1. Find the eigenvalues and eigenvectors of ... Suppose an electron
is in the spin state ... If Sy is measured, what is the probability of
the result ℏ/2? Get solution
2. Consider the 2 × 2 matrix defined by...where a0 is a real number and a is a three-dimensional vector with real components.a. Prove that U is unitary and unimodular.b. In general, a 2 × 2 unitary unimodular matrix represents a rotation in three dimensions. Find the axis and angle of rotation appropriate for U in terms of a0, a1a2, and a3. Get solution
3. The spin-dependent Hamiltonian of an electron-positron system in the presence of a uniform magnetic field in the z-direction can be written as...Suppose the spin function of the system is given by ...a. Is this an eigenfunction of H in the limit A →0, eB/mc ≠ 0? If it is, what is the energy eigenvalue? If it is not, what is the expectation value of H?b. Same problem when eB/mc → 0, A ≠ 0. Get solution
4. Consider a spin 1 particle. Evaluate the matrix elements of... Get solution
5. Let the Hamiltonian of a rigid body be...where K is the angular momentum in the body frame. From this expression obtain the Heisenberg equation of motion for K and then find Euler’s equation of motion in the correspondence limit. Get solution
6. Let ... where (α, β, γ) are the Eulerian angles. In order that U represent a rotation (α, β, γ), what are the commutation rules satisfied by the Gk?Relate G to the angular momentum operators. Get solution
7. What is the meaning of the following equation:...where the three components of A are matrices? From this equation show that matrix elements ⟨m|Ak|n⟩ transform like vectors. Get solution
8. Consider a sequence of Euler rotations represented by...Because of the group properties of rotations, we expect that this sequence of operations is equivalent to a single rotation about some axis by an angle θ. Find θ. Get solution
10. a. Prove that the time evolution of the density operator p (in the Schrödinger picture) is given by...b. Suppose we have a pure ensemble at t = 0. Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrödinger equation. Get solution
11. Consider an ensemble of spin 1 systems. The density matrix is now a 3 × 3 matrix. How many independent (real) parameters are needed to characterize the density matrix? What must we know in addition to [Sx], [Sy], and [Sz] to characterize the ensemble completely? Get solution
12. An angular-momentum eigenstate |j, m = mmax = j⟩ is rotated by an infinitesimal angle ε about the y-axis. Without using the explicit form of the ... function, obtain an expression for the probability for the new rotated state to be found in the original state up to terms of order ε2. Get solution
13. Show that the 3 × 3 matrices G, (i = 1, 2, 3) whose elements are given by...where j and k are the row and column indices, satisfy the angular momentum commutation relations. What is the physical (or geometric) significance of the transformation matrix that connects Gi, to the more usual 3 × 3 representations of the angular-momentum operator Ji with J3 taken to be diagonal? Relate your result to...under infinitesimal rotations. (Note: This problem may be helpful in understanding the photon spin.) Get solution
14. a. Let J be angular momentum. It may stand for orbital L, spin S, or Jtotal.) Using the fact that Jx, Jy, Jz (J± = Jx + iJy) satisfy the usual angular-momentum commutation relations, prove...b. Using (a) (or otherwise), derive the “famous” expression for the coefficient c_ that appears in... Get solution
15. The wave function of a particle subjected to a spherically symmetrical potential V(r) is given by...a. Is ψ an eigenfunction of L2? If so, what is the l-value? If not, what are the possible values of l we may obtain when L2 is measured?b. What are the probabilities for the particle to be found in various ml states?c. Suppose it is known somehow that ψ(x) is an energy eigenfunction with eigenvalue E. Indicate how we may find V(r). Get solution
16. A particle in a spherically symmetrical potential is known to be in an eigenstate of L2 and Lz with eigenvalues ℏ2l(l + 1) and mℏ, respectively. Prove that the expectation values between |lm⟩ states satisfy...Interpret this result semiclassically. Get solution
17. Suppose a half-integer l-value, say ..., were allowed for orbital angular momentum. From...we may deduce, as usual,...Now try to construct Y1/2,−1/2(θ, ϕ); by (a) applying L_to Y1/2,−1/2(θ, ϕ); and (b) using L_Y1/2,−1/2(θ, ϕ) = 0. Show that the two procedures lead to contradictory results. (This gives an argument against half-integer l-values for orbital angular momentum.) Get solution
18. Consider an orbital angular-momentum eigenstate |l = 2, m = 0⟩. Suppose this state is rotated by an angle β about the y-axis. Find the probability for the new state to be found in m = 0, ±1, and ±2. (The spherical harmonics for l = 0, 1, and 2 given in Appendix A may be useful.) Get solution
19. What is the physical significance of the operators...in Schwinger’s scheme for angular momentum? Give the nonvanishing matrix elements of K±. Get solution
20. We are to add angular momenta j1 = 1 and j2 = 1 to form j = 2, 1, and 0 states. Using either the ladder operator method or the recursion relation, express all (nine) {j, m}eigenkets in terms of |j1j2;m1m2⟩. Write your answer as...where + and 0 stand for m1,2 = 1,0, respectively. Get solution
21. a. Evaluate...for any j (integer or half-integer); then check your answer for ...b. Prove, for any j,...[Hint: This can be proved in many ways. You may, for instance, examine the rotational properties of ... using the spherical (irreducible) tensor language.] Get solution
22. a. Consider a system with j = 1. Explicitly write...in 3 × 3 matrix form.b. Show that for j = 1 only, it is legitimate to replace ... by...c. Using (b), prove... Get solution
23. Express the matrix element ... in terms of a series in... Get solution
24. Consider a system made up of two spin ... particles. Observer A specializes in measuring the spin components of one of the particles (s1z, s1x, and so on), while observer B measures the spin components of the other partible. Suppose the system is known to be in a spin-singlet state, that is, Stotal= 0.a. What is the probability for observer A to obtain s1z= ℏ/2 when observer B makes no measurement? Same problem for s1x= ℏ/2.b. Observer B determines the spin of particle 2 to be in the s2z=ℏ/2 state with certainty. What can we then conclude about the outcome of observer A’s measurement if (i) A measures s1z and (ii) A measures s1x? Justify your answer. Get solution
25. Consider a spherical tensor of rank 1 (that is, a vector)...Using the expression for d(j = 1) given in Problem 22, evaluate...and show that your results are just what you expect from the transformation properties of Vx, y, z under rotations about the y-axis. Get solution
26. a. Construct a spherical tensor of rank 1 out of two different vectors U = (Ux, Uy, Uz) and V = (Vx, Vy, Vz). Explicitly write ... in terms of Ux, y, z and Vx, y, z.b. Construct a spherical tensor of rank 2 out of two different vectors U and V. Write down explicitly ... in terms of Ux, y, z and Vx, y, z. Get solution
27. Consider a spinless particle bound to a fixed center by a central force potential.a. Relate, as much as possible, the matrix elements...using only the Wigner-Eckart theorem. Make sure to state under what conditions the matrix elements are nonvanishing.b. Do the same problem using wave functions ... Get solution
28. a. Write xy, xz, and (x2− y2) as components of a spherical (irreducible) tensor of rank 2.b. The expectation value...is known as the quadrupole moment. Evaluate...(where m′ = j, j −1, j −2,...) in terms of Q and appropriate Clebsch-Gordan coefficients. Get solution
29. A spin ... nucleus situated at the origin is subjected to an external inhomogeneous electric field. The basic electric quadrupole interaction may by taken to be...where ϕ is the electrostatic potential satisfying Laplace’s equation and the coordinate axes are so chosen that...Show that the interaction energy can be written as...and express A and B in terms of (∂2ϕ/∂x2)0 and so on. Determine the energy eigenkets (in terms of |m⟩, where m = ± ..., ± ...) and the corresponding energy eigenvalues. Is there any degeneracy? Get solution
2. Consider the 2 × 2 matrix defined by...where a0 is a real number and a is a three-dimensional vector with real components.a. Prove that U is unitary and unimodular.b. In general, a 2 × 2 unitary unimodular matrix represents a rotation in three dimensions. Find the axis and angle of rotation appropriate for U in terms of a0, a1a2, and a3. Get solution
3. The spin-dependent Hamiltonian of an electron-positron system in the presence of a uniform magnetic field in the z-direction can be written as...Suppose the spin function of the system is given by ...a. Is this an eigenfunction of H in the limit A →0, eB/mc ≠ 0? If it is, what is the energy eigenvalue? If it is not, what is the expectation value of H?b. Same problem when eB/mc → 0, A ≠ 0. Get solution
4. Consider a spin 1 particle. Evaluate the matrix elements of... Get solution
5. Let the Hamiltonian of a rigid body be...where K is the angular momentum in the body frame. From this expression obtain the Heisenberg equation of motion for K and then find Euler’s equation of motion in the correspondence limit. Get solution
6. Let ... where (α, β, γ) are the Eulerian angles. In order that U represent a rotation (α, β, γ), what are the commutation rules satisfied by the Gk?Relate G to the angular momentum operators. Get solution
7. What is the meaning of the following equation:...where the three components of A are matrices? From this equation show that matrix elements ⟨m|Ak|n⟩ transform like vectors. Get solution
8. Consider a sequence of Euler rotations represented by...Because of the group properties of rotations, we expect that this sequence of operations is equivalent to a single rotation about some axis by an angle θ. Find θ. Get solution
10. a. Prove that the time evolution of the density operator p (in the Schrödinger picture) is given by...b. Suppose we have a pure ensemble at t = 0. Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrödinger equation. Get solution
11. Consider an ensemble of spin 1 systems. The density matrix is now a 3 × 3 matrix. How many independent (real) parameters are needed to characterize the density matrix? What must we know in addition to [Sx], [Sy], and [Sz] to characterize the ensemble completely? Get solution
12. An angular-momentum eigenstate |j, m = mmax = j⟩ is rotated by an infinitesimal angle ε about the y-axis. Without using the explicit form of the ... function, obtain an expression for the probability for the new rotated state to be found in the original state up to terms of order ε2. Get solution
13. Show that the 3 × 3 matrices G, (i = 1, 2, 3) whose elements are given by...where j and k are the row and column indices, satisfy the angular momentum commutation relations. What is the physical (or geometric) significance of the transformation matrix that connects Gi, to the more usual 3 × 3 representations of the angular-momentum operator Ji with J3 taken to be diagonal? Relate your result to...under infinitesimal rotations. (Note: This problem may be helpful in understanding the photon spin.) Get solution
14. a. Let J be angular momentum. It may stand for orbital L, spin S, or Jtotal.) Using the fact that Jx, Jy, Jz (J± = Jx + iJy) satisfy the usual angular-momentum commutation relations, prove...b. Using (a) (or otherwise), derive the “famous” expression for the coefficient c_ that appears in... Get solution
15. The wave function of a particle subjected to a spherically symmetrical potential V(r) is given by...a. Is ψ an eigenfunction of L2? If so, what is the l-value? If not, what are the possible values of l we may obtain when L2 is measured?b. What are the probabilities for the particle to be found in various ml states?c. Suppose it is known somehow that ψ(x) is an energy eigenfunction with eigenvalue E. Indicate how we may find V(r). Get solution
16. A particle in a spherically symmetrical potential is known to be in an eigenstate of L2 and Lz with eigenvalues ℏ2l(l + 1) and mℏ, respectively. Prove that the expectation values between |lm⟩ states satisfy...Interpret this result semiclassically. Get solution
17. Suppose a half-integer l-value, say ..., were allowed for orbital angular momentum. From...we may deduce, as usual,...Now try to construct Y1/2,−1/2(θ, ϕ); by (a) applying L_to Y1/2,−1/2(θ, ϕ); and (b) using L_Y1/2,−1/2(θ, ϕ) = 0. Show that the two procedures lead to contradictory results. (This gives an argument against half-integer l-values for orbital angular momentum.) Get solution
18. Consider an orbital angular-momentum eigenstate |l = 2, m = 0⟩. Suppose this state is rotated by an angle β about the y-axis. Find the probability for the new state to be found in m = 0, ±1, and ±2. (The spherical harmonics for l = 0, 1, and 2 given in Appendix A may be useful.) Get solution
19. What is the physical significance of the operators...in Schwinger’s scheme for angular momentum? Give the nonvanishing matrix elements of K±. Get solution
20. We are to add angular momenta j1 = 1 and j2 = 1 to form j = 2, 1, and 0 states. Using either the ladder operator method or the recursion relation, express all (nine) {j, m}eigenkets in terms of |j1j2;m1m2⟩. Write your answer as...where + and 0 stand for m1,2 = 1,0, respectively. Get solution
21. a. Evaluate...for any j (integer or half-integer); then check your answer for ...b. Prove, for any j,...[Hint: This can be proved in many ways. You may, for instance, examine the rotational properties of ... using the spherical (irreducible) tensor language.] Get solution
22. a. Consider a system with j = 1. Explicitly write...in 3 × 3 matrix form.b. Show that for j = 1 only, it is legitimate to replace ... by...c. Using (b), prove... Get solution
23. Express the matrix element ... in terms of a series in... Get solution
24. Consider a system made up of two spin ... particles. Observer A specializes in measuring the spin components of one of the particles (s1z, s1x, and so on), while observer B measures the spin components of the other partible. Suppose the system is known to be in a spin-singlet state, that is, Stotal= 0.a. What is the probability for observer A to obtain s1z= ℏ/2 when observer B makes no measurement? Same problem for s1x= ℏ/2.b. Observer B determines the spin of particle 2 to be in the s2z=ℏ/2 state with certainty. What can we then conclude about the outcome of observer A’s measurement if (i) A measures s1z and (ii) A measures s1x? Justify your answer. Get solution
25. Consider a spherical tensor of rank 1 (that is, a vector)...Using the expression for d(j = 1) given in Problem 22, evaluate...and show that your results are just what you expect from the transformation properties of Vx, y, z under rotations about the y-axis. Get solution
26. a. Construct a spherical tensor of rank 1 out of two different vectors U = (Ux, Uy, Uz) and V = (Vx, Vy, Vz). Explicitly write ... in terms of Ux, y, z and Vx, y, z.b. Construct a spherical tensor of rank 2 out of two different vectors U and V. Write down explicitly ... in terms of Ux, y, z and Vx, y, z. Get solution
27. Consider a spinless particle bound to a fixed center by a central force potential.a. Relate, as much as possible, the matrix elements...using only the Wigner-Eckart theorem. Make sure to state under what conditions the matrix elements are nonvanishing.b. Do the same problem using wave functions ... Get solution
28. a. Write xy, xz, and (x2− y2) as components of a spherical (irreducible) tensor of rank 2.b. The expectation value...is known as the quadrupole moment. Evaluate...(where m′ = j, j −1, j −2,...) in terms of Q and appropriate Clebsch-Gordan coefficients. Get solution
29. A spin ... nucleus situated at the origin is subjected to an external inhomogeneous electric field. The basic electric quadrupole interaction may by taken to be...where ϕ is the electrostatic potential satisfying Laplace’s equation and the coordinate axes are so chosen that...Show that the interaction energy can be written as...and express A and B in terms of (∂2ϕ/∂x2)0 and so on. Determine the energy eigenkets (in terms of |m⟩, where m = ± ..., ± ...) and the corresponding energy eigenvalues. Is there any degeneracy? Get solution
