Theory of Angular Momentum | Week 2

Course Name: Theory of Angular Momentum

Course Link: Theory of Angular Momentum

These are answers of Theory of Angular Momentum Coursera Week 2 Quiz

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Question 1
Spinor
Use the following information for Questions 1-3:
An electron is in a state specified by spinor(ab​).
When z-component of spin is measured, what is the probability of measuring +ℏ/2​?
Express your answers in terms of a, b, imaginary unit i and reduced Planck’s constant hbar.

Answer: a^2

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Question 2
Spinor
When x-component of spin is measured, what is the probability of measuring +ℏ/2​?
Express your answers in terms of a, b, imaginary unit i and reduced Planck’s constant hbar.

Answer: (a+b)^2/2

Question 3
Spinor
When y-component of spin is measured, what is the probability of measuring +ℏ/2?
Express your answers in terms of a, b, imaginary unit i and reduced Planck’s constant hbar.

Answer:

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These are answers of Theory of Angular Momentum Coursera Week 2 Quiz

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Question 4
Spin 1/2 system
Use this information to answer Question 4-9:
An electron was found in the eigenstate of S_^{^}x​ with eigenvalue of +ℏ/2​.
Using the two eigenkets of S_^{^}z​ as the basis, express the spin state in the two-component spinor representation, (a b​).
Enter the expression for the first element a in terms of imaginary unit, i, and reduced Planck’s constant hbar. Note that your expression may not include all of these parameters.

Answer: 1/sqrt(2)

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These are answers of Theory of Angular Momentum Coursera Week 2 Quiz

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Question 5
Spin 1/2 system
Enter the expression for the second element b in terms of imaginary unit, i, and reduced Planck’s constant hbar. Note that your expression may not include all of these parameters.

Answer: 1/sqrt(2)

Question 6
Spin 1/2 system
We now rotate the state by 180 degrees about z -axis. Express the resulting state in spinor, (c d​).
Enter the expression for the first element c in terms of imaginary unit, i, and reduced Planck’s constant hbar. Note that your expression may not include all of these parameters.

Answer: -1/sqrt(2)*i

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These are answers of Theory of Angular Momentum Coursera Week 2 Quiz

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Question 7
Spin 1/2 system
Enter the expression for the second element d in terms of imaginary unit, i, and reduced Planck’s constant hbar. Note that your expression may not include all of these parameters.

Answer: 1/sqrt(2)*i

Question 8
Spin 1/2 system
Suppose we measured the x-component of spin again after the rotation. What is the probability of measuring +ℏ/2?

Answer: 0.000163

Question 9
Spin 1/2 system
This time, we rotate the original state given in Question 4 by 360 degrees about z-axis.
What is the probability of measuring +ℏ2+2ℏ​ for the x-component of spin after the rotation?

Answer: 1.00003

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These are answers of Theory of Angular Momentum Coursera Week 2 Quiz

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Question 10
Spinor representation
Use the following information for Questions 10-11:
n^{^} is the unit vector in the yz plane that bisect the angle between y and z axes. That is, n^{^} makes a 45 degree angle with respect to both y and z axes.
Write down the 2×2 matrix representation for σ⋅n^{^} and find the spinor for the eigenvector of σ⋅n^{^} with eigenvalue −1. Here σ is a vector whose x-, y-, z-components are given by the Pauli’s spin matrices.
Express your answer in a two-dimensional vector form, (a ib​), where a and b are real numbers and a>b. What is the value of a?

Answer: 0.3827

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These are answers of Theory of Angular Momentum Coursera Week 2 Quiz

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Question 11
Spinor representation
What is the value of b?

Answer: -0.9239

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These are answers of Theory of Angular Momentum Coursera Week 2 Quiz

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