1. (a) A particle with spin 1 has orbital angular momentum l = 0.
What are the possible values for the total angular momentum quantum
number j?(b) The same particle has l = 3. What are the possible values
for j? Get solution
2. (a) A particle has spin 3/2 and orbital angular momentum l = 1. What are the possible values for the total angular momentum quantum number j?(b) For each value of j in part (a), determine the possible values of mj. Get solution
3. Determine (using the matrix representation) which of the following operators are Hermitian and which are not: Sx, Sy, Sz, S+, S−. Get solution
4. Derive the eigenvalues and the corresponding normalized eigenvectors of Sy given in Equations (8.24) and (8.25). Get solution
5. A particle has spin 1, so that ms = − 1, 0, or 1. Derive the matrices which correspond to Sx, Sy, and Sz. Get solution
6. (a) A particle has s = 3/2. The operator S++ is defined to be the square of the raising operator: S++ = (S+)2, where S+ is the usual raising operator:...Derive the matrix corresponding to the operator S++.(b) What is the matrix corresponding to the adjoint operator (S++)†? Get solution
7. Let the operator Q be given by Q = S+S−, where S+ and S− are the usual raising and lowering operators:...Derive the matrix corresponding to the operator Q for a spin 1 particle. Determine whether or not Q is Hermitian. Get solution
8. Using the matrix representation of the spin operators, verify the results for [Sx, Sy], [Sy, Sz], and [SZ, Sx] given in Equations (8.1)–(8.3). Get solution
9. A large number of spin-1/2 particles are run through a Stem-Gerlach machine. When they emerge, all of the particles have the same spin wave function ... (in the usual basis in which ... represents spin in the +z direction, and ... represents spin in the −z direction). The spin of these particles is measured in the z direction. On average, 9/25 of the particles have spin in the +z direction, and 16/25 have spin in the − z direction.(a) Determine a possible normalized spin wave function ....(b) Is there a single unique solution to part (a), a finite number of different solutions, or an infinite number of different solutions? (Multiplying the entire wave vector by a constant does not count as a different solution.) Get solution
10. A Stem-Gerlach experiment is set up with the axis of the inhomogeneous magnetic field in the x-yplane, at an angle θ relative to the x-axis. Call this direction r:...The spin operator in the r direction is...(a) For a spin-1/2 particle, calculate the matrix corresponding to Sr. Calculate the eigenvalues and corresponding eigenvectors. Normalize the eigenvectors and express them in the form a| ↑〉 + b|↓〉, where a and b are constants.(b) Suppose a measurement of the spin of the particle in the r direction is made and it is determined that the spin is in the positive r direction, i.e., Sr|ψ〉 = (+ħ/2)|ψ〉. Now a second measurement is made to determine msx (the component of the spin in the x direction). What is the probability that msx = −1/2? Suppose that instead of measuring msx, the z component of the spin ms is measured. What is the probability that ms = +1/2?(c) Suppose that the particle has spin in the positive r direction as in part (b). The z component of the spin is measured and it is discovered that ms = +1/2. Now a third measurement is made to determine msx. What is the probability that msx = −1/2? Get solution
11. A spin-1/2 particle is in the state ....(a) Verify that the wave function is correctly normalized.(b) A measurement is made of the x component of the spin. What is the probability that the spin will be in the − x direction?(c) Suppose a measurement is made of the spin in the z direction and it is discovered that the particle has ms = −1/2. Now a second measurement is made to determine the spin in the x direction. What is the probability that the spin will be in the +x direction? Get solution
12. An electron is precessing in a magnetic field. The wave function for the electron is...(a) Describe the plane of rotation of this particle.(b) In what direction is it rotating in this plane? Get solution
13. A magnetic field pointing in the − z direction produces a Hamiltonian H = −ωSz, where ω is a constant with units of frequency. A spin-1/2 particle is placed in this magnetic field. At t = 0, the particle is pointing in the +y direction.(a) Derive an expression for the spin vector ... as a function of time.(b) At t = π/ω, a measurement is made of the spin in the x direction. What is the probability that the spin is in the +x direction?(c) Suppose that at t = π/ω, a measurement is made of the spin in the x direction, and it is found that the spin is in the +x direction. Then at the time t = 2π/ω, another measurement is made of the spin in the x direction. What is the probability that the spin is in the +x direction? Get solution
14. An electron is precessing in a magnetic field aligned along the + z-axis. At t = 0, the spin of the electron is in the positive x direction. The wave function is...For t > 0, calculate the probability of finding the electron in the state(a) ms = +ħ/2(b) msx = +ħ/2 Get solution
15. A spin-1/2 particle is placed in a magnetic field pointing in the +x direction which produces a Hamiltonian H = ωSx, where ω is a constant with units of frequency. At t = 0, the particle is pointing in the +z direction. Derive an expression for the spin vector ... as a function of time. Get solution
16. A magnetic field pointing in the +z direction produces a Hamiltonian H = ωSz, where ω is a constant with units of frequency. A spin-1 particle is placed in this magnetic field. The matrix corresponding to Sz for a spin-1 particle is...At t = 0, the particle is pointing in the +x direction with normalized spin vector...Derive an expression for the spin vector ... as a function of time. Get solution
17. Consider a system of two particles: particle 1 has spin 1, and particle 2 has spin 1 /2. Let S be the total angular momentum operator for the two particles, where the eigenvalues of S2 and Sz are ħ2s(s + 1) and ħms, respectively. The particles are in the state s = 3/2 and ms = 1/2.(a) Calculate the wave function |s = 3/2 ms = 1/2) as a linear combination of the wave functions |m1s m2s〉, where m1s is the z component of the spin of particle 1, and is the z component of the spin of particle 2.(b) Find the probabilities that the z component of the spin of particle 1 isi. m1s= +1ii. m1s = 0iii. m1s = −1 Get solution
18. Suppose that particle 1 (with spin 1) and particle 2 (with spin 1/2) interact via the Hamiltonian operator...where λ is a constant. Calculate the energy of the state |s ms〉. Get solution
19. Two spin-1/2 particles are fixed in space with the Hamiltonian...where a and b are constants, and as usual, S2 is the total spin operator squared and Sz is the operator which gives the z component of the total spin. What are the energy levels of this system? Get solution
2. (a) A particle has spin 3/2 and orbital angular momentum l = 1. What are the possible values for the total angular momentum quantum number j?(b) For each value of j in part (a), determine the possible values of mj. Get solution
3. Determine (using the matrix representation) which of the following operators are Hermitian and which are not: Sx, Sy, Sz, S+, S−. Get solution
4. Derive the eigenvalues and the corresponding normalized eigenvectors of Sy given in Equations (8.24) and (8.25). Get solution
5. A particle has spin 1, so that ms = − 1, 0, or 1. Derive the matrices which correspond to Sx, Sy, and Sz. Get solution
6. (a) A particle has s = 3/2. The operator S++ is defined to be the square of the raising operator: S++ = (S+)2, where S+ is the usual raising operator:...Derive the matrix corresponding to the operator S++.(b) What is the matrix corresponding to the adjoint operator (S++)†? Get solution
7. Let the operator Q be given by Q = S+S−, where S+ and S− are the usual raising and lowering operators:...Derive the matrix corresponding to the operator Q for a spin 1 particle. Determine whether or not Q is Hermitian. Get solution
8. Using the matrix representation of the spin operators, verify the results for [Sx, Sy], [Sy, Sz], and [SZ, Sx] given in Equations (8.1)–(8.3). Get solution
9. A large number of spin-1/2 particles are run through a Stem-Gerlach machine. When they emerge, all of the particles have the same spin wave function ... (in the usual basis in which ... represents spin in the +z direction, and ... represents spin in the −z direction). The spin of these particles is measured in the z direction. On average, 9/25 of the particles have spin in the +z direction, and 16/25 have spin in the − z direction.(a) Determine a possible normalized spin wave function ....(b) Is there a single unique solution to part (a), a finite number of different solutions, or an infinite number of different solutions? (Multiplying the entire wave vector by a constant does not count as a different solution.) Get solution
10. A Stem-Gerlach experiment is set up with the axis of the inhomogeneous magnetic field in the x-yplane, at an angle θ relative to the x-axis. Call this direction r:...The spin operator in the r direction is...(a) For a spin-1/2 particle, calculate the matrix corresponding to Sr. Calculate the eigenvalues and corresponding eigenvectors. Normalize the eigenvectors and express them in the form a| ↑〉 + b|↓〉, where a and b are constants.(b) Suppose a measurement of the spin of the particle in the r direction is made and it is determined that the spin is in the positive r direction, i.e., Sr|ψ〉 = (+ħ/2)|ψ〉. Now a second measurement is made to determine msx (the component of the spin in the x direction). What is the probability that msx = −1/2? Suppose that instead of measuring msx, the z component of the spin ms is measured. What is the probability that ms = +1/2?(c) Suppose that the particle has spin in the positive r direction as in part (b). The z component of the spin is measured and it is discovered that ms = +1/2. Now a third measurement is made to determine msx. What is the probability that msx = −1/2? Get solution
11. A spin-1/2 particle is in the state ....(a) Verify that the wave function is correctly normalized.(b) A measurement is made of the x component of the spin. What is the probability that the spin will be in the − x direction?(c) Suppose a measurement is made of the spin in the z direction and it is discovered that the particle has ms = −1/2. Now a second measurement is made to determine the spin in the x direction. What is the probability that the spin will be in the +x direction? Get solution
12. An electron is precessing in a magnetic field. The wave function for the electron is...(a) Describe the plane of rotation of this particle.(b) In what direction is it rotating in this plane? Get solution
13. A magnetic field pointing in the − z direction produces a Hamiltonian H = −ωSz, where ω is a constant with units of frequency. A spin-1/2 particle is placed in this magnetic field. At t = 0, the particle is pointing in the +y direction.(a) Derive an expression for the spin vector ... as a function of time.(b) At t = π/ω, a measurement is made of the spin in the x direction. What is the probability that the spin is in the +x direction?(c) Suppose that at t = π/ω, a measurement is made of the spin in the x direction, and it is found that the spin is in the +x direction. Then at the time t = 2π/ω, another measurement is made of the spin in the x direction. What is the probability that the spin is in the +x direction? Get solution
14. An electron is precessing in a magnetic field aligned along the + z-axis. At t = 0, the spin of the electron is in the positive x direction. The wave function is...For t > 0, calculate the probability of finding the electron in the state(a) ms = +ħ/2(b) msx = +ħ/2 Get solution
15. A spin-1/2 particle is placed in a magnetic field pointing in the +x direction which produces a Hamiltonian H = ωSx, where ω is a constant with units of frequency. At t = 0, the particle is pointing in the +z direction. Derive an expression for the spin vector ... as a function of time. Get solution
16. A magnetic field pointing in the +z direction produces a Hamiltonian H = ωSz, where ω is a constant with units of frequency. A spin-1 particle is placed in this magnetic field. The matrix corresponding to Sz for a spin-1 particle is...At t = 0, the particle is pointing in the +x direction with normalized spin vector...Derive an expression for the spin vector ... as a function of time. Get solution
17. Consider a system of two particles: particle 1 has spin 1, and particle 2 has spin 1 /2. Let S be the total angular momentum operator for the two particles, where the eigenvalues of S2 and Sz are ħ2s(s + 1) and ħms, respectively. The particles are in the state s = 3/2 and ms = 1/2.(a) Calculate the wave function |s = 3/2 ms = 1/2) as a linear combination of the wave functions |m1s m2s〉, where m1s is the z component of the spin of particle 1, and is the z component of the spin of particle 2.(b) Find the probabilities that the z component of the spin of particle 1 isi. m1s= +1ii. m1s = 0iii. m1s = −1 Get solution
18. Suppose that particle 1 (with spin 1) and particle 2 (with spin 1/2) interact via the Hamiltonian operator...where λ is a constant. Calculate the energy of the state |s ms〉. Get solution
19. Two spin-1/2 particles are fixed in space with the Hamiltonian...where a and b are constants, and as usual, S2 is the total spin operator squared and Sz is the operator which gives the z component of the total spin. What are the energy levels of this system? Get solution
