These are answers of Theory of Angular Momentum Coursera Week 2 Quiz
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
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:
These are answers of Theory of Angular Momentum Coursera Week 2 Quiz
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)
These are answers of Theory of Angular Momentum Coursera Week 2 Quiz
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.
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
These are answers of Theory of Angular Momentum Coursera Week 2 Quiz
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
These are answers of Theory of Angular Momentum Coursera Week 2 Quiz
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?
