes d′ a real number.

Answer: i/sqrt(2)

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 15Rotation of spin 1 systemWhat is the first element f′?Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.Note that you can always multiply a phase factor e**

^{iθ}to a vector and it would still represent the same quantum state. This is the inherent ambiguity arising from the normalization condition. For Q13-15, to avoid confusion with the autograder, choose a phase factor that makes d′ a real number.

Answer: -1/2

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 16Angular momentum eigenstatesUse the following information for Questions 16-17:A particle in a central potential V(r) was found to be in an eigenstate of L**

^{^2}and L

_{z}with eigenvalues l(l+1)ℏ

^{2}and mℏ, respectively.Calculate the expectation value of L

^{^}

_{x}operator, ⟨L

^{^}

_{x}⟩.Express your answer in terms of quantum numbers l and m, imaginary unit i, constant pi and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.

Answer: 0

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

**Question 17Calculate the expectation value of L ^{^2}_{x} operator, ⟨L^{^2}_{x}⟩.Express your answer in terms of quantum numbers l and m, imaginary unit i, constant pi and reduced Planck’s constant hbar. Your expression does not have to include all these parameters.**

Answer: 1/2hbar^2( (l^2+l)-m^2)

**These are answers of Theory of Angular Momentum Coursera Week 3 Quiz**

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