1. Prove... Get solution
2. Suppose a 2 × 2 matrix X (not necessarily Hermitian, nor unitary) is written as...Where a0 and a1,2,3 are numbers.a. How are a0 and ak (k = 1, 2, 3) related to tr(X) and tr(σk X)?b. Obtain a0 and ak in terms of the matrix elements Xij. Get solution
3. Show that the determinant of a 2 × 2 matrix σ · a is invariant under...Find ... in terms of ak when ... is in the positive z-direction and interpret your result. Get solution
4. Using the rules of bra-ket algebra, prove of evaluate the following:a. tr(XY) = tr(YX), where X and Y are operators;b. (XY)† = Y†X†, where X and Y are operators;c. exp[if(A)] = ? in ket-bra form, where A is a Hermitian operator whose eigenvalues are known;d. ... where ... Get solution
5. a. Consider two kets |α⟩ and |β⟩. Suppose ⟨a′|α⟩, ⟨a″|α⟩,… and ⟨a′|β⟩, ⟨a″|β⟩,… are all known, where |a′⟩, |a″⟩,… from a complete set of base kets. Find the matrix representation of the operator |α⟩⟨β| in that basis.b. we now consider a spin ... system and let |α⟩ and |β⟩ be |sz = ℏ/2⟩, respectively. Write down explicitly the square matrix that corresponds to |α⟩⟨β| in the usual (sz diagonal) basis. Get solution
6. Suppose |i⟩ and |j⟩ are eigenkets of some Hermitian operator A. Under what condition can we conclude that |i⟩ + |j⟩ is also an eigenket of A? Justify your answer. Get solution
7. Consider a ket space spanned by the eigenkets {|a′⟩} of a Hermitian operator A. There is no degeneracy.a. Prove that...is the null operator.b. What is the significance of...c. Illustrate (a) and (b) using A set equal to Sz of a spin ... system. Get solution
8. Using the orthonormality of |+ ⟩ and | − ⟩, prove...where... Get solution
9. Construct ... such that...Where ... is characterized by the angles shown in the figure. Express your answer as a linear combination of |+ ⟩ and |− ⟩. [Note: The answer is...But do not just verify that this answer satisfies the above eigenvalue equation. Rather, treat the problem as a straightforward eigenvalue...problem. Also do not use rotation operators, which we will introduce later in this book.] Get solution
10. The Hamiltonian operator for a two-state system is given by...Where a is a number with the dimension of energy. Find the energy eigenvalues and the corresponding energy eigenkets (as linear combination of |1⟩ and |2⟩). Get solution
11. A two-state system is characterized by the Hamiltonian...Where H11, H22, and H12 are real numbers with the dimension of energy, and |1⟩ and |2⟩ are eigenkets of some observable (≠ H). Find the energy eigenkets and corresponding energy eigenvalues. Make sure that your answer makes good sense for H12 = 0. (You need not solve this problem from scratch. The following fact may be used without proof:...with ... given by...where β and α are the polar and azimuthal angles, respectively, that characterize ....) Get solution
12. A spin ... system is known to be in an eigenstate of ... with eigenvalue ℏ/2, where ... is a unit vector lying in the xz-plane that makes an angle γ with the positive z-axiz.a. Suppose Sx is measured. What is the probability of getting + ℏ/2?b. Evaluate the dispersion in Sx, that is,...(For your own peace of mind check your answers for the special cases γ = 0, π/2, and π.) Get solution
13. A beam of spin ... atoms goes through a series of Stern-Gerlach-type measurements as follows:a. The first measurement accepts sz = ℏ/2 atoms and rejects sz = −ℏ/2 atoms.b. The second measurement accepts sn = ℏ/2 atoms and rejects sn = −ℏ/2 atoms, where sn is the eigenvalue of the operator ... with ... making an angle β in the xz-plane with respect to the z-axiz.c. The third measurement accepts sz = −ℏ/2 atom and rejects sz = ℏ/2 atoms.What is the intensity of the final sz = −ℏ/2 beam when the sz = ℏ/2 beam surviving the first measurement is normalized to unity? How must we orient the second measuring apparatus if we are to maximize the intensity of the final sz = −ℏ/2 beam? Get solution
14. A certain observable in quantum mechanics has a 3 × 3 matrix representation as follow:...a. Find the normalized eigenvectors of this observable and the corresponding eigenvalues. Is there any degeneracy?b. Give a physical example where all this is relevant. Get solution
15. Let A and B be observables. Suppose the simultaneous eigenkets of A and B {|a′, b′⟩} from a complete orthonormal set of base kets. Can we always conclude that...If your answer is yes, prove the assertion. If your answer is no, give a counterexample. Get solution
16. Two Hermitian operators anticommute:...Is it possible to have a simultaneous (that is, common) eigenket of A and B? Prove or illustrate your assertion. Get solution
17. Two observables A1 and A2, which do not involve time explicitly, are known not to commute,...yet we also know that A1 and A2 both commute with Hamiltonian:...Prove that the energy eigenstates are, in general, degenerate. Are there exceptions? As an example, you may think of the central-force problem H = p2/2m + V(r), with A1 → Lz, A2 → Lx. Get solution
18. a. The simplest way to derive the Schwarz inequality goes as follows. First, observe...for any complex number λ; then choose λ in such a way that the preceding inequality reduces to the Schwarz inequality.b. Show that equality sign in the generalized uncertainty relation holds if the state in question satisfies...with λ purely imaginary.c. Explicit calculations using the usual rules of wave mechanics show that the wave function for a Gaussian wave packet given by...Satisfies the minimum uncertainty relation...Prove that the requirement...is indeed satisfied for such a Gaussian wave packet, in agreement with (b). Get solution
19. a. Compute...where the expectation value is taken for the Sz +state. Using your result, check the generalized uncertainty relation...With A → Sx, B → Sy.b. Check the uncertainty relation with A → Sx, B → Sy for the Sx + state. Get solution
20. Find the linear combination of |+ ⟩ and |− ⟩ kets that maximizes the uncertainty product...Verify explicitly that for the linear combination you found, the uncertainly relation for Sx and Sy is not violated. Get solution
21. Evaluate the x-p uncertainty product ⟨(Δx)2⟩⟨(Δp)2⟩ for a one-dimensional particle confined between two rigid walls...Do this for both the ground and excited states. Get solution
22. Estimate the rough order of magnitude of the length of time that an ice pick can be balanced on its point if the only limitation is that set by the Heisenberg uncertainty principle. Assume that the point is sharp and that the point and the surface on which it rests are hard. You may make approximations which do not alter the general order of magnitude of the result. Assume reasonable values for the dimensions and weight of the ice pick. Obtain an approximate numerical result and express it in seconds. Get solution
23. Consider a three-dimensional ket space. If a certain set of orthonormal kets—say, |1⟩, |2⟩, and |3⟩—are used as the base kets, the operators A and B are represented by...with a and b both real.a. Obviously A exhibits a degenerate spectrum. Does B also exhibit a degenerate spectrum?b. Show that A and B commute.c. Find a new set of orthonormal kets which are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket? Get solution
24. a. Prove that ... acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the x-axis by angle − π/2. (The minus sign signifies that the rotation is clockwise.)b. Construct the matrix representation of Sz when the eigenkets of Sy are used as base vectors. Get solution
25. Some authors define an operator to be real when every member of its matrix elements ⟨b′|A|b″⟩ is real in some representation ({|b′⟩} basis in this case). Is this concept representation independent, that is, do the matrix elements remain real even if some basis other than {|b′⟩}is used? Check your assertion using familiar operators such as Sy and Sz (see Problem 24) or x and px. Get solution
26. Construct the transformation matrix that connects the Sz diagonal basis to the Sx diagonal basis. Show that your result is consistent with the general relation... Get solution
27. a. Suppose that f(A) is a function of a Hermitian operator A with the property A|a′⟩ = a′|a′⟩. Evaluate ⟨b″|f(A)|b′⟩ when the transformation matrix from the a′ basis to the b′ basis is known.b. Using the continuum analogue of the result obtained in (a), evaluate...Simplify your expression as far as you can. Note that r is ..., where x, y, and z are operators. Get solution
28. a. Let x and px be the coordinate and linear momentum in one dimension. Evaluate the classical Poisson bracket...b. Let x and px be the corresponding quantum-mechanical operators this time. Evaluate the commutator...c. Using the result obtained in (b), prove that...is an eigenstate of the coordinate operator x. What is the corresponding eigenvalue? Get solution
29. a. On page 247, Gottfried (1966) states that...can be “easily derived” from the fundamental commutation relations for all functions of F and G that can be expressed as power series in their arguments. Verify this statement.b. Evaluate [x2, p2]. Compare your result with the classical Poisson bracket [x2, p2]classical. Get solution
30. The translation operator for a finite (spatial) displacement is given by...where p is the momentum operator.a. Evaluate...b. Using (a) (or otherwise), demonstrate how the expectation value ⟨x⟩ changes under translation. Get solution
31. In the main text we discussed the effect of ... on the position and momentum eigenkets and on a more general state ket |α⟩. We can also study the behavior of expectation values ⟨x⟩ and ⟨p⟩ under infinitesimal translation. Using (1.6.25), (1.6.45), and ... only, prove ⟨x⟩ → ⟨x⟩ + dx′, ⟨p⟩ → ⟨p⟩ under infinitesimal translation. Get solution
32. a. Verify (1.7.39a) and (1.7.39b) for the expectation value of p and p2 from the Gaussian wave packet (1.7.35).b. Evaluate the expectation value of p and p2 using the momentum-space wave function (1.7.42). Get solution
33. a. Prove the following:...where ϕα(p′)= ⟨p′|α⟩ and ϕβ(p′)= ⟨p′|β⟩ are momentum-space wave functions.b. What is the physical significance of...where x is the position operator and Ξ is some number with the dimension of momentum? Justify your answer. Get solution
2. Suppose a 2 × 2 matrix X (not necessarily Hermitian, nor unitary) is written as...Where a0 and a1,2,3 are numbers.a. How are a0 and ak (k = 1, 2, 3) related to tr(X) and tr(σk X)?b. Obtain a0 and ak in terms of the matrix elements Xij. Get solution
3. Show that the determinant of a 2 × 2 matrix σ · a is invariant under...Find ... in terms of ak when ... is in the positive z-direction and interpret your result. Get solution
4. Using the rules of bra-ket algebra, prove of evaluate the following:a. tr(XY) = tr(YX), where X and Y are operators;b. (XY)† = Y†X†, where X and Y are operators;c. exp[if(A)] = ? in ket-bra form, where A is a Hermitian operator whose eigenvalues are known;d. ... where ... Get solution
5. a. Consider two kets |α⟩ and |β⟩. Suppose ⟨a′|α⟩, ⟨a″|α⟩,… and ⟨a′|β⟩, ⟨a″|β⟩,… are all known, where |a′⟩, |a″⟩,… from a complete set of base kets. Find the matrix representation of the operator |α⟩⟨β| in that basis.b. we now consider a spin ... system and let |α⟩ and |β⟩ be |sz = ℏ/2⟩, respectively. Write down explicitly the square matrix that corresponds to |α⟩⟨β| in the usual (sz diagonal) basis. Get solution
6. Suppose |i⟩ and |j⟩ are eigenkets of some Hermitian operator A. Under what condition can we conclude that |i⟩ + |j⟩ is also an eigenket of A? Justify your answer. Get solution
7. Consider a ket space spanned by the eigenkets {|a′⟩} of a Hermitian operator A. There is no degeneracy.a. Prove that...is the null operator.b. What is the significance of...c. Illustrate (a) and (b) using A set equal to Sz of a spin ... system. Get solution
8. Using the orthonormality of |+ ⟩ and | − ⟩, prove...where... Get solution
9. Construct ... such that...Where ... is characterized by the angles shown in the figure. Express your answer as a linear combination of |+ ⟩ and |− ⟩. [Note: The answer is...But do not just verify that this answer satisfies the above eigenvalue equation. Rather, treat the problem as a straightforward eigenvalue...problem. Also do not use rotation operators, which we will introduce later in this book.] Get solution
10. The Hamiltonian operator for a two-state system is given by...Where a is a number with the dimension of energy. Find the energy eigenvalues and the corresponding energy eigenkets (as linear combination of |1⟩ and |2⟩). Get solution
11. A two-state system is characterized by the Hamiltonian...Where H11, H22, and H12 are real numbers with the dimension of energy, and |1⟩ and |2⟩ are eigenkets of some observable (≠ H). Find the energy eigenkets and corresponding energy eigenvalues. Make sure that your answer makes good sense for H12 = 0. (You need not solve this problem from scratch. The following fact may be used without proof:...with ... given by...where β and α are the polar and azimuthal angles, respectively, that characterize ....) Get solution
12. A spin ... system is known to be in an eigenstate of ... with eigenvalue ℏ/2, where ... is a unit vector lying in the xz-plane that makes an angle γ with the positive z-axiz.a. Suppose Sx is measured. What is the probability of getting + ℏ/2?b. Evaluate the dispersion in Sx, that is,...(For your own peace of mind check your answers for the special cases γ = 0, π/2, and π.) Get solution
13. A beam of spin ... atoms goes through a series of Stern-Gerlach-type measurements as follows:a. The first measurement accepts sz = ℏ/2 atoms and rejects sz = −ℏ/2 atoms.b. The second measurement accepts sn = ℏ/2 atoms and rejects sn = −ℏ/2 atoms, where sn is the eigenvalue of the operator ... with ... making an angle β in the xz-plane with respect to the z-axiz.c. The third measurement accepts sz = −ℏ/2 atom and rejects sz = ℏ/2 atoms.What is the intensity of the final sz = −ℏ/2 beam when the sz = ℏ/2 beam surviving the first measurement is normalized to unity? How must we orient the second measuring apparatus if we are to maximize the intensity of the final sz = −ℏ/2 beam? Get solution
14. A certain observable in quantum mechanics has a 3 × 3 matrix representation as follow:...a. Find the normalized eigenvectors of this observable and the corresponding eigenvalues. Is there any degeneracy?b. Give a physical example where all this is relevant. Get solution
15. Let A and B be observables. Suppose the simultaneous eigenkets of A and B {|a′, b′⟩} from a complete orthonormal set of base kets. Can we always conclude that...If your answer is yes, prove the assertion. If your answer is no, give a counterexample. Get solution
16. Two Hermitian operators anticommute:...Is it possible to have a simultaneous (that is, common) eigenket of A and B? Prove or illustrate your assertion. Get solution
17. Two observables A1 and A2, which do not involve time explicitly, are known not to commute,...yet we also know that A1 and A2 both commute with Hamiltonian:...Prove that the energy eigenstates are, in general, degenerate. Are there exceptions? As an example, you may think of the central-force problem H = p2/2m + V(r), with A1 → Lz, A2 → Lx. Get solution
18. a. The simplest way to derive the Schwarz inequality goes as follows. First, observe...for any complex number λ; then choose λ in such a way that the preceding inequality reduces to the Schwarz inequality.b. Show that equality sign in the generalized uncertainty relation holds if the state in question satisfies...with λ purely imaginary.c. Explicit calculations using the usual rules of wave mechanics show that the wave function for a Gaussian wave packet given by...Satisfies the minimum uncertainty relation...Prove that the requirement...is indeed satisfied for such a Gaussian wave packet, in agreement with (b). Get solution
19. a. Compute...where the expectation value is taken for the Sz +state. Using your result, check the generalized uncertainty relation...With A → Sx, B → Sy.b. Check the uncertainty relation with A → Sx, B → Sy for the Sx + state. Get solution
20. Find the linear combination of |+ ⟩ and |− ⟩ kets that maximizes the uncertainty product...Verify explicitly that for the linear combination you found, the uncertainly relation for Sx and Sy is not violated. Get solution
21. Evaluate the x-p uncertainty product ⟨(Δx)2⟩⟨(Δp)2⟩ for a one-dimensional particle confined between two rigid walls...Do this for both the ground and excited states. Get solution
22. Estimate the rough order of magnitude of the length of time that an ice pick can be balanced on its point if the only limitation is that set by the Heisenberg uncertainty principle. Assume that the point is sharp and that the point and the surface on which it rests are hard. You may make approximations which do not alter the general order of magnitude of the result. Assume reasonable values for the dimensions and weight of the ice pick. Obtain an approximate numerical result and express it in seconds. Get solution
23. Consider a three-dimensional ket space. If a certain set of orthonormal kets—say, |1⟩, |2⟩, and |3⟩—are used as the base kets, the operators A and B are represented by...with a and b both real.a. Obviously A exhibits a degenerate spectrum. Does B also exhibit a degenerate spectrum?b. Show that A and B commute.c. Find a new set of orthonormal kets which are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket? Get solution
24. a. Prove that ... acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the x-axis by angle − π/2. (The minus sign signifies that the rotation is clockwise.)b. Construct the matrix representation of Sz when the eigenkets of Sy are used as base vectors. Get solution
25. Some authors define an operator to be real when every member of its matrix elements ⟨b′|A|b″⟩ is real in some representation ({|b′⟩} basis in this case). Is this concept representation independent, that is, do the matrix elements remain real even if some basis other than {|b′⟩}is used? Check your assertion using familiar operators such as Sy and Sz (see Problem 24) or x and px. Get solution
26. Construct the transformation matrix that connects the Sz diagonal basis to the Sx diagonal basis. Show that your result is consistent with the general relation... Get solution
27. a. Suppose that f(A) is a function of a Hermitian operator A with the property A|a′⟩ = a′|a′⟩. Evaluate ⟨b″|f(A)|b′⟩ when the transformation matrix from the a′ basis to the b′ basis is known.b. Using the continuum analogue of the result obtained in (a), evaluate...Simplify your expression as far as you can. Note that r is ..., where x, y, and z are operators. Get solution
28. a. Let x and px be the coordinate and linear momentum in one dimension. Evaluate the classical Poisson bracket...b. Let x and px be the corresponding quantum-mechanical operators this time. Evaluate the commutator...c. Using the result obtained in (b), prove that...is an eigenstate of the coordinate operator x. What is the corresponding eigenvalue? Get solution
29. a. On page 247, Gottfried (1966) states that...can be “easily derived” from the fundamental commutation relations for all functions of F and G that can be expressed as power series in their arguments. Verify this statement.b. Evaluate [x2, p2]. Compare your result with the classical Poisson bracket [x2, p2]classical. Get solution
30. The translation operator for a finite (spatial) displacement is given by...where p is the momentum operator.a. Evaluate...b. Using (a) (or otherwise), demonstrate how the expectation value ⟨x⟩ changes under translation. Get solution
31. In the main text we discussed the effect of ... on the position and momentum eigenkets and on a more general state ket |α⟩. We can also study the behavior of expectation values ⟨x⟩ and ⟨p⟩ under infinitesimal translation. Using (1.6.25), (1.6.45), and ... only, prove ⟨x⟩ → ⟨x⟩ + dx′, ⟨p⟩ → ⟨p⟩ under infinitesimal translation. Get solution
32. a. Verify (1.7.39a) and (1.7.39b) for the expectation value of p and p2 from the Gaussian wave packet (1.7.35).b. Evaluate the expectation value of p and p2 using the momentum-space wave function (1.7.42). Get solution
33. a. Prove the following:...where ϕα(p′)= ⟨p′|α⟩ and ϕβ(p′)= ⟨p′|β⟩ are momentum-space wave functions.b. What is the physical significance of...where x is the position operator and Ξ is some number with the dimension of momentum? Justify your answer. Get solution
