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 1
Rotation of spin 1/2 system
Use 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 2
Rotation of spin 1/2 system
What 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 3
Rotation of spin 1/2 system
We 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 4
Rotation of spin 1/2 system
What 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 5
Rotation of spin 1/2 system
This time, consider the matrix representation of S^y using the Pauli matrices. Find the eigenvector (c′ d′) of the matrix Sy 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 eiθ 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 6
Rotation of spin 1/2 system
This time, consider the matrix representation of S^y using the Pauli matrices. Find the eigenvector (c′ d′) of the matrix Sy 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 eiθ 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 7
Rotation of spin 1 system
Use 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 8
Rotation of spin 1 system
What 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 9
Rotation of spin 1 system
What 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 10
Rotation of spin 1 system
We 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 11
Rotation of spin 1 system
What 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 12
Rotation of spin 1 system
What 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 13
Rotation of spin 1 system
This 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^zoperators.
Find the eigenvector (d′ e′ f′) of the matrix Jy 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 eiθ 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 14
Rotation of spin 1 system
What 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 eiθ 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 15
Rotation of spin 1 system
What 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 eiθ 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 16
Angular momentum eigenstates
Use 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 Lz with eigenvalues l(l+1)ℏ2 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 17
Calculate the expectation value of L^2x operator, ⟨L^2x⟩.
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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