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. Find, by explicit construction using Pauli matrices, the eigenvalues for the Hamiltonian...for a spin ... particle in the presence of a magnetic field .... Get solution
3. 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, a1, a2, and a3. Get solution
4. 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) Solve the same problem when eB / mc → 0, A ≠ 0. Get solution
5. Consider a spin 1 particle. Evaluate the matrix elements of... Get solution
6. 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
7. Let ..., where (α, β, γ) are the Eulerian angles. In order that U represent a rotation (α, β, γ), what are the commutation rules that must be satisfied by the Gk? Relate G to the angular-momentum operators. Get solution
8. 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
9. 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) Consider a pure ensemble of identically prepared spin ... systems. Suppose the expectation values ⟨Sx⟩ and ⟨Sz⟩ and the sign of ⟨Sy⟩ are known. Show how we may determine the state vector. Why is it unnecessary to know the magnitude of ⟨Sy⟩?(b) Consider a mixed ensemble of spin ... systems. Suppose the ensemble averages [Sx], [Sy], and [Sz] are all known. Show how we may construct the 2 × 2 density matrix that characterizes the ensemble. Get solution
11. (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
12. 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
13. 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
14. Show that the 3 × 3 matrices Gi(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
15. (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
16. Show that the orbital angular-momentum operator L commutes with both the operators p2 and x2; that is, prove (3.7.2). Get solution
17. 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 that 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
18. 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
19. 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
20. 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 Section B.5 in Appendix B may be useful.) Get solution
21. The goal of this problem is to determine degenerate eigenstates of the threedimensional isotropic harmonic oscillator written as eigenstates of L2 and Lz, in terms of the Cartesian eigenstates |nxnynz).(a) Show that the angular-momentum operators are given by...where summation is implied over repeated indices, εijk is the totally antisymmetric symbol, and ... counts the total number of quanta.(b) Use these relations to express the states |qlm⟩ = |01m⟩, m = 0, ±1, in terms of the three eigenstates |nxnynz⟩ that are degenerate in energy. Write down the representation of your answer in coordinate space, and check that the angular and radial dependences are correct.(c) Repeat for |qlm⟩ = |200⟩.(d) Repeat for |qlm⟩ = |02m⟩, with m = 0, 1, and 2. Get solution
22. Follow these steps to show that solutions to Kummer’s Equation (3.7.46) can be written in terms of Laguerre polynomials Ln(x), which are defined according to a generating function as...where 0 t (a) Prove that Ln(0) = n! and L0(x) = 1.(b) Differentiate g(x, t) with respect to x, show that...and find the first few Laguerre polynomials.(c) Differentiate g(x, t) with respect to t and show that...(d) Now show that Kummer’s Equation is solved by deriving...and associate n with the principal quantum number for the hydrogen atom. Get solution
23. What is the physical significance of the operators...in Schwinger’s scheme for angular momentum? Give the nonvanishing matrix elements of K±. Get solution
24. 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 |j1 j2;m1m2⟩. Write your answer as...where + and 0 stand for m1,2 = 1, 0, respectively. Get solution
25. (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
26. (a) Consider a system with j = 1. Explicitly write⟨j = 1, m′|Jy|j = 1, m⟩in 3 × 3 matrix form.(b) Show that for j = 1 only, it is legitimate to replace e−iJyβ/ħ by...(c) Using (b), prove... Get solution
27. Express the matrix element ⟨α2β2γ2|J32|α1β1γ1⟩ in terms of a series in... Get solution
28. 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 particle. 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? Solve the 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 (i) if A measures s1z; (ii) if A measures s1x ? Justify your answer. Get solution
29. Consider a spherical tensor of rank 1 (that is, a vector)...Using the expression for d(j=1) given in Problem 3.26, 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
30. (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 Ux,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
31. 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
32. (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
33. 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 chosen such 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. Find, by explicit construction using Pauli matrices, the eigenvalues for the Hamiltonian...for a spin ... particle in the presence of a magnetic field .... Get solution
3. 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, a1, a2, and a3. Get solution
4. 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) Solve the same problem when eB / mc → 0, A ≠ 0. Get solution
5. Consider a spin 1 particle. Evaluate the matrix elements of... Get solution
6. 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
7. Let ..., where (α, β, γ) are the Eulerian angles. In order that U represent a rotation (α, β, γ), what are the commutation rules that must be satisfied by the Gk? Relate G to the angular-momentum operators. Get solution
8. 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
9. 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) Consider a pure ensemble of identically prepared spin ... systems. Suppose the expectation values ⟨Sx⟩ and ⟨Sz⟩ and the sign of ⟨Sy⟩ are known. Show how we may determine the state vector. Why is it unnecessary to know the magnitude of ⟨Sy⟩?(b) Consider a mixed ensemble of spin ... systems. Suppose the ensemble averages [Sx], [Sy], and [Sz] are all known. Show how we may construct the 2 × 2 density matrix that characterizes the ensemble. Get solution
11. (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
12. 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
13. 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
14. Show that the 3 × 3 matrices Gi(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
15. (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
16. Show that the orbital angular-momentum operator L commutes with both the operators p2 and x2; that is, prove (3.7.2). Get solution
17. 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 that 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
18. 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
19. 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
20. 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 Section B.5 in Appendix B may be useful.) Get solution
21. The goal of this problem is to determine degenerate eigenstates of the threedimensional isotropic harmonic oscillator written as eigenstates of L2 and Lz, in terms of the Cartesian eigenstates |nxnynz).(a) Show that the angular-momentum operators are given by...where summation is implied over repeated indices, εijk is the totally antisymmetric symbol, and ... counts the total number of quanta.(b) Use these relations to express the states |qlm⟩ = |01m⟩, m = 0, ±1, in terms of the three eigenstates |nxnynz⟩ that are degenerate in energy. Write down the representation of your answer in coordinate space, and check that the angular and radial dependences are correct.(c) Repeat for |qlm⟩ = |200⟩.(d) Repeat for |qlm⟩ = |02m⟩, with m = 0, 1, and 2. Get solution
22. Follow these steps to show that solutions to Kummer’s Equation (3.7.46) can be written in terms of Laguerre polynomials Ln(x), which are defined according to a generating function as...where 0 t (a) Prove that Ln(0) = n! and L0(x) = 1.(b) Differentiate g(x, t) with respect to x, show that...and find the first few Laguerre polynomials.(c) Differentiate g(x, t) with respect to t and show that...(d) Now show that Kummer’s Equation is solved by deriving...and associate n with the principal quantum number for the hydrogen atom. Get solution
23. What is the physical significance of the operators...in Schwinger’s scheme for angular momentum? Give the nonvanishing matrix elements of K±. Get solution
24. 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 |j1 j2;m1m2⟩. Write your answer as...where + and 0 stand for m1,2 = 1, 0, respectively. Get solution
25. (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
26. (a) Consider a system with j = 1. Explicitly write⟨j = 1, m′|Jy|j = 1, m⟩in 3 × 3 matrix form.(b) Show that for j = 1 only, it is legitimate to replace e−iJyβ/ħ by...(c) Using (b), prove... Get solution
27. Express the matrix element ⟨α2β2γ2|J32|α1β1γ1⟩ in terms of a series in... Get solution
28. 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 particle. 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? Solve the 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 (i) if A measures s1z; (ii) if A measures s1x ? Justify your answer. Get solution
29. Consider a spherical tensor of rank 1 (that is, a vector)...Using the expression for d(j=1) given in Problem 3.26, 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
30. (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 Ux,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
31. 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
32. (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
33. 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 chosen such 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
