### Chapter #7 Solutions - Modern Quantum Mechanics, Revised Edition - J. J. Sakurai - 1st Edition

1. The Lippmann-Schwinger formalism can also be applied to a one-dimensional transmission-reflection problem with a finite-range potential, V(x) ≠ 0 for 0 x| a only.a. Suppose we have an incident wave coming from the left: ...... How must we handle the singular 1/(E − H0) operator if we are to have a transmitted wave only for x > a and a reflected wave and the original wave for x ? Is the E → E + iε prescription still correct? Obtain an expression for the appropriate Green’s function and write an integral equation for ⟨x|ψ(+)⟩.b. Consider the special case of an attractive δ-function potential...Solve the integral equation to obtain the transmission and reflection amplitudes. Check your results with Gottfried 1966, 52.c. The one-dimensional δ-function potential with γ > 0 admits one (and only one) bound state for any value of γ. Show that the transmission and reflection amplitudes you computed have bound-state poles at the expected positions when k is regarded as a complex variable. Get solution

2. Prove...in each of the following ways.a. By integrating the differential cross section computed using the first-order Born approximation.b. By applying the optical theorem to the forward-scattering amplitude in the second-oider Born approximation. [Note that f(0) is real if the first-order Born approximation is used.] Get solution

3. Consider a potential...where V0 may be positive or negative. Using the method of partial waves, show that for |V0| ≪ E = ℏ2k2/2m and kR ≪ 1 the differential cross section is isotropic and that the total cross section is given by...Suppose the energy is raised slightly. Show that the angular distribution can then be written as...Obtain an approximate expression for B/A. Get solution

4. A spinless particle is scattered by a weak Yukawa potential...where µ > 0 but V0 can be positive or negative. It was shown in the text that the first-order Born amplitude is given by...a. Using f(1)(θ) and assuming |δl| ≪ 1, obtain an expression for δl in terms of a Legendre function of the second kind,...b. Use the expansion formula...to prove each assertion.(i) δl is negative (positive) when the potential is repulsive (attractive).(ii) When the de Broglie wavelength is much longer than the range of the potential, δl is proportional to k2l + 1. Find the proportionality constant. Get solution

5. Check explicitly the x − px uncertainty relation for the ground state of a particle confined inside a hard sphere: V = ∞ for r > a, V = 0 for r (Hint: Take advantage of spherical symmetry.) Get solution

6. Consider the scattering of a particle by an impenetrable sphere...a. Derive an expression for the s-wave (l = 0) phase shift. (You need not know the detailed properties of the spherical Bessel functions to be able to do this simple problem!)b. What is the total cross section σ[σ = ∫(dσ/dΩ)dΩ] in the extreme low-energy limit k → 0? Compare your answer with the geometric cross section πa2. You may assume without proof:... Get solution

7. Use δl =Δ(b)|b = l/k to obtain the phase shift δl for scattering at high energies by (a) the Gaussian potential, V = V0exp( − r2/a2), and (b) the Yukawa potential, V = V0exp(−µr)/µr. Verify the assertion that δl goes to zero very rapidly with increasing l (k fixed) for l ≫ kR, where R is the “range” of the potential. [The formula for Δ(b) is given in (7.4.14)]. Get solution

8. a. Prove...where r (r>) stands for the smaller (larger) of r and r'.b. For spherically symmetric potentials, the Lippmann-Schwinger equa­tion can be written for spherical waves:...Using (a), show that this equation, written in the x-representation, leads to an equation for the radial function, Al(k; r), as follows:...By taking r very large, also obtain... Get solution

9. Consider scattering by a repulsive δ-shell potential:...a. Set up an equation that determines the s-wave phase shift δ0 as a function of k (E = ℏk2/2m).b. Assume now that γ is very large,...Show that if tan kR is not close to zero, the s-wave phase shift resembles the hard-sphere result discussed in the text. Show also that for tan kR close to (but not exactly equal to) zero, resonance behavior is possible; that is, cot δ0 goes through zero from the positive side as k increases. Determine approximately the positions of the resonances keeping terms of order 1/γ; compare them with the bound-state energies for a particle confined inside a spherical wall of the same radius,...Also obtain an approximate expression for the resonance width Γdenned by...and notice, in particular, that the resonances become extremely sharp as γ becomes large, (Note: For a different, more sophisticated approach to this problem see Gottfried 1966, 131−141, who discusses, the analytic properties of the Dl-function defined by Al = jl/Dl.) Get solution

10. A spinless particle is scattered by a time-dependent potential...Show that if the potential is treated to first order in the transition amplitude, the energy of the scattered particle is increased or decreased by ℏω. Obtain dσ/dΩ. Discuss qualitatively what happens if the higher-order terms are taken into account. Get solution

11. Show that the differential cross section for the elastic scattering of a fast electron by the ground state of the hydrogen atom is given by...(Ignore the effect of identity.) Get solution

12. Let the energy of a particle moving in a central field be E(J1J2J3), where (J1, J2, J3) are the three action variables. How does the functional form of E specialize for the Coulomb potential? Using the recipe of the action-angle method, compare the degeneracy of the central field and the Coulomb problems and relate it to the vector A.If the Hamiltonian is...how are these statements changed?Describe the corresponding degeneracies of the central field and Coulomb problems in quantum theory in terms of the usual quantum numbers (n, l, m) and also in terms of the quantum numbers (k, m, n). Here the second set, (k, m, n), labels the wave functions ...How are the wave functions ... related to Laguerre times spherical harmonics? Get solution

### Chapter #6 Solutions - Modern Quantum Mechanics, Revised Edition - J. J. Sakurai - 1st Edition

1. a. N identical spin ... particles are subjected to a one-dimensional simple harmonic oscillator potential. What is the ground-state energy? What is the Fermi energy?b. What are the ground state and Fermi energies if we ignore the mutual interactions and assume N to be very large? Get solution

2. It is obvious that two nonidentical spin1 particles with no orbital angular momenta (that is, s-states for both) can form j = 0, j = 1, and j = 2. Suppose, however, that the two particles are identical. What restrictions do we get? Get solution

3. Discuss what would happen to the energy levels of a helium atom if the electron were a spinless boson. Be as quantitative as you can. Get solution

4. Three spin 0 particles are situated at the corners of an equilateral triangle. Let us define the z-axis to go through the center and in the direction normal to the plane of the triangle. The whole system is free to rotate about the z-axis. Using statistics considerations, obtain restrictions on the magnetic quantum numbers corresponding to Jz.... Get solution

5. Consider three weakly interacting, identical spin 1 particles.a. Suppose the space part of the state vector is known to be symmetric under interchange of any pair. Using notation |+⟩|0⟩| + ⟩ for particle 1 in ms = +1, particle 2 in ms = 0, particle 3 in ms = +1, and so on, construct the normalized spin states in the following three cases:(i) All three of them in | + ⟩.(ii) Two of them in | + ⟩, one in |0⟩.(iii) All three in different spin states.What is the total spin in each case?b. Attempt to do the same problem when the space part is antisymmetric under interchange of any pair. Get solution

6. Suppose the electron were a spin ... particle obeying Fermi-Dirac statistics. Write the configuration of a hypothetical Ne (Z = 10) atom made up of such “electrons” [that is, the analog of (1s)2(2s)2(2p)6]. Show that the configuration is highly degenerate. What is the ground state (the lowest term) of the hypothetical Ne atom in spectroscopic notation (2S+1LJ, where S, L, and J stand for the total spin, the total orbital angular momentum, and the total angular momentum, respectively) when exchange splitting and spin-orbit splitting are taken into account? Get solution

7. Two identical spin ... fermions move in one dimension under the influence of the infinite-wall potential V = ∞ for x x > L, and V = 0 for 0 ≤ x ≤ L.a. Write the ground-state wave function and the ground-state energy when the two particles are constrained to a triplet spin state (ortho state).b. Repeat (a) when they are in a singlet spin state (para state).c. Let us now suppose that the two particles interact mutually via a very short-range attractive potential that can be approximated by...Assuming that perturbation theory is valid even with such a singular potential, discuss semiquantitatively what happens to the energy levels obtained in (a) and (b). Get solution

### Chapter #5 Solutions - Modern Quantum Mechanics, Revised Edition - J. J. Sakurai - 1st Edition

1. A simple harmonic oscillator (in one dimension) is subjected to a perturbationλH1 = bxwhere b is a real constant.a. Calculate the energy shift of the ground state to lowest nonvanishing order.b. Solve this problem exactly and compare with your result obtained in (a).[you may assume without proof... Get solution

2. In nondegenerate time-independent perturbation theory, what is the probability of finding in a perturbed energy eigenstate (|k〉) the corresponding unperturbed eigenstate (|k(0)〉)? Solve this up to terms of order g2. Get solution

3. Consider a particle in a two-dimensional potential...Write the energy eigenfunctions for the ground and first excited states. We now add a time-independent perturbation of the from...Obtain the zeroth-order energy eigenfunctions and the first-order energy shifts for ground and first excited states. Get solution

4. Consider an isotropic harmonic oscillator in two dimensions. The Hamiltonian is given by...a. What are the energies of the three lowest-lying states? Is there any degeneracy?b. We now apply a perturbation...Where δ is a dimensionless real number much smaller than unity. Find the zeroth-order energy eigenket and the corresponding energy to first order [that is, the unperturbed energy obtained in (a) plus the first-order energy shift] for each of the three lowest-lying states.c. Solve the H0 + V problem exactly. Compare with the perturbation results obtained in (b).[you may use ...] Get solution

5. Establish (5.1.54) for the one-dimensional harmonic oscillator given by (5.1.50) with an additional perturbation .... Show that all other matrix elements Vk0 vanish. Get solution

6. A slightly anisotropic three-dimensional harmonic oscillator has ωz ≈ ωx = ωy. Acharged particle moves in the field of this oscillator and is at the same time exposed to a uniform magnetic field in the x- direction. Assuming that the Zeeman splitting is comparable to the splitting produced by the anisotropy, but small compared to ℏω, calculate to first order the energies of the components of the first excited state. Discuss various limiting cases. (From Merzbacher, Quantum Mechanics, 2/e, © 1970. Reprinted by permission of Ellis Horwood, Ltd.) Get solution

7. A one-electron atom whose ground state is nondegenerate is placed in a uniform electric field in the z-direction. Obtain an approximate expression for the induced electric dipole moment of the ground state by considering the expectation value of ez with respect to the perturbed state vector computed to first order. Show that the same expression can also be obtained from the energy shift ∆ = − α(E|2/2 of the ground state computed to second order. (Note: α stands for the polarizability.) Ignore spin. Get solution

8. Evaluate the matrix elements (or expectation values) given below. If any vanishes, explain why it vanishes using simple symmetry (or other) arguments.a. 〈n = 2, l = 1, m = 0 |x| n = 2, l = 0, m = 0〉.b. 〈n = 2, l = 1, m = 0|pz| n = 2, l = 0, m = 0〉.[In (a) and (b), |nlm〉 stands for the energy eigenket of a nonrelativistic hydrogen atom with spin ingnored.]c. 〈Lz〉 for an electron in a central field with ...d. 〈singlet, ... triplet, ms = 0〉 for an s-state positronium.e. 〈s(1)·s(2) 〉 for the ground state of a hydrogen molecule Get solution

9. A p-orbital electron characterized by |n, l = 1, m = ±1,0〉 (ignore spin) is subjected to a potential...a. Obtain the “correct” zeroth-order energy eigenstates that diagonalize the perturbation. You need not evaluate the energy shifts in detail, but show that the original threefold degeneracy is now completely removed.b. Because V is invariant under time reversal and because there is no longer any degeneracy, we expect each of the energy eigenstates obtained in (a) to go into itself (up to a phase factor or sign) under time reversal. Check this point explicitly. Get solution

10. Consider a spinless particle in a two-dimensional infinite square well:...a. What are the energy eigenvalues for the three lowest states? Is there any degeneracy?b. We now add a potential...Taking this as a weak perturbation, answer the following:(i) Is the energy shift due to the perturbation linear or quadratic in λ for each of the three states?(ii) Obtain expressions for the energy shifts of the three lowest states accurate to order λ. (You need not evaluate integrals that may appear.)(iii) Draw an energy diagram with and without the perturbation for the three energy states. Make sure to specify which unperturbed state is connected to winch perturbed state. Get solution

11. The Hamiltonian matrix for a two-state system can be written as...Clearly the energy eigenfuctions for the unperturbed problem (λ = 0) are given by...a. Solve this problem exactly to find the energy eigenfunctions ψ2 and the energy eigenvalues E1 and E2.b. Assuming that ... solve the same problem using time-independent perturbation theory up to first order in the energy eigenfunctions and up to second order in the energy eigenvalues. Compare with the exact results obtained in (a).c. Suppose the two unperturbed energies are “almost degenerate,” that is,...Show that the exact results obtained in (a) closely resemble what you would expect by applying degenerate perturbation theory to this problem with ... set exactly equal to ... Get solution

12. (This is a tricky problem because the degeneracy between the first and the second state is not removed in first order. See also Gottfried 1966, 397, problem 1.) This problem is from Schiff 1968, 295, problem 4. A system that has three unperturbed states can be represented by the perturbed has three unperturbed states can be represented by the perturbed Hamiltonian matrix...where E2> E1 The quantities a and b are to be regarded as perturbations that are of the same order and are small Compared with E2–E1 Use the second-order nondegenerate perturbation theory to calculate the perturbed eigenvalues. (Is this procedure correct?) Then diagonalize the matrix to find the exact eigenvalues. Finally, use the second-order degenerate perturbation theory. Compare the three results obtained. Get solution

13. Compute the Stark effect for the 2S1/2 and 2P1/2 levels of hydrogen for a field ε sufficiently weak so that eεa0 is small compared to the fine structure, but take the Lamb shift δ (δ = 1057 MHz) into account (that is, ignore 2P3/2 in this calculation). Show that for eεa0 ≪: δ, the energy shifts are quadratic in ε, whereas for eεa0 ≫ δ they are linear in ε, (The radial integral you need is 〈2s|r|2p〉 = ...) Briefly discuss the consequences (if any) of time reversal for this problem. This problem is from Gottfried 1966, Problem 7-3. Get solution

14. Work out the Stark effect to lowest nonvanishing order for the n = 3 level of the hydrogen atom. Ignoring the spin-orbit force and relativistic correction (Lamb shift), obtain not only the energy shifts to lowest nonvanishing order but also the corresponding zeroth-order eigenket. Get solution

15. Suppose the electron had a very small intrinsic electric dipole moment analogous to the spin magnetic moment (that is,µel proportional to σ). Treating the hypothetical −µel·E interaction as a small perturbation, discuss qualitatively how the energy levels of the Na atom (Z = 11) would be altered in the absence of any external electromagnetic field. Are the level shifts first order or second order? State explicitly which states get mixed with each other. Obtain an expression for the energy shift of the lowest level that is affected by the perturbation. Assume throughout that only the valence electron is subjected to the hypothetical interaction. Get solution

16. Consider a particle bound to a fixed center by a spherically symmetric potential V(r).a. prove...For all s states, ground and excited.b. Check this relation for the ground state of a three-dimensional isotropic oscillator, the hydrogen atom, and so on.(Note: This relation has actually been found to be useful in guessing the from of the potential between a quark and an antiquark.) Get solution

17. a. Suppose the Hamiltonian of a rigid rotator in a magnetic field perpendicular to the axis is of the form (Merzbacher 1970, Problem 17-1)...if terms quadratic in the field are neglected. Assuming B ≫ C, use perturbation theory to lowest nonvanishing order to get approximate energy eigenvalues.b. Consider the matrix elements...Of a one-electron (for example, alkali) atom. Write the selection rules for ∆l, ∆m1, ∆ms. Justify your answer. Get solution

18. Work out the quadratic Zeeman effect for the ground-state hydrogen atom ... due to the usually neglected e2A2/ 2meC2-term in the Hamiltonian taken to first order. Write the energy shift as...And obtain an expression for diamagnetic susceptibility, x (The following definite integral may be useful:... Get solution

19. (Merzbacher 1970, 448, problem 11.) For the he wave function, use...With Zeff = ... as obtained by the variational method. The measured value of the diamagnetic susceptibility is 1.88 × 10−6 cm3/mole.Using the Hamiltonian for an atomic electron in a magnetic field, determine, for a state of zero angular momentum, the energy change to order B2 if the system is in a uniform magnetic field represented by the vector potential ...Defining the atomic diamagnetic susceptibility x by ... calculate x for a helium atom in the ground state and compare the result with measured value. Get solution

20. Estimate the ground-state energy of a one-dimenensional simple harmonic oscillator using...as a trial function with β to be varied. (you may use)... Get solution

21. Estimate the lowest eigenvalue (λ) of the differential equation...Using the variational method with...as a trial function. (Caution: dψ / dx is discontinuous at x = 0) Numerical data that may be useful for this problem are:31/3 = 1.442, 51/3 = 1.710, 32/3 = 2.080, π2/3 = 2.145.The exact value of the lowest eigenvalue can be shown to be 1.019. Get solution

22. Consider a one-dimensional simple harmonic oscillator whose classical angular frequency is ω0. For t t > 0 there is also a time-dependent potentialV(t) = F0xcosωtwhere F0 is constant in both space and time. Obtain an expression for the expectation value 〈x〉 as a function of time using time-dependent perturbation theory to lowest nonvanishing order. Is this procedure valid for ω ≃ ω0? [You may use 〈n'|x|n〉 = ... ... Get solution

23. A one-dimensional harmonic oscillator is in its ground state for t 0. For t ≥ 0 it is subjected to a time-dependent but spatially uniform force (not potential!) in the x-direction,...a. Using time-dependent perturbation theory to first order, obtain the probability of finding the oscillator in its first excited state for t > 0. Show that the t → ∞ (... finite) limit of your expression is independent of time. Is this reasonable or surprising?b. Can we find higher excited states?[You may use... Get solution

24. Consider a particle bound in a simple harmonic oscillator potential. Initially (t 0), it is in the ground state. At t = 0 a perturbation of the form...is switched on. Using time-dependent perturbation theory, calculate the probability that, after a sufficiently long time (t ≫ ...), the system will have made a transition to a given excited state. Consider all final states. Get solution

25. The unperturbed Hamiltonian of a two-state system is represented by...There is, in addition, a time-dependent perturbation...a. At t = 0 the system is known to be in the first state, represented by...Using time-dependent perturbation theory and assuming that ... ... is not close to ± ℏω, derive an expression for the probability that the system be found in the secound state represented by...as a function of t(t > 0).b. Why is this procedure not valid when ... is close to ±ℏω? Get solution

26. A one-dimensional simple harmonic oscillator of angular frequency ω is acted upon by a spatially uniform but time-dependent force (not potential)...At t = −∞, the oscillator is known to be in the ground state. Using the time-dependent perturbation theory to first order, calculate the probability that oscillator is found in the first excited state at t = + ∞.Challenge for experts: F(t) is so normalized that the impulse...imparted to the oscillator is always the same‒that is, independent of ...: yet for ... ≫ 1/ω, the probability for excitation is essentially negligible. Is this reasonable? [Matrix element of x ... .... Get solution

27. Consider a particle in one dimension moving under the influence of some time-independent potential. The energy levels and the corresponding eigenfunctions for this problem are assumed to be known. We now subject the particle to a traveling pulse represented by a time-dependent potential,...a. Suppose at t = − ∞ the particle is known to be in the ground state whose energy eigenfunction is 〈x\i〉 = ui(x). Obtain the probability for finding the system in some excited state with energy eigenfunction 〈x\f〉 = uf(x) at t = + ∞.b. Interpret your result in (a) physically by regarding the δ-function pulse as a superposition of harmonic perturbations; recall...Emphasize the role played by energy conservation, which holds even quantum mechanically as long as the perturbation has been on for a very long time. Get solution

28. A hydrogen atom in its ground state [(n, l, m) = (1,0,0)] is placed between the plates of a capacitor. A time-dependent but spatial uniform electric field (not potential!) is applied as follows:...Using first-order time-dependent perturbation theory, compute the probability for the atom to be found at t ≫ ... in each of the three 2p states: (n, l, m) = (2,1, ±1 or 0). Repeat the problem for the 2s state: (n, l, m) = (2,0,0). You need not attempt to evaluate radial integrals, but perform all other integrations (with respect to angles and time). Get solution

29. Consider a composite system made up of two spin ... objects. For t t > 0, the Hamiltonian is given by...Suppose the system is in |+ − > for t ≤ 0. Find, as a function of time, the probability for being found in each of the following states |+ + 〉, |+− 〉, |−+〉,and|−−〉:a. By solving the problem exactly.b. By solving the problem assuming the validity of first-order time-dependent perturbation theory with H as a perturbation switched on at t = 0. Under what condition does (b) give the correct results? Get solution

30. Consider a two-level system with E1E2. There is a time-dependent potential that connects the two levels as follows:...At t = 0, it is known that only the lower level is populated—that is, c1(0) = l, c2(0) = 0.a. Find |c1(t)|2 and |c2(t)|2 for t>0 by exactly solving the coupled differential equation...b. Do the same problem using time-dependent perturbation theory to lowest nonvanishing order. Compare the two approaches for small values of γ. Treat the following two cases separately: (i) ω very different from ω 21and (ii) ω close to ω21.Answer for (a): (Rabi's formula)... Get solution

31. Show that the slow-turn-on of perturbation V→ Veᵑt (see Baym 1969, 257) can generate contribution from the second term in (5.6.36). Get solution

32. a. Consider the positronium problem you solved in chapter 3, problem 3. In the presence of a uniform and static magnetic field B along the z-axis, the Hamiltonian is given by...Solve this problem to obtain the energy levels of all four states using degenerate time-independent perturbation theory (instead of diagon-alizing the Hamiltonian matrix). Regard the first and the second terms in the expression for H as H0 and V, respectively. Compare your results with the exact expressions...where triplet (singlet) m = 0 stands for the state that becomes a pure triplet (singlet) with m = 0 as B → 0.b. We now attempt to cause transitions (via stimulated emission and absorption) between the two m = 0 states by introducing an oscillating magnetic field of the “right” frequency. Should we orient the magnetic field along the z-axis or along the x- (or y-) axis? Justify your choice. (The original static field is assumed to be along the z-axis throughout.)c. Calculate the eigenvectors to first order.Repeat Problem 32 above, but with the atomic hydrogen Hamiltonian...Where in the hyperfine term AS1, S2, S1 is the electron spin, while s2 is the proton spin. [Note the problem here has less symmetry than that of the positronium case]. Get solution

33. Consider the spontaneous emission of a photon by an excited atom.The process is known to be an E1 transition. Suppose the magnetic quantum number of the atom decreases by one unit. What is the angular distribution of the emitted photon? Also discuss the polarization of the photon with attention to angular-momentum conservation for the whole (atom plus photon) system. Get solution

34. Consider an atom made up of an electron and a singly charged (Z= 1) triton (3H). Initially the system is in its ground state (n =1, l=0). Suppose the nuclear charge suddenly increases by one unit (realistically by emitting an electron and an antineutrino). This means that the triton nucleus turns into a helium (Z = 2) nucleus of mass 3 (3He). Obtain the probability for the system to be found in the ground state of the resulting helium ion. The hydrogenic wave function is given by... Get solution

35. The ground state of a hydrogen atom (n =1,l = 0) is subjected to a time-dependent potential as follows:...Using time-dependent perturbation theory, obtain an expression for the transition rate at which the electron is emitted with momentum p. Show, in particular, how you may compute the angular distribution of the ejected electron (in terms of θ and ϕ defined with respect to the z-axis). Discuss briefly the similarities and the differences between this problem and the (more realistic) photoelectric effect. (Note: For the initial wave function see Problem 34. If you have a normalization problem, the final wave function may be taken to be...with L very large, but you should be able to show that the observable effects are independent of L.) Get solution

36. Derive an expression for the density of free particle states in two-dimensions, that is, the two-dimensional analog of...Your answer should be written as a function ok K (or E) times dE dϕ, where ϕ is the polar angle that characterizes the momentum direction in two dimensions. Get solution

37. A particle of mass m constrained to move in one dimension is confined within 0 x L by an infinite-wall potential...Obtain an expression for the density of states (that is, the number of states per unit energy interval) for high energies as a function of E. (Check your dimension!) Get solution

38. Linearly polarized light of angular frequency ω is incident on a one-electron “atom” whose wave function can be approximated by the ground state of a three-dimensional isotropic harmonic oscillator of angular frequency ω0 Show that the differential cross section for the ejection of a photoelectron is given by...Provided the ejected electron of momentum ℏKf can be regarded as being in a plane-wave state. (The coordinate system used is shown in figure 5.10.) Get solution

39. Find the probability |ϕ(pˊ)|2d3 pˊ of the particular pˊ for the ground-state fourier transforms. To perform the angular integration choose the z-axis in the direction of p.) Get solution

40. Obtain an expression for ... for the hydrogen atom. Verify that it is equal to 1.6 × 10−9 S. Get solution

### Chapter #4 Solutions - Modern Quantum Mechanics, Revised Edition - J. J. Sakurai - 1st Edition

1. Calculate the three lowest energy levels, together with their degeneracies, for the following systems (assume equal mass distinguishable particles):a. Three noninteracting spin ... particles in a box of length L.b. Four noninteracting spin ... particles in a box of length L. Get solution

2. Let ... denote the translation operator (displacement vector d); ... the rotation operator (... and ϕ are the axis and angle of rotation, respectively); and π the parity operator. Which, if any, of the following pairs commute? Why?a. ... and ... (d and d′ in different directions).b. ... and ... (... and ... in different directions).c. ... and π.d. ... and π. Get solution

3. A quantum-mechanical state Ψ is known to be a simultaneous eigenstate of two Hermitian operators A and B which anticommute,...What can you say about the eigenvalues of A and B for state Ψ? Illustrate your point using the parity operator (which can be chosen to satisfy π = π−1 = π†) and the momentum operator. Get solution

4. A spin ... particle is bound to a fixed center by a spherically symmetrical potential.a. Write down the spin angular function ...b. Express ... in terms of some other ...c. Show that your result in (b) is understandable in view of the transformation properties of the operator S · x under rotations and under space inversion (parity). Get solution

5. Because of weak (neutral-current) interactions there is a parity-violating potential between the atomic electron and the nucleus as follows:...where S and p are the spin and momentum operators of the electron, and the nucleus is assumed to be situated at the origin. As a result, the ground state of an alkali atom, usually characterized by |n, l, j, m⟩ actually contains very tiny contributions from other eigenstates as follows:...On the basis of symmetry considerations alone, what can you say about (n′ l′, j′, m′), which give rise to nonvanishing contributions? Suppose the radial wave functions and the energy levels are all known. Indicate how you may calculate Cn′l′j′m′. Do we get further restrictions on (n′, l′, j′, m′)? Get solution

6. Consider a symmetric rectangular double-well potential:...Assuming that V0 is very high compared to the quantized energies of low-lying states, obtain an approximate expression for the energy splitting between the two lowest-lying states. Get solution

7. a. Let ψ|(x, t) be the wave function of a spinless particle corresponding to a plane wave in three dimensions. Show that ψ*(x, −t) is the wave function for the plane wave with the momentum direction reversed.b. Let ... be the two-component eigenspinor of ... with eigenvalue +1. Using the explicit form of ... (in terms of the polar and azimuthal angles β and γ that characterize ... verify that ... is the two-component eigenspinor with the spin direction reversed. Get solution

8. a. Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function for a spinless nondegenerate system at any given instant of time can always be chosen to be real.b. The wave function for a plane-wave state at t = 0 is given by a complex function eip · x/ℏ. Why does this not violate time-reversal invariance? Get solution

9. Let ϕ(p′) be the momentum-space wave function for state |α⟩, that is, ϕ(p′) = ⟨p′|α⟩. Is the momentum-space wave function for the time-reversed state Ө|α⟩ given by ϕ(p′), ϕ(−p′), ϕ*(p′), or ϕ*(−p′)? Justify your answer. Get solution

10. a. What is the time-reversed state corresponding to ...b. Using the properties of time reversal and rotations, prove...c. Prove Ө|j, m⟩ = i2m|j, −m⟩. Get solution

11. Suppose a spinless particle is bound to a fixed center by a potential V(x) so asymmetrical that no energy level is degenerate. Using time-reversal invariance prove...for any energy eigenstate. (This is known as quenching of orbital angular momentum.) If the wave function of such a nondegenerate eigenstate is expanded as...what kind of phase restrictions do we obtain on Fim(r)? Get solution

12. The Hamiltonian for a spin 1 system is given by...Solve this problem exactly to find the normalized energy eigenstates and eigenvalues. (A spin-dependent Hamiltonian of this kind actually appears in crystal physics.) Is this Hamiltonian invariant under time reversal? How do the normalized eigenstates you obtained transform under time reversal? Get solution

### Chapter #3 Solutions - Modern Quantum Mechanics, Revised Edition - J. J. Sakurai - 1st Edition

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

### Chapter #2 Solutions - Modern Quantum Mechanics, Revised Edition - J. J. Sakurai - 1st Edition

1. Consider the spin-precession problem discussed in the text. It can also be solved in the Heisenberg picture. Using the Hamiltonian...write the Heisenberg equations of motion for the time-dependent opertors Sx(t), Sy(t), and Sz(t). Solve them to obtain SX,y,z as functions of time. Get solution

2. Look again at the Hamiltonian of Chapter 1, Problem 11. Suppose the typist made an error and wrote H as...What principle is now violated? Illustrate your point explicitly by attempting to solve the most general time-dependent problem using an illegal Hamiltonian of this kind. (You may assume H11 = H22 = 0 for simplicity.) Get solution

3. An electron is subject to a uniform, time-independent magnetic field of strength B in the positive z-direction. At t = 0 the electron 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 β with the z-axis.a. Obtain the probability for finding the electron in the sx = ℏ/2 state as a function of time.b. Find the expectation value of Sx as a function of time.c. For your own peace of mind show that your answers make good sense in the extreme cases (i) β → 0 and (ii) β → π/2. Get solution

4. Let x(t) be the coordinate operator for a free particle in one dimension in the Heisenberg picture. Evaluate... Get solution

5. Consider a particle in one dimension whose Hamiltonian is given by...By calculating [[H, x], x]prove...where |a′⟩ is an energy eigenket with eigenvalue .... Get solution

6. Consider a particle in three dimensions whose Hamiltonian is given by...By calculating [x · p, H] obtain...To identify the preceding relation with the quantum-mechanical analogue of the virial theorem it is essential that the left-hand side vanish. Under what condition would this happen? Get solution

7. Consider a free-particle wave packet in one dimension. At t = 0 it satisfies the minimum uncertainty relation...In addition, we know...Using the Heisenbergpicture, obtain ⟨(Δx)2⟩, as a function of t(t ≥ 0) when ⟨(Δx)2⟩t = 0 is given. (Hint: Take advantage of the property of the minimum-uncertainty wave packet you worked out in Chapter 1, Problem 18). Get solution

8. Let |a′⟩ and |a″⟩ be eigenstates of a Hermitian operator A with eigenvalues a′ and a″, respectively (a′ ≠ a″). The Hamiltonian operator is given by...Where δ is just a real number.a. Clearly, |a′⟩ and |a″⟩ are not eigenstates of the Hamiltonian. Write down the eigenstates of the Hamiltonian. What are their energy eigenvalues?b. Suppose the system is known to be in state |a′⟩ at t = 0. Write down the state vector in the Schrödinger picture for t > 0.c. What is the probability for finding the system in |a″⟩ for t > 0 if the system is known to be in state |a′⟩ at t = 0?d. Can you think of a physical situation corresponding to this problem? Get solution

9. A box containing a particle is divided into a right and left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state is represented by the position eigenket |R⟩(|L⟩),where we have neglected spatial variations within each half of the box. The most general state vector can then be written as...where ⟨R|α⟩ and ⟨L|α⟩ can be regarded as “wave functions.” The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian...where Δ is a real number with the dimension of energy.a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues?b. In the Schrodinger picture the base kets |R⟩ and |L⟩ are fixed, and the state vector moves with time. Suppose the system is represented by |α⟩ as given above at t = 0. Find the state vector |α, t0 = 0; t⟩ for t > 0 by applying the appropriate time-evolution operator to |α⟩.c. Suppose at t = 0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time?d. Write down the coupled Schrödinger equations for the wave functions ⟨R|α, t0=0; t⟩ and ⟨L|α, t0 = 0; t⟩. Show that the solutions to the coupled Schrödinger equations are just what you expect from (b).e. Suppose the printer made an error and wrote H as...By explicitly solving the most general time-evolution problem with this Hamiltonian, show that probability conservation is violated. Get solution

10. Using the one-dimensional simple harmonic oscillator as an example, illustrate the difference between the Heisenberg picture and the Schrödinger picture. Discuss in particular how (a) the dynamic variables x and p and (b) the most general state vector evolve with time in each of the two pictures. Get solution

11. Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose at t = 0 the state vector is given by...where p is the momentum operator and a is some number with dimension of length. Using the Heisenberg picture, evaluate the expectation value ⟨x⟩ for t ≥ 0. Get solution

12. a. Write down the wave function (in coordinate space) for the state specified in Problem 11 at t = 0. You may use...b. Obtain a simple expression for the probability that the state is found in the ground state at t = 0. Does this probability change for t > 0? Get solution

13. Consider a one-dimensional simple harmonic oscillator.a. Using...evaluate ⟨m|x|n⟩, ⟨m|p|n⟩, ⟨m|{x, p}|n⟩, ⟨m|x2|n⟩, and ⟨m|p2|n⟩.b. Check that the virial theorem holds for the expectation values of the kinetic and the potential energy taken with respect to an energy eigenstate. Get solution

14. a. Using...prove...b. Consider a one-dimensional simple harmonic oscillator. Starting with the Schrödinger equation for the state vector, derive the Schrödinger equation for the momentum-space wave function. (Make sure to distinguish the operator p from the eigenvalue p′.) Can you guess the energy eigenfunctions in momentum space? Get solution

15. Consider a function, known as the correlation function, denned by...where x(t) is the position operator in the Heisenberg picture. Evaluate the correlation function explicitly for the ground state of a one-dimensional simple harmonic oscillator. Get solution

16. Consider again a one-dimensional simple harmonic oscillator. Do the following algebraically, that is, without using wave functions.a. Construct a linear combination of |0⟩ and |1⟩ such that ⟨x⟩ is as large as possible.b. Suppose the oscillator is in the state constructed in (a) at t = 0. What is the state vector for t > 0 in the Schrödinger picture? Evaluate the expectation value ⟨x⟩ as a function of time for t > 0 using (i) the Schrödinger picture and (ii) the Heisenberg picture.c. Evaluate ⟨(Δx)2⟩ as a function of time using either picture. Get solution

17. Show for the one-dimensional simple harmonic oscillator...where x is the position operator. Get solution

18. A coherent state of a one-dimensional simple harmonic oscillator is defined to be an eigenstate of the (non-Hermitian) annihilation operator a:...where λis, in general, a complex number.a. Prove that...is a normalized coherent state.b. Prove the minimum uncertainty relation for such a state.c. Write |λ⟩ as...Show that the distribution of |f(n)|2 with respect to n is of the Poisson form. Find the most probable value of n, hence of E.d. Show that a coherent state can also be obtained by applying the translation (finite-displacement) operator e−ipl/ℏ (where p is the momentum operator, and l is the displacement distance) to the ground state. (See also Gottfried 1966, 262–64.) Get solution

19. Let...where a± and a†± are the annihilation and creation operators of two independent simple harmonic oscillator satisfying the usual simple harmonic oscillator commutation relations. Prove... Get solution

20. Consider a particle of mass m subject to a one-dimensional potential of the following form:...a. What is the ground-state energy?b. What is the expectation value ⟨x2⟩ for the ground state? Get solution

21. A particle in one dimension is trapped between two rigid walls:...At t = 0 it is known to be exactly at x = L/2 with certainty. What are the relative probabilities for the particle to be found in various energy eigenstates? Write down the wave function for t ≥ 0. (You need not worry about absolute normalization, convergence, and other mathematical subtleties.) Get solution

22. Consider a particle in one dimension bound to a fixed center by a δ-function potential of the form...Find the wave function and the binding energy of the ground state. Are there excited bound states? Get solution

23. A particle of mass m in one dimension is bound to a fixed center by an attractive δ-function potential:...At t = 0, the potential is suddenly switched off (that is, V = 0 for t > 0). Find the wave function for t > 0. (Be quantitative! But you need not attempt to evaluate an integral that may appear.) Get solution

24. A particle in one dimension (− ∞ x ...a. Is the energy spectrum continuous or discrete? Write down an approximate expression for the energy eigenfunction specified by E. Also sketch it crudely.b. Discuss briefly what changes are needed if V is replaced by... Get solution

25. Consider an electron confined to the interior of a hollow cylindrical shell whose axis coincides with the z-axis. The wave function is required to vanish on the inner and outer walls, ρ = ρa and ρb, and also at the top and bottom, z = 0 and L.a. Find the energy eigenfunctions. (Do not bother with normalization.) Show that the energy eigenvalues are given by...where kmn is the nth root of the transcendental equation...b. Repeat the same problem when there is a uniform magnetic field ... for 0 ρ a. Note that the energy eigenvalues are influenced by the magnetic field even though the electron never “touches” the magnetic field.c. Compare, in particular, the ground state of the B = 0 problem with that of the B ≠ 0 problem. Show that if we require the ground-state energy to be unchanged in the presence of B, we obtain “flux quantization”... Get solution

26. Consider a particle moving in one dimension under the influence of a potential V(x). Suppose its wave function can be written as exp[iS(x, t)/ℏ]. Prove that S(x, t) satisfies the classical Hamilton-Jacobi equation to the extent that ℏ can be regarded as small in some sense. Show how one may obtain the correct wave function for a plane wave by starting with the solution of the classical Hamilton-Jacobi equation with V(x) set equal to zero. Why do we get the exact wave function in this particular case? Get solution

27. Using spherical coordinates, obtain an expression for j for the ground and excited states of the hydrogen atom. Show, in particular, that for ml ≠ 0 states, there is a circulating flux in the sense that j is in the direction of increasing or decreasing ϕ, depending on whether ml is positive or negative. Get solution

28. Derive (2.5.16) and obtain the three-dimensional generalization of (2.5.16). Get solution

29. Define the partition function as...as in (2.5.20)−(2.5.22). Show that the ground-state energy is obtained by taking...Illustrate this for a particle in a one-dimensional box. Get solution

30. The propagator in momentum space analogous to (2.5.26) is given by ⟨p″, t|p′, t0). Derive an explicit expression for ⟨p″, t|p′, t0) for the free-particle case. Get solution

31. a. Write down an expression for the classical action for a simple harmonic oscillator for a finite time interval.b. Construct ⟨xn, tn|xn−1, tn−1⟩ for a simple harmonic oscillator using Feynman’s prescription for tn − tn−1 = Δt small. Keeping only terms up to order (Δt)2, show that it is in complete agreement with the t − t0 → 0 limit of the propagator given by (2.5.26). Get solution

32. State the Schwinger action principle (see Finkelstein 1973, 155). Obtain the solution for ⟨x2t2|x1t1⟩ by integrating the Schwinger principle and compare it with the corresponding Feynman expression for ⟨x2t1|x1t1⟩. Describe the classical limits of these two expressions. Get solution

33. Show that the wave-mechanics approach to the gravity-induced problem discussed in Section 2.6 also leads to phase-difference expression (2.6.17). Get solution

34. a. Verify (2.6.25) and (2.6.27).b. Verify continuity equation (2.6.30) with j given by (2.6.31). Get solution

35. Consider the Hamiltonian of a spinless particle of charge e. In the presence of a static magnetic field, the interaction terms can be generated by...where A is the appropriate vector potential. Suppose, for simplicity, that the magnetic field B is uniform in the positive z-direction. Prove that the above prescription indeed leads to the correct expression for the interaction of the orbital magnetic moment (e/2mc)L with the magnetic field B. Show that there is also an extra term proportional to B2(x2 + y2), and comment briefly on its physical significance. Get solution

36. An electron moves in the presence of a uniform magnetic field in the z-direction ...a. Evaluate...where...b. By comparing the Hamiltonian and the commutation relation obtained in (a) with those of the one-dimensional oscillator problem, show how we can immediately write the energy eigenvalues as...where ℏk is the continuous eigenvalue of the pz operator and n is a nonnegative integer including zero. Get solution

37. Consider the neutron interferometer....Prove that the difference in the magnetic fields that produce two successive maxima in the counting rates in given by...where gn(= −1.91) is the neutron magnetic moment in units of −eℏ/2mnc. [If you had solved this problem in 1967, you could have punished it in physical Review Letters!] Get solution

### Chapter #1 Solutions - Modern Quantum Mechanics, Revised Edition - J. J. Sakurai - 1st Edition

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

### Chapter #7 Solutions - Modern Quantum Mechanics, Revised Edition - J. J. Sakurai - 1st Edition

1. The Lippmann-Schwinger formalism can also be applied to a one-dimensional transmission-reflection problem with a finite-range pote...