1. (a) Show that the set of all square-integrable functions is a
vector space (refer to Section A.l for the definition). Hint: The main
problem is to show that the sum of two square-integrable functions is
itself square-integrable. Use Equation 3.7. Is the set of all normalized
functions a vector space?(b) Show that the integral in Equation 3.6
satisfies the conditions for an inner product (Section A.2). Get solution
2. (a) For what range of v is the function f(x) = xv in Hilbert space, on the interval (0, 1)? Assume v is real, but not necessarily positive.(b) For the specific case v = 1/2, is f(x) in this Hilbert space? What about xf(x)? How about (d/dx)f(x)? Get solution
3. Show that if ... for all functions h (in Hilbert space), then ... for all f and g (i.e., the two definitions of "hermitian"—Equations 3.16 and 3.17—are equivalent). Hint: First let h = f + g, and then let h = f + ig. Get solution
4. (a) Show that the sum of two hermitian operators is hermitian.(b) Suppose ... is hermitian, and α is a complex number. Under what condition (on α) is α ... hermitian?(c) When is the product of two hermitian operators hermitian?(d) Show that the position operator (... = x) and the hamiltonian operator (... = −(h2/2m)d2/dx2 + V(x)) are hermitian. Get solution
5. The hermitian conjugate (or adjoint) of an operator ... is the operator ... such that... [3.20](A hermitian operator, then, is equal to its hermitian conjugate: ....)(a) Find the hermitian conjugates of x, i, and d/dx.(b) Construct the hermitian conjugate of the harmonic oscillator raising operator, a+ (Equation 2.47).(c) Show that .... Get solution
6. Consider the operator ...where (as in Example 3.1) is the azimuthal angle in polar coordinates, and the functions are subject to Equation 3.26. Is ... Hermitian? Find its eigenfunctions and eigenvalues. What is the spectrum of ... Is the spectrum degenerate? ... ... Get solution
7. (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ..., with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of ..., with eigenvalue q.(b) Check that f(x) = exp(x) and g(x) = exp(–x) are eigenfunctions of the operator d2/dx2, with the same eigenvalue. Construct two linear combinations of f and g that are orthogonal eigenfunctions on the interval (–1, 1). Get solution
8. (a) Check that the eigenvalues of the hermitian operator in Example 3.1 are real. Show that the eigenfunctions (for distinct eigenvalues) are orthogonal.(b) Do the same for the operator in Problem 3.6. Get solution
9. (a) Cite a Hamiltonian from Chapter 2 (other than the harmonic oscillator) that has only a discrete spectrum.(b) Cite a Hamiltonian from Chapter 2 (other than the free particle) that has only a continuous spectrum.(c) Cite a Hamiltonian from Chapter 2 (other than the finite square well) that has both a discrete and a continuous part to its spectrum. Get solution
10. Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not? Get solution
11. Find the momentum-space wave function, ф(p, t), for a particle in the ground state of the harmonic oscillator. What is the probability (to 2 significant digits) that a measurement of p on a particle in this state would yield a value outside the classical range (for the same energy)? Hint: Look in a math table under "Normal Distribution" or "Error Function" for the numerical part—or use Mathematica. Get solution
12. Show that... [3.57]Hint: Notice that x exp(ipx/h) = —ih(d/dp)exp(ipx/h).In momentum space, then, the position operator is iћ∂/∂p. More generally,... [3.58]In principle you can do all calculations in momentum space just as well (though not always aseasily) as in position space. Get solution
13. (a) Prove the following commutator identity:... [3.64](b) Show that...(c) Show more generally that... [3.65]for any function f(x). Get solution
14. Prove the famous “(your name) uncertainty principle,” relating the uncertainty in position (A = x) to the uncertainty in energy (B = p2/2m + V):...For stationary states this doesn’t tell you much—why not? Get solution
15. Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if... and ... have a complete set of common eigenfunctions, then [..., ...] f = 0 for any function in Hilbert space. Get solution
16. Solve Equation 1 for Ψ (x). Note that ...x... and ...p... areconstants.Equation 1... Get solution
17. Apply Equation 1 to the following special cases: (a) Q = 1; (b) Q = H; (c) Q = x; (d) Q = p. In each case, comment on the result, with particular reference to Equations 2, 3, 1.33 and conservation of energy (comments following Equation 2.39).Equation 1...Equation 2...Equation 3... Get solution
18. Test the energy-time uncertainty principle for the wave function in Problem 2.5 and observable x, by calculating ... and ... exactly. (Reference: Problem 2.5) A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: ... ... Get solution
19. Test the energy-time uncertainty principle for the free particle wave packet in Problem 2.43 and the observable x, by calculating σH, σx and d...x.../dt exactly. Get solution
20. Show that the energy-time uncertainty principle reduces to the “your name” uncertainty principle (Problem 1), when the observable in question is x.Problem 1Prove the famous “(your name) uncertainty principle,” relating the uncertainty in position (A = x) to the uncertainty in energy (B = p2/2m + V):...For stationary states this doesn’t tell you much—why not? Get solution
21. Show that projection operators are idempotent: .... Determine the eigenvalues of ..., and characterize its eigenvectors. Get solution
22. Consider a three-dimensional vector space spanned by an orthonormal basis ... are given by ... Get solution
23. The Hamiltonian for a certain two-level system is...where ..., ... is an orthonormal basis and ϵ is a number with the dimensions of energy. Find its eigenvalues and eigenvectors (as linear combinations of ... and ...). What is the matrix H representing ... with respect to this basis? Get solution
24. Let ... be an operator with a complete set of orthonormal eigenvectors:...Show that ... can be written in terms of its spectral decomposition:...Hint: An operator is characterized by its action on all possible vectors, so what you must show is that...for any vector .... Get solution
25. ... ... Get solution
26. An anti-hermitian (or skew-hermitian) operator is equal to minus its hermitian conjugate:... [3.95](a) Show that the expectation value of an anti-hermitian operator is imaginary.(b) Show that the commutator of two hermitian operators is anti-hermitian. How about the commutator of two anti-hermitian operators? Get solution
27. Sequential measurements. An operator ..., representing observable A, has two normalized eigenstates ψ1 and ψ2, with eigenvalues a1 and a2, respectively. Operator ..., representing observable B, has two normalized eigenstates ϕ1 and ϕ2, with eigenvalues b1 and b2. The eigenstates are related by...(a) Observable A is measured, and the value a1 is obtained. What is the state of the system (immediately) after this measurement?(b) If B is now measured, what are the possible results, and what are their probabilities?(c) Right after the measurement of B, A is measured again. What is the probability of getting a1? (Note that the answer would be quite different if I had told you the outcome of the B measurement.) Get solution
28. Find the momentum-space wave function ϕn(p, t) for the nth stationary state of the infinite square well. Graph ... and ..., as functions of p (pay particular attention to the points...). Use ϕn(p, t) to calculate the expectation value of p2. Compare your answer to Problem 2.4. Get solution
29. Consider the wave function...where n is some positive integer. This function is purely sinusoidal (with wavelength k) on the interval —nkp, 0). Sketch the graphs of ... and..., and determine their widths, wx and wp (the distance between zeros on either side of the main peak). Note what happens to each width as n → ∞. Using wx and wp as estimates of ∆x and ∆p, check that the uncertainty principle is satisfied. Warning: If you try calculating σp, you're in for a rude surprise. Can you diagnose the problem? Get solution
30. Suppose...for constants A and a.(a) Determine A, by normalizing ....(b) Find ..., ..., and σx (at time t = 0)(c) Find the moment space wave function Φ(p, 0), and check that it is normalized.(d) Use Φ(p, 0) to calculate ..., ..., and σp (at time t = 0)(e) Check the Heisenberg uncertainty principle for this state. Get solution
31. Virial theorem. Use Equation 1 to show that... [3.96]where T is the kinetic energy (H = T + V). In a stationary state the left side is zero (why?) so... [3.97]This is called the virial theorem. Use it to prove that ...T... = ...V... for stationary states of the harmonic oscillator, and check that this is consistent with the results you got in Problems 2.11 and 2.12.Equation 1... Get solution
32. In an interesting version of the energy-time uncertainty principle31 Δt = τ/π, where τ is the time it takes Ψ(x, t) to evolve into a state orthogonal to Ψ(x,0). Test this out, using a wave function that is an equal admixture of two (orthonormal) stationary states of some (arbitrary) potential: Ψ(x, 0) = (1/√2)[ψ1(x)+ψ2(x)]. Get solution
33. ... ... Get solution
34. A harmonic oscillator is in a state such that a measurement of the energy would yield either (1/2)ћω or (3/2)ћω, with equal probability. What is the largest possible value of ...p... in such a state? If it assumes this maximal value at time t = 0, what is Ψ(x, t)? Get solution
35. ... ... ... Get solution
36. ... ... ... ... Get solution
37. The Hamiltonian for a certain three-level system is represented by the matrix...where a, b, and c are real numbers (assume a — c ≠ ± b).(a) If the system starts out in the state...what is ...(b) If the system starts out in the state...what is .... Get solution
38. The Hamiltonian for a certain three-level system is represented by the matrix...Two other observables, A and B, are represented by the matrices...where ω, λ, and μ are positive real numbers.(a) Find the eigenvalues and (normalized) eigenvectors of H, A, and B.(b) Suppose the system starts out in the generic state...with |c1|2 + |c2|2 + |C3|2 = 1. Find the expectation values (at t = 0) of H, A, and B.(c) What is ... If you measured the energy of this state (at time t), what values might you get, and what is the probability of each? Answer the same questions for A and for B. Get solution
39. ... ... ... Get solution
40. (a) Write down the time-dependent “Schrodinger equation” in momentum space, for a free particle, and solve it. Answer: exp(‒ip2t/2mћ) Φ (p, 0).(b) Find Φ(p, 0) for the traveling gaussian wave packet (Problem 2.43), and construct Φ(p, t) for this case. Also construct |Φ(p, t)|2, and note that it is independent of time.(c) Calculate (p) and (p2) by evaluating the appropriate integrals involving Φ, and compare your answers to Problem 2.43.(d) Show that (H) = (p)2/2m + (H)0 (where the subscript 0 denotes the stationary gaussian), and comment on this result. Get solution
2. (a) For what range of v is the function f(x) = xv in Hilbert space, on the interval (0, 1)? Assume v is real, but not necessarily positive.(b) For the specific case v = 1/2, is f(x) in this Hilbert space? What about xf(x)? How about (d/dx)f(x)? Get solution
3. Show that if ... for all functions h (in Hilbert space), then ... for all f and g (i.e., the two definitions of "hermitian"—Equations 3.16 and 3.17—are equivalent). Hint: First let h = f + g, and then let h = f + ig. Get solution
4. (a) Show that the sum of two hermitian operators is hermitian.(b) Suppose ... is hermitian, and α is a complex number. Under what condition (on α) is α ... hermitian?(c) When is the product of two hermitian operators hermitian?(d) Show that the position operator (... = x) and the hamiltonian operator (... = −(h2/2m)d2/dx2 + V(x)) are hermitian. Get solution
5. The hermitian conjugate (or adjoint) of an operator ... is the operator ... such that... [3.20](A hermitian operator, then, is equal to its hermitian conjugate: ....)(a) Find the hermitian conjugates of x, i, and d/dx.(b) Construct the hermitian conjugate of the harmonic oscillator raising operator, a+ (Equation 2.47).(c) Show that .... Get solution
6. Consider the operator ...where (as in Example 3.1) is the azimuthal angle in polar coordinates, and the functions are subject to Equation 3.26. Is ... Hermitian? Find its eigenfunctions and eigenvalues. What is the spectrum of ... Is the spectrum degenerate? ... ... Get solution
7. (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ..., with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of ..., with eigenvalue q.(b) Check that f(x) = exp(x) and g(x) = exp(–x) are eigenfunctions of the operator d2/dx2, with the same eigenvalue. Construct two linear combinations of f and g that are orthogonal eigenfunctions on the interval (–1, 1). Get solution
8. (a) Check that the eigenvalues of the hermitian operator in Example 3.1 are real. Show that the eigenfunctions (for distinct eigenvalues) are orthogonal.(b) Do the same for the operator in Problem 3.6. Get solution
9. (a) Cite a Hamiltonian from Chapter 2 (other than the harmonic oscillator) that has only a discrete spectrum.(b) Cite a Hamiltonian from Chapter 2 (other than the free particle) that has only a continuous spectrum.(c) Cite a Hamiltonian from Chapter 2 (other than the finite square well) that has both a discrete and a continuous part to its spectrum. Get solution
10. Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not? Get solution
11. Find the momentum-space wave function, ф(p, t), for a particle in the ground state of the harmonic oscillator. What is the probability (to 2 significant digits) that a measurement of p on a particle in this state would yield a value outside the classical range (for the same energy)? Hint: Look in a math table under "Normal Distribution" or "Error Function" for the numerical part—or use Mathematica. Get solution
12. Show that... [3.57]Hint: Notice that x exp(ipx/h) = —ih(d/dp)exp(ipx/h).In momentum space, then, the position operator is iћ∂/∂p. More generally,... [3.58]In principle you can do all calculations in momentum space just as well (though not always aseasily) as in position space. Get solution
13. (a) Prove the following commutator identity:... [3.64](b) Show that...(c) Show more generally that... [3.65]for any function f(x). Get solution
14. Prove the famous “(your name) uncertainty principle,” relating the uncertainty in position (A = x) to the uncertainty in energy (B = p2/2m + V):...For stationary states this doesn’t tell you much—why not? Get solution
15. Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if... and ... have a complete set of common eigenfunctions, then [..., ...] f = 0 for any function in Hilbert space. Get solution
16. Solve Equation 1 for Ψ (x). Note that ...x... and ...p... areconstants.Equation 1... Get solution
17. Apply Equation 1 to the following special cases: (a) Q = 1; (b) Q = H; (c) Q = x; (d) Q = p. In each case, comment on the result, with particular reference to Equations 2, 3, 1.33 and conservation of energy (comments following Equation 2.39).Equation 1...Equation 2...Equation 3... Get solution
18. Test the energy-time uncertainty principle for the wave function in Problem 2.5 and observable x, by calculating ... and ... exactly. (Reference: Problem 2.5) A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: ... ... Get solution
19. Test the energy-time uncertainty principle for the free particle wave packet in Problem 2.43 and the observable x, by calculating σH, σx and d...x.../dt exactly. Get solution
20. Show that the energy-time uncertainty principle reduces to the “your name” uncertainty principle (Problem 1), when the observable in question is x.Problem 1Prove the famous “(your name) uncertainty principle,” relating the uncertainty in position (A = x) to the uncertainty in energy (B = p2/2m + V):...For stationary states this doesn’t tell you much—why not? Get solution
21. Show that projection operators are idempotent: .... Determine the eigenvalues of ..., and characterize its eigenvectors. Get solution
22. Consider a three-dimensional vector space spanned by an orthonormal basis ... are given by ... Get solution
23. The Hamiltonian for a certain two-level system is...where ..., ... is an orthonormal basis and ϵ is a number with the dimensions of energy. Find its eigenvalues and eigenvectors (as linear combinations of ... and ...). What is the matrix H representing ... with respect to this basis? Get solution
24. Let ... be an operator with a complete set of orthonormal eigenvectors:...Show that ... can be written in terms of its spectral decomposition:...Hint: An operator is characterized by its action on all possible vectors, so what you must show is that...for any vector .... Get solution
25. ... ... Get solution
26. An anti-hermitian (or skew-hermitian) operator is equal to minus its hermitian conjugate:... [3.95](a) Show that the expectation value of an anti-hermitian operator is imaginary.(b) Show that the commutator of two hermitian operators is anti-hermitian. How about the commutator of two anti-hermitian operators? Get solution
27. Sequential measurements. An operator ..., representing observable A, has two normalized eigenstates ψ1 and ψ2, with eigenvalues a1 and a2, respectively. Operator ..., representing observable B, has two normalized eigenstates ϕ1 and ϕ2, with eigenvalues b1 and b2. The eigenstates are related by...(a) Observable A is measured, and the value a1 is obtained. What is the state of the system (immediately) after this measurement?(b) If B is now measured, what are the possible results, and what are their probabilities?(c) Right after the measurement of B, A is measured again. What is the probability of getting a1? (Note that the answer would be quite different if I had told you the outcome of the B measurement.) Get solution
28. Find the momentum-space wave function ϕn(p, t) for the nth stationary state of the infinite square well. Graph ... and ..., as functions of p (pay particular attention to the points...). Use ϕn(p, t) to calculate the expectation value of p2. Compare your answer to Problem 2.4. Get solution
29. Consider the wave function...where n is some positive integer. This function is purely sinusoidal (with wavelength k) on the interval —nkp, 0). Sketch the graphs of ... and..., and determine their widths, wx and wp (the distance between zeros on either side of the main peak). Note what happens to each width as n → ∞. Using wx and wp as estimates of ∆x and ∆p, check that the uncertainty principle is satisfied. Warning: If you try calculating σp, you're in for a rude surprise. Can you diagnose the problem? Get solution
30. Suppose...for constants A and a.(a) Determine A, by normalizing ....(b) Find ..., ..., and σx (at time t = 0)(c) Find the moment space wave function Φ(p, 0), and check that it is normalized.(d) Use Φ(p, 0) to calculate ..., ..., and σp (at time t = 0)(e) Check the Heisenberg uncertainty principle for this state. Get solution
31. Virial theorem. Use Equation 1 to show that... [3.96]where T is the kinetic energy (H = T + V). In a stationary state the left side is zero (why?) so... [3.97]This is called the virial theorem. Use it to prove that ...T... = ...V... for stationary states of the harmonic oscillator, and check that this is consistent with the results you got in Problems 2.11 and 2.12.Equation 1... Get solution
32. In an interesting version of the energy-time uncertainty principle31 Δt = τ/π, where τ is the time it takes Ψ(x, t) to evolve into a state orthogonal to Ψ(x,0). Test this out, using a wave function that is an equal admixture of two (orthonormal) stationary states of some (arbitrary) potential: Ψ(x, 0) = (1/√2)[ψ1(x)+ψ2(x)]. Get solution
33. ... ... Get solution
34. A harmonic oscillator is in a state such that a measurement of the energy would yield either (1/2)ћω or (3/2)ћω, with equal probability. What is the largest possible value of ...p... in such a state? If it assumes this maximal value at time t = 0, what is Ψ(x, t)? Get solution
35. ... ... ... Get solution
36. ... ... ... ... Get solution
37. The Hamiltonian for a certain three-level system is represented by the matrix...where a, b, and c are real numbers (assume a — c ≠ ± b).(a) If the system starts out in the state...what is ...(b) If the system starts out in the state...what is .... Get solution
38. The Hamiltonian for a certain three-level system is represented by the matrix...Two other observables, A and B, are represented by the matrices...where ω, λ, and μ are positive real numbers.(a) Find the eigenvalues and (normalized) eigenvectors of H, A, and B.(b) Suppose the system starts out in the generic state...with |c1|2 + |c2|2 + |C3|2 = 1. Find the expectation values (at t = 0) of H, A, and B.(c) What is ... If you measured the energy of this state (at time t), what values might you get, and what is the probability of each? Answer the same questions for A and for B. Get solution
39. ... ... ... Get solution
40. (a) Write down the time-dependent “Schrodinger equation” in momentum space, for a free particle, and solve it. Answer: exp(‒ip2t/2mћ) Φ (p, 0).(b) Find Φ(p, 0) for the traveling gaussian wave packet (Problem 2.43), and construct Φ(p, t) for this case. Also construct |Φ(p, t)|2, and note that it is independent of time.(c) Calculate (p) and (p2) by evaluating the appropriate integrals involving Φ, and compare your answers to Problem 2.43.(d) Show that (H) = (p)2/2m + (H)0 (where the subscript 0 denotes the stationary gaussian), and comment on this result. Get solution
