1. (a) Work out all of the canonical commutation relations for
components of the operators r and p: [x, y], [x, py], [x, px], [py, pz],
and so on. Answer:... [4.10]where the indices stand for x, y, or z, and
rx = x, ry = y, and rz = z.(b) Confirm Ehrenfest's theorem for
3-dimensions:... [4.11](Each of these, of course, stands for three
equations—one for each component.) Hint: First check that Equation 3.71
is valid in three dimensions.(C) Formulate Heisenberg's uncertainty
principle in three dimensions. Answer:... [4.12]but there is no
restriction on, say,... Get solution
2. Use separation of variables in cartesian coordinates to solve the infinite cubical well (or "particle in a box"):...(a) Find the stationary states, and the corresponding energies.(b) Call the distinct energies E1, E2, E3... in order of increasing energy. Find E1, E2, E3, E4, E5, and E6. Determine their degeneracies (that is, the number of different states that share the same energy). Comment: In one dimension degenerate bound states do not occur (see Problem 2.45), but in three dimensions they are very common.(c) What is the degeneracy of E14, and why is this case interesting? Get solution
3. Use Equations 4.27, 4.28, and 4.32, to construct ... and .... Check that they are normalized and orthogonal. (Reference: Equations) ... ... ... Get solution
4. Show that ... satisfies the (Reference: Equation) ... Get solution
5. Use Equation 4.32 to construct ...(ѳ,Φ) and...(ѳ,Φ). (You can take ... from Table 4.2, but you'll have to work out ... from Equations 4.27 and 4.28.) Check that they satisfy the angular equation (Equation 4.18), for the appropriate values of I and m. Get solution
6. Starting from the Rodrigues formula, derive the orthonormality condition for Legendre polynomials:... [4.34]Hint: Use integration by parts Get solution
7. (a) From the definition (Equation 4.46), construct n1(x) and n2(x).(b) Expand the sines and cosines to obtain approximate formulas for n1(x) and n2(x), valid when x Get solution
8. (a) Check that Ar1(kr) satisfies the radial equation with V(r) = 0 and l = 1.(b) Determine graphically the allowed energies for the infinite spherical well, when l = 1. Show that for large n, En1 ≈ (h2∏2/2ma2)(n + 1/2)2. Hint: First show that j1(x) = 0 ... x = tan x. Plot x and tan x on the same graph, and locate the points of intersection Get solution
9. A particle of mass m is placed in a finite spherical well:...Find the ground state, by solving the radial equation with l = 0. Show that there is no bound state if .... Get solution
10. Work out the radial wave functions ... using the recursion formula (Equation 4.76). Don’t bother to normalize them. ... Get solution
11. a. Normalize ... (Equation 4.82), and construct the function ... b. Normalize ... (Equation 4.83), and construct the function ... ... ... Get solution
12. (a) Using Equation 4.88, work out the first four Laguerre polynomials. (b) Using Equations 4.86, 4.87, and 4.88, find..., for the case n = 5, l = 2. (c) Find ... again (for the case n = 5, l = 2), but this time get it from the recursion formula (Equation 4.76). (Reference: Equations) ... [4.76] ... [4.86] ... [4.87] ... [4.88] Get solution
13. (a) Find (r) and (r2) for an electron in the ground state of hydrogen. Express your answer in terms of Bohr radius.(b) Find {x} and {x2} for and electron in the ground state of hydrogen. Hint: This requires no new integration -note that r2 = x2 + y2 + z2, and exploit the symmetry of the ground state.(c) Find (x2) in the state n = 2, 1 = 1, m = 1. Warning: This state is not symmetrical in x, y, z. Use x = r sin ѳ cos Φ Get solution
14. What is the most probable value of r, in the ground state of hydrogen? (The answer is not zero!) Hint: First you must figure out the probability that the electron would be found between r and r + dr Get solution
15. A hydrogen atom starts out in the following linear combination of the stationary states n = 2, l = 1, m = 1 and n = 2, l = 1, m = -1...(a) Construct Ψ(r, t). Simplify it as much as you can.(b) Find the expectation value of potential energy, ...V.... (Does it depend on t? Give both the formula and the actual number, in electron volts. Get solution
16. A hydrogenic atom consists of a single electron orbiting a nucleus with Z protons (Z =1 would be hydrogen itself, Z = 2 is ionized helium, Z = 3 ... is doubly ionized lithium, and so on). Determine the Bohr energies ... the binding energy E1 (Z), the Bohr radius a (Z), and the Rydberg constant R(Z) for a hydrogenic atom. (Express your answers as appropriate multiples of the hydrogen values.) Where in the electromagnetic spectrum would the Lyman series fall, for Z = 2 and Z = 3? Hint: There's nothing much to calculate here—in the potential (Equation 4.52) ... so all you have to do is make the same substitution in all the final results. ... Get solution
17. Consider the earth-sun system as a gravitational analog to the hydrogen atom.(a) What is the potential energy function (replacing Equation 4.52)? (Let m be the mass of the earth, and M the mass of the sun.)(b) What is the "Bohr radius," ag, for this system? Work out the actual number.(c) Write down the gravitational "Bohr formula," and, by equating En to the classical energy of a planet in a circular orbit of radius r0, show that n = .... From this, estimate the quantum number n of the earth.(d) Suppose the earth made a transition to the next lower level (n — 1). How much energy (in Joules) would be released? What would the wavelength of the emitted photon (or, more likely, graviton) be? (Express your answer in light years—is the remarkable answer20 a coincidence?) Get solution
18. The raising and lowering operators change the value of m by one unit:... [4.120]where ... is some constant. Question: What is ... if the eigenfunctions are to be normalized? Hint: First show that ... is the hermitian conjugate of L± (since Lx and Ly are observables, you may assume they are hermitian ... but prove it if you like); then use Equation 4.112. Answer:... [4.121]Note what happens at the top and bottom of the ladder (i.e., when you apply L+ to ... or L- to ...). Get solution
19. (a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators:... [4.122](b) Use these results to obtain [Lz, Lv] = ihLy directly from Equation 4.96.(c) Evaluate the commutators [Lz, r2] and [Lz, p2] (where, of course, r2 = x2 + y2 + z2 and p2 =... + ... +...).(d) Show that the Hamiltonian H = (p2/2m) + V commutes with all three components of L, provided that V depends only on r. (Thus H, L2, and Lz are mutually compatible observables.) Get solution
20. (a) Prove that for a particle in a potentialV(r) the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque:...Where...(This is the rotational analog to Ehrenfest's theorem.)(b) Show that ... for any spherically symmetric potential. (This is one form of the quantum statement of conservation of angular momentum.) Get solution
21. (a) Derive Equation 4.131 from Equation 4.130. Hint: Use a test function; otherwise you're likely to drop some terms.(b) Derive Equation 4.132 from Equations 4.129 and 4.131. Hint: Use Equation 4.112. Get solution
22. (a) What is (No calculation allowed!) (b) Use the result of (a), together with Equation 4.130 and the fact that ... ... to determine ...up to a normalization constant. (c) Determine the normalization constant by direct integration. Compare your final answer to what you got in problem 4.5. ... ... Get solution
23. In problem 4.3 you showed that ... Apply the raising operator of find ... use equation 4.121 to get the normalization. ... problem 4.3: ... ... ... ... Get solution
24. Two particles of mass m are attached to the ends of a massless rigid rod of length a. The system is free to rotate in three dimensions about the center (but the centre point itself is fixed).(a) show that the allowed energies of this rigid rod are... for n = 0, 1, 2,...Hint: First express the (classical) energy in terms of the total angular momentum.(b) What are the normalized eigenfunctions for this system? What is the degeneracy of the nth energy level? Get solution
25. If the electron were a classical solid sphere, with radius... [4.138](the so-calledclassical electron radius, obtained by assuming the electron's mass is attributable to energy stored in its electric field, via the Einstein formula E = mc2), and its angular momentum is (1/2)..., then how fast (in m/s) would a point on the "equator" be moving? Does this model make sense? (Actually, the radius of the electron is known experimentally to be much less than rc, but this only makes matters worse.) Get solution
26. (a) Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134. (b) Show that the pauli spin matrices (Equation 4.148) satisfy the product rule ... Where the indices stand for x, y, or z and ...is the Levi-Civita symbol: +1 if jkl = 123, 231, or 312; -1 if jkl = 132, 213, or 321; 0 otherwise. ... ... ... ... Get solution
27. An electron is in the spin state...(a) Determine the normalization constant A.(b) Find the expectation values of Sx, Sy, and Sz.(c) Find the "uncertainties" ...,..., and ... (Note: These sigmas are standard deviations, not Pauli matrices!)(d) Confirm that your results are consistent with all three uncertainty principles (Equation 4.100 and its cyclic permutations—only with S in place of L, of course). Get solution
28. For the most general normalized spinor ... (Equation 4.139), compute ...... ... Get solution
29. (a) Find the eigenspinors of Sy(b) If you measured Sy on a particle in the general state ... (Equation 4.139) what values might you get, and what is the probability of each? Check the the probabilities add up to 1. Note: a and h need not be real!(c) If you measured ..., what values might you get, and with what probabilities? Get solution
30. Construct the matrix Sr representing the component of spin angular momentum along an arbitrary direction .... Use spherical coordinates, for which... [4.155]Note: You're always free to multiply by an arbitrary phase factor—say, eiΦ —so your answer may not look exactly the same as mine Get solution
31. Construct the spin matrices(Sx, Sy, and Sz) for a particle of spin 1. Hint: How many eigenstates of Sz are there? Determine the action of Sz, S+, and S- on each of these states. Follow the procedure used in the text for spin 1/2. Get solution
32. In Example 4.3: (a) If you measured the component of spin angular momentum along the x direction, at time t, what is the probability that you would get ... (b) Same question, but for the y component. (c) Same, for the z component. ... ... Get solution
33. An electron is at rest in an oscillating magnetic field...where Bo and ... are constants.(a) Construct the Hamiltonian matrix for this system.(b) The electron starts out (at t = 0) in the spin-up state with respect to the x-axis (that is: ... (0) = .... Determine ... (t) at any subsequent time. Beware: This is a time-dependent Hamiltonian, so you cannot get x(0 in the usual way from stationary states. Fortunately, in this case you can solve the time- dependent Schrodinger equation (Equation 4.162) directly.(c) Find the probability of getting —.../2, if you measure Sx. Answer:...(d) What is the minimum field (Bo) required to force a complete flip in Sx? Get solution
34. (a) Apply ... (Equation 4.177), and confirm that you get ... (b) Apply ... (Equation 4.178), and confirm that you get (c) Show that ... (Equation 4.177) are eigenstates of ... with the appropriate eigenvalue. ... ... Get solution
35. Quarks carry spin 1/2. Three quarks bind together to make a baryon (such as the proton or neutron); two quarks (or more precisely a quark and an antiquark) bind together to make a meson (such as the pion or the kaon). Assume the quarks are in the ground state (so the orbital angular momentum is zero).(a) What spins are possible for baryons?(b) What spins are possible for mesons? Get solution
36. (a) A particle of spin 1 and a particle of spin 2 are at rest in a configuration such that the total spin is 3, and its z component is h. If you measured the z component of the angular momentum of the spin-2 particle, what values might you get, and what is the probability of each one?(b) An electron with spin down is in the state of the hydrogen atom. If you could measure the total angular momentum squared of the electron alone (not including the proton spin), what values might you get, and what is the probability of each? Get solution
37. Determine the commutator of ... Generalize your result to show that ... Comment: Because ...does not commute with ... we cannot hope to find states that are simultaneous eigenvectors of both. In order to form eigenstates of ...we need linear combinations of eigenstates of ... This is precisely what the Clebsch- Gordan coefficients (in Equation 4.185) do for us. On the other hand, it follows by obvious inference from Equation 4.187 that the sum ... does commute with ... which is a special case of something we already knew (see Equation 4.103) ... ... ... Get solution
38. Consider the three-dimensional harmonic oscillator, for which the potential is... [4.188](a) Show that separation of variables in cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies. Answer:... [4.189](b) Determine the degeneracy d(n) of En Get solution
39. Because the three-dimensional harmonic oscillator potential (Equation 4.188) is spherically symmetric, the Schrodinger equation can be handled by separation of variables in spherical coordinates, as well as cartesian coordinates. Use the power series method to solve the radial equation. Find the recursion formula for the coefficients, and determine the allowed energies. Check your answer against Equation 4.189. ... Get solution
40. (a)Prove the three-dimensional virial theorem:... [4.190](for stationary states). Hint: Refer to Problem 3.31.(b) Apply the virial theorem to the case of hydrogen, and show that...T... = -En; ...V... = 3En [4.191](c) Apply the virial theorem to the three-dimensional harmonic oscillator (Problem 4.38), and show that in this case...T... = ...V... = En/2 [4.192] Get solution
41. [Attempt this problem only if you are familiar with vector calculus.] 'Define the (three-dimensional) probability current by generalization of Problem 1.14:... [4.193](a) Show that J satisfies the continuity equation... [4.194]which expresses local conservation of probability. It follows (from the divergence theorem) that... [4.194]where ... is a (fixed) volume and S is its boundary surface. In words: The flow of probability out through the surface is equal to the decrease in probability of finding the particle in the volume.(b) Find J for hydrogen in the state n = 2, l = 1, m = 1. Answer:...(c) If we interpret mJ as the flow of mass, the angular momentum is...Use this to calculate Lz for the state Ψ211, and comment on the result Get solution
42. The (time independent) momentum space wave function is three dimensions is defined by the natural generalization of Equation 3.54:... [4.196](a) Find the momentum space wave function for the ground state of hydrogen (Equation 4.80). Hint: Use spherical coordinates, setting the polar axis along the direction of p. Dotheѳ integral first. Answer... [4.197](b) Check that Φ (p) is normalized.(c) Use Φ(p) to calculate (p2), in the ground state of hydrogen.(d) What is the expectation value of the kinetic energy in this state? Express your answer as a multiple of E1, and check that it is consistent with the virial theorem (Equation 4.191). Get solution
43. (a) Construct the spatial wave function (Ψ) for hydrogen in the state n = 3,l = 2, m = 1. Express your answer as a function of r, ѳ, Φ, and a (the Bohr radius) only—no other variables (ρ, z, etc.) or functions (Y, v, etc.), or constants (A, co, etc.), or derivatives, allowed (∏ is okay, and e, and 2, etc.).(b) Check that this wave function is properly normalized, by carrying out the appropriate integrals over r, ѳ, and Φ.(c) Find the expectation value of rs in this state. For what range of s (positive and negative) is the result finite? Get solution
44. (a) Construct the wave function for hydrogen in the state n = 4, l = 3, m = 3. Express your answer as a function of the spherical coordinates r, ѳ, and Φ.(b) Find the expectation value of r in this state. (As always, look up any nontrivial integrals.)(c) If you could somehow measure the observable on an atom in this state, what value (or values) could you get, and what is the probability of each? Get solution
45. What is the probability that an electron in the ground state of hydrogen will be found inside the nucleus?(a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus.(b) Expand your result as a power series in the small number ε = 2b/a, and show that the lowest-order term is the cubic: P ≈ (4/3)(b/a)3. This should be a suitable approximation, provided that b a (which it is).(c) Alternatively, we might assume that ψ(r) is essentially constant over the (tiny) volume of the nucleus, so that P ≈ (4/3)πb3|ψ(0)|2. Check that you get the same answer this way.(d) Use b ≈ 10‒15 m and a ≈ 0.5 × 10‒10 m to get a numerical estimate for P. Roughly speaking, this represents the “fraction of its time that the electron spends inside the nucleus.” Get solution
46. (a) Use the recursion formula (Equation 4.76) to confirm that when l = n ‒ 1 the radial wave function takes the form(b) ...and determine the normalization constant Nn by direct integration.(b) Calculate ...r... and ...r2... for states of the form ψn(n-1)m.(c) Show that the “uncertainty” in r (σr) is ...r.../y/√2n + 1 for such states. Note that the fractional spread in r decreases, with increasing n (in this sense the system “begins to look classical,” with identifiable circular “orbits,” for large n). Sketch the radial wave functions for several values of n, to illustrate this point. Get solution
47. Coincident spectral lines. According to the Rydberg formula (Equation 4.93) the wavelength of a line in the hydrogen spectrum is determined by the principal quantum numbers of the initial and final states. Find two distinct pairs ... that yield the same ... For example, {6851, 6409} and { 15283, 11687} will do it, but you're not allowed to use those! ... Get solution
48. Consider the observables A = x2 and B = LZ.(a) Construct the uncertainty principle for σAσB.(b) Evaluate σB in the hydrogen state ψnlm.(c) What can you conclude about ...xy... in this state? Get solution
49. An electron is in the spin state...(a) Determine the constant A by normalizing χ.(b) If you measured Sz on this electron, what values could you get, and what is the probability of each? What is the expectation value of Sz?(c) If you measured Sx on this electron, what values could you get, and what is the probability of each? What is the expectation value of Sx?(d) If you measured Sy on this electron, what values could you get, and what is the probability of each? What is the expectation value of Sy? Get solution
50. Suppose two spin ½ particles are known to be in the singlet configuration (Equation 4.178).Let ... be the component of the spin angular momentum of particle number 1 in the direction defined by the unit vector ...Similarly, let ... be the component of 2’s angular momentum in the direction ... show that ... Where ... is the angle between ... ... Get solution
51. (a) Work out the Clebsch-Gordan coefficients for the case ... = anything. Hint: You're looking for the coefficients A and B in ... where the signs are determined by ... (b) Check this general result against three or four entries in Table 4.8. (References: Table 4.8 & Equations) ... ... [4.136] ... ... ... Get solution
52. Find the matrix representing Sxfor a particle of spin 3/2 (using, as always, the basis of eigenstates of Sz). Solve the characteristics equation to determine the eigenvalues of Sx. Get solution
53. Work out the spin matrices for arbitrary spin s, generalizing spin 1/2 (Equations 4.145 and 4.147), spin 1 (Problem 4.31), and spin 3/2 (Problem 4.52).Answer:...where... Get solution
54. Work out the normalization factor for the spherical harmonics, as follows. From section 4.12 we know that ... The problem is to determine the factor ... (which I quoted, but did not derive, in Equation 4.32). use Equations 4.120, 4.121, and 4.130 to obtain a recursion relation giving ... in terms of ...solve it by induction on m to get ... up to an overall constant, ... Finally, use the result of Problem 4.22 to fix the constant. You may find the following formula for the derivative of an associated Legendrefunction useful: ... ... ... ... ... Get solution
55. The electron in a hydrogen atom occupies the combined spin and position state...(a) If you measured the orbital angular momentum squared (L2), what values might you get, and what is the probability of each?(b) Same for the z component of orbital angular momentum (Lz).(c) Same for the spin angular momentum squared (S2).(d) Same for the z component of spin angular momentum (Sz)Let J ≡ L + S be the total angular momentum.(e) If you measured J2, what values might you get, and what is the probability of each?(f) Same for Jz.(g) If you measured the position of the particle, what is the probability density for finding it at r, θ, ф?(h) If you measured both the z component of the spin and the distance from the origin (note that these are compatible observables), what is the probability density for finding the particle with spin up and at radius r? Get solution
56. (a) For a function f(ф) that can be expanded in a Taylor series, show that...(where φ is an arbitrary angle). For this reason, ... is called the generator of rotations about the z-axis Hint: Use Equation 4.129, and refer to Problem 3.39.More generally, ... is the generator of rotations about the direction ..., in the sense that exp(...) effects a rotation through angle φ (in the right-hand sense) about the axis .... In the case of spin, the generator of rotations is .... In particular, for spin 1/2... [4.200]tells us how spinors rotate.(b) Construct the (2 × 2) matrix representing rotation by 180° about the x-axis, and show that it converts “spin up” (χ+) into "spin down" (χ‒), as you would expect.(c) Construct the matrix representing rotation by 90° about the y-axis, and check what it does to χ+(d) Construct the matrix representing rotation by 360° about the z-axis. If the answer is not quite what you expected, discuss its implications.(e) Show that... [4.201] Get solution
57. The fundamental commutation relations for angular momentum (Equation 4.99) allow for half-integer (as well as integer) eigenvalues. But for orbital angular momentum only the integer values occur. There must be some extra constraint in the specific form L = r x p that excludes half-integer values.` Let a be some convenient constant with the dimensions of length (the Bohr radius, say, if we're talking about hydrogen), and define the operators ... ... ... Get solution
58. Deduce the condition for minimum uncertainty in ... (that is, equality in the expression... for a particle of spin ½ in the generic state (Equation 4.139). Answer: With no loss of generality we can pick a to be real; then the condition for minimum uncertainty is that b is either pure real or else pure imaginary. ... Get solution
59. ... ... ... Get solution
60. [Refer to Problem 4.59 for background.] Suppose...where B0 and K are constants.(a) Find the fields E and B.(b) Find the allowed energies, for a particle of mass m and charge q, in these fields. Answer:... [4.209]where to... and ... Comment: If K = 0 this is the quantum analog to cyclotron motion; ω1 is the classical cyclotron frequency, and it's a free particle in the z direction. The allowed energies, ... are called Landau Levels.46 Get solution
61. [Refer to problem 4.59 for background.] In classical electrodynamics the potentials A and ... are not uniquely determined; the physical quantities are the fields, E and B. ... Refer to problem 4.59: ... Get solution
2. Use separation of variables in cartesian coordinates to solve the infinite cubical well (or "particle in a box"):...(a) Find the stationary states, and the corresponding energies.(b) Call the distinct energies E1, E2, E3... in order of increasing energy. Find E1, E2, E3, E4, E5, and E6. Determine their degeneracies (that is, the number of different states that share the same energy). Comment: In one dimension degenerate bound states do not occur (see Problem 2.45), but in three dimensions they are very common.(c) What is the degeneracy of E14, and why is this case interesting? Get solution
3. Use Equations 4.27, 4.28, and 4.32, to construct ... and .... Check that they are normalized and orthogonal. (Reference: Equations) ... ... ... Get solution
4. Show that ... satisfies the (Reference: Equation) ... Get solution
5. Use Equation 4.32 to construct ...(ѳ,Φ) and...(ѳ,Φ). (You can take ... from Table 4.2, but you'll have to work out ... from Equations 4.27 and 4.28.) Check that they satisfy the angular equation (Equation 4.18), for the appropriate values of I and m. Get solution
6. Starting from the Rodrigues formula, derive the orthonormality condition for Legendre polynomials:... [4.34]Hint: Use integration by parts Get solution
7. (a) From the definition (Equation 4.46), construct n1(x) and n2(x).(b) Expand the sines and cosines to obtain approximate formulas for n1(x) and n2(x), valid when x Get solution
8. (a) Check that Ar1(kr) satisfies the radial equation with V(r) = 0 and l = 1.(b) Determine graphically the allowed energies for the infinite spherical well, when l = 1. Show that for large n, En1 ≈ (h2∏2/2ma2)(n + 1/2)2. Hint: First show that j1(x) = 0 ... x = tan x. Plot x and tan x on the same graph, and locate the points of intersection Get solution
9. A particle of mass m is placed in a finite spherical well:...Find the ground state, by solving the radial equation with l = 0. Show that there is no bound state if .... Get solution
10. Work out the radial wave functions ... using the recursion formula (Equation 4.76). Don’t bother to normalize them. ... Get solution
11. a. Normalize ... (Equation 4.82), and construct the function ... b. Normalize ... (Equation 4.83), and construct the function ... ... ... Get solution
12. (a) Using Equation 4.88, work out the first four Laguerre polynomials. (b) Using Equations 4.86, 4.87, and 4.88, find..., for the case n = 5, l = 2. (c) Find ... again (for the case n = 5, l = 2), but this time get it from the recursion formula (Equation 4.76). (Reference: Equations) ... [4.76] ... [4.86] ... [4.87] ... [4.88] Get solution
13. (a) Find (r) and (r2) for an electron in the ground state of hydrogen. Express your answer in terms of Bohr radius.(b) Find {x} and {x2} for and electron in the ground state of hydrogen. Hint: This requires no new integration -note that r2 = x2 + y2 + z2, and exploit the symmetry of the ground state.(c) Find (x2) in the state n = 2, 1 = 1, m = 1. Warning: This state is not symmetrical in x, y, z. Use x = r sin ѳ cos Φ Get solution
14. What is the most probable value of r, in the ground state of hydrogen? (The answer is not zero!) Hint: First you must figure out the probability that the electron would be found between r and r + dr Get solution
15. A hydrogen atom starts out in the following linear combination of the stationary states n = 2, l = 1, m = 1 and n = 2, l = 1, m = -1...(a) Construct Ψ(r, t). Simplify it as much as you can.(b) Find the expectation value of potential energy, ...V.... (Does it depend on t? Give both the formula and the actual number, in electron volts. Get solution
16. A hydrogenic atom consists of a single electron orbiting a nucleus with Z protons (Z =1 would be hydrogen itself, Z = 2 is ionized helium, Z = 3 ... is doubly ionized lithium, and so on). Determine the Bohr energies ... the binding energy E1 (Z), the Bohr radius a (Z), and the Rydberg constant R(Z) for a hydrogenic atom. (Express your answers as appropriate multiples of the hydrogen values.) Where in the electromagnetic spectrum would the Lyman series fall, for Z = 2 and Z = 3? Hint: There's nothing much to calculate here—in the potential (Equation 4.52) ... so all you have to do is make the same substitution in all the final results. ... Get solution
17. Consider the earth-sun system as a gravitational analog to the hydrogen atom.(a) What is the potential energy function (replacing Equation 4.52)? (Let m be the mass of the earth, and M the mass of the sun.)(b) What is the "Bohr radius," ag, for this system? Work out the actual number.(c) Write down the gravitational "Bohr formula," and, by equating En to the classical energy of a planet in a circular orbit of radius r0, show that n = .... From this, estimate the quantum number n of the earth.(d) Suppose the earth made a transition to the next lower level (n — 1). How much energy (in Joules) would be released? What would the wavelength of the emitted photon (or, more likely, graviton) be? (Express your answer in light years—is the remarkable answer20 a coincidence?) Get solution
18. The raising and lowering operators change the value of m by one unit:... [4.120]where ... is some constant. Question: What is ... if the eigenfunctions are to be normalized? Hint: First show that ... is the hermitian conjugate of L± (since Lx and Ly are observables, you may assume they are hermitian ... but prove it if you like); then use Equation 4.112. Answer:... [4.121]Note what happens at the top and bottom of the ladder (i.e., when you apply L+ to ... or L- to ...). Get solution
19. (a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators:... [4.122](b) Use these results to obtain [Lz, Lv] = ihLy directly from Equation 4.96.(c) Evaluate the commutators [Lz, r2] and [Lz, p2] (where, of course, r2 = x2 + y2 + z2 and p2 =... + ... +...).(d) Show that the Hamiltonian H = (p2/2m) + V commutes with all three components of L, provided that V depends only on r. (Thus H, L2, and Lz are mutually compatible observables.) Get solution
20. (a) Prove that for a particle in a potentialV(r) the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque:...Where...(This is the rotational analog to Ehrenfest's theorem.)(b) Show that ... for any spherically symmetric potential. (This is one form of the quantum statement of conservation of angular momentum.) Get solution
21. (a) Derive Equation 4.131 from Equation 4.130. Hint: Use a test function; otherwise you're likely to drop some terms.(b) Derive Equation 4.132 from Equations 4.129 and 4.131. Hint: Use Equation 4.112. Get solution
22. (a) What is (No calculation allowed!) (b) Use the result of (a), together with Equation 4.130 and the fact that ... ... to determine ...up to a normalization constant. (c) Determine the normalization constant by direct integration. Compare your final answer to what you got in problem 4.5. ... ... Get solution
23. In problem 4.3 you showed that ... Apply the raising operator of find ... use equation 4.121 to get the normalization. ... problem 4.3: ... ... ... ... Get solution
24. Two particles of mass m are attached to the ends of a massless rigid rod of length a. The system is free to rotate in three dimensions about the center (but the centre point itself is fixed).(a) show that the allowed energies of this rigid rod are... for n = 0, 1, 2,...Hint: First express the (classical) energy in terms of the total angular momentum.(b) What are the normalized eigenfunctions for this system? What is the degeneracy of the nth energy level? Get solution
25. If the electron were a classical solid sphere, with radius... [4.138](the so-calledclassical electron radius, obtained by assuming the electron's mass is attributable to energy stored in its electric field, via the Einstein formula E = mc2), and its angular momentum is (1/2)..., then how fast (in m/s) would a point on the "equator" be moving? Does this model make sense? (Actually, the radius of the electron is known experimentally to be much less than rc, but this only makes matters worse.) Get solution
26. (a) Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134. (b) Show that the pauli spin matrices (Equation 4.148) satisfy the product rule ... Where the indices stand for x, y, or z and ...is the Levi-Civita symbol: +1 if jkl = 123, 231, or 312; -1 if jkl = 132, 213, or 321; 0 otherwise. ... ... ... ... Get solution
27. An electron is in the spin state...(a) Determine the normalization constant A.(b) Find the expectation values of Sx, Sy, and Sz.(c) Find the "uncertainties" ...,..., and ... (Note: These sigmas are standard deviations, not Pauli matrices!)(d) Confirm that your results are consistent with all three uncertainty principles (Equation 4.100 and its cyclic permutations—only with S in place of L, of course). Get solution
28. For the most general normalized spinor ... (Equation 4.139), compute ...... ... Get solution
29. (a) Find the eigenspinors of Sy(b) If you measured Sy on a particle in the general state ... (Equation 4.139) what values might you get, and what is the probability of each? Check the the probabilities add up to 1. Note: a and h need not be real!(c) If you measured ..., what values might you get, and with what probabilities? Get solution
30. Construct the matrix Sr representing the component of spin angular momentum along an arbitrary direction .... Use spherical coordinates, for which... [4.155]Note: You're always free to multiply by an arbitrary phase factor—say, eiΦ —so your answer may not look exactly the same as mine Get solution
31. Construct the spin matrices(Sx, Sy, and Sz) for a particle of spin 1. Hint: How many eigenstates of Sz are there? Determine the action of Sz, S+, and S- on each of these states. Follow the procedure used in the text for spin 1/2. Get solution
32. In Example 4.3: (a) If you measured the component of spin angular momentum along the x direction, at time t, what is the probability that you would get ... (b) Same question, but for the y component. (c) Same, for the z component. ... ... Get solution
33. An electron is at rest in an oscillating magnetic field...where Bo and ... are constants.(a) Construct the Hamiltonian matrix for this system.(b) The electron starts out (at t = 0) in the spin-up state with respect to the x-axis (that is: ... (0) = .... Determine ... (t) at any subsequent time. Beware: This is a time-dependent Hamiltonian, so you cannot get x(0 in the usual way from stationary states. Fortunately, in this case you can solve the time- dependent Schrodinger equation (Equation 4.162) directly.(c) Find the probability of getting —.../2, if you measure Sx. Answer:...(d) What is the minimum field (Bo) required to force a complete flip in Sx? Get solution
34. (a) Apply ... (Equation 4.177), and confirm that you get ... (b) Apply ... (Equation 4.178), and confirm that you get (c) Show that ... (Equation 4.177) are eigenstates of ... with the appropriate eigenvalue. ... ... Get solution
35. Quarks carry spin 1/2. Three quarks bind together to make a baryon (such as the proton or neutron); two quarks (or more precisely a quark and an antiquark) bind together to make a meson (such as the pion or the kaon). Assume the quarks are in the ground state (so the orbital angular momentum is zero).(a) What spins are possible for baryons?(b) What spins are possible for mesons? Get solution
36. (a) A particle of spin 1 and a particle of spin 2 are at rest in a configuration such that the total spin is 3, and its z component is h. If you measured the z component of the angular momentum of the spin-2 particle, what values might you get, and what is the probability of each one?(b) An electron with spin down is in the state of the hydrogen atom. If you could measure the total angular momentum squared of the electron alone (not including the proton spin), what values might you get, and what is the probability of each? Get solution
37. Determine the commutator of ... Generalize your result to show that ... Comment: Because ...does not commute with ... we cannot hope to find states that are simultaneous eigenvectors of both. In order to form eigenstates of ...we need linear combinations of eigenstates of ... This is precisely what the Clebsch- Gordan coefficients (in Equation 4.185) do for us. On the other hand, it follows by obvious inference from Equation 4.187 that the sum ... does commute with ... which is a special case of something we already knew (see Equation 4.103) ... ... ... Get solution
38. Consider the three-dimensional harmonic oscillator, for which the potential is... [4.188](a) Show that separation of variables in cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies. Answer:... [4.189](b) Determine the degeneracy d(n) of En Get solution
39. Because the three-dimensional harmonic oscillator potential (Equation 4.188) is spherically symmetric, the Schrodinger equation can be handled by separation of variables in spherical coordinates, as well as cartesian coordinates. Use the power series method to solve the radial equation. Find the recursion formula for the coefficients, and determine the allowed energies. Check your answer against Equation 4.189. ... Get solution
40. (a)Prove the three-dimensional virial theorem:... [4.190](for stationary states). Hint: Refer to Problem 3.31.(b) Apply the virial theorem to the case of hydrogen, and show that...T... = -En; ...V... = 3En [4.191](c) Apply the virial theorem to the three-dimensional harmonic oscillator (Problem 4.38), and show that in this case...T... = ...V... = En/2 [4.192] Get solution
41. [Attempt this problem only if you are familiar with vector calculus.] 'Define the (three-dimensional) probability current by generalization of Problem 1.14:... [4.193](a) Show that J satisfies the continuity equation... [4.194]which expresses local conservation of probability. It follows (from the divergence theorem) that... [4.194]where ... is a (fixed) volume and S is its boundary surface. In words: The flow of probability out through the surface is equal to the decrease in probability of finding the particle in the volume.(b) Find J for hydrogen in the state n = 2, l = 1, m = 1. Answer:...(c) If we interpret mJ as the flow of mass, the angular momentum is...Use this to calculate Lz for the state Ψ211, and comment on the result Get solution
42. The (time independent) momentum space wave function is three dimensions is defined by the natural generalization of Equation 3.54:... [4.196](a) Find the momentum space wave function for the ground state of hydrogen (Equation 4.80). Hint: Use spherical coordinates, setting the polar axis along the direction of p. Dotheѳ integral first. Answer... [4.197](b) Check that Φ (p) is normalized.(c) Use Φ(p) to calculate (p2), in the ground state of hydrogen.(d) What is the expectation value of the kinetic energy in this state? Express your answer as a multiple of E1, and check that it is consistent with the virial theorem (Equation 4.191). Get solution
43. (a) Construct the spatial wave function (Ψ) for hydrogen in the state n = 3,l = 2, m = 1. Express your answer as a function of r, ѳ, Φ, and a (the Bohr radius) only—no other variables (ρ, z, etc.) or functions (Y, v, etc.), or constants (A, co, etc.), or derivatives, allowed (∏ is okay, and e, and 2, etc.).(b) Check that this wave function is properly normalized, by carrying out the appropriate integrals over r, ѳ, and Φ.(c) Find the expectation value of rs in this state. For what range of s (positive and negative) is the result finite? Get solution
44. (a) Construct the wave function for hydrogen in the state n = 4, l = 3, m = 3. Express your answer as a function of the spherical coordinates r, ѳ, and Φ.(b) Find the expectation value of r in this state. (As always, look up any nontrivial integrals.)(c) If you could somehow measure the observable on an atom in this state, what value (or values) could you get, and what is the probability of each? Get solution
45. What is the probability that an electron in the ground state of hydrogen will be found inside the nucleus?(a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus.(b) Expand your result as a power series in the small number ε = 2b/a, and show that the lowest-order term is the cubic: P ≈ (4/3)(b/a)3. This should be a suitable approximation, provided that b a (which it is).(c) Alternatively, we might assume that ψ(r) is essentially constant over the (tiny) volume of the nucleus, so that P ≈ (4/3)πb3|ψ(0)|2. Check that you get the same answer this way.(d) Use b ≈ 10‒15 m and a ≈ 0.5 × 10‒10 m to get a numerical estimate for P. Roughly speaking, this represents the “fraction of its time that the electron spends inside the nucleus.” Get solution
46. (a) Use the recursion formula (Equation 4.76) to confirm that when l = n ‒ 1 the radial wave function takes the form(b) ...and determine the normalization constant Nn by direct integration.(b) Calculate ...r... and ...r2... for states of the form ψn(n-1)m.(c) Show that the “uncertainty” in r (σr) is ...r.../y/√2n + 1 for such states. Note that the fractional spread in r decreases, with increasing n (in this sense the system “begins to look classical,” with identifiable circular “orbits,” for large n). Sketch the radial wave functions for several values of n, to illustrate this point. Get solution
47. Coincident spectral lines. According to the Rydberg formula (Equation 4.93) the wavelength of a line in the hydrogen spectrum is determined by the principal quantum numbers of the initial and final states. Find two distinct pairs ... that yield the same ... For example, {6851, 6409} and { 15283, 11687} will do it, but you're not allowed to use those! ... Get solution
48. Consider the observables A = x2 and B = LZ.(a) Construct the uncertainty principle for σAσB.(b) Evaluate σB in the hydrogen state ψnlm.(c) What can you conclude about ...xy... in this state? Get solution
49. An electron is in the spin state...(a) Determine the constant A by normalizing χ.(b) If you measured Sz on this electron, what values could you get, and what is the probability of each? What is the expectation value of Sz?(c) If you measured Sx on this electron, what values could you get, and what is the probability of each? What is the expectation value of Sx?(d) If you measured Sy on this electron, what values could you get, and what is the probability of each? What is the expectation value of Sy? Get solution
50. Suppose two spin ½ particles are known to be in the singlet configuration (Equation 4.178).Let ... be the component of the spin angular momentum of particle number 1 in the direction defined by the unit vector ...Similarly, let ... be the component of 2’s angular momentum in the direction ... show that ... Where ... is the angle between ... ... Get solution
51. (a) Work out the Clebsch-Gordan coefficients for the case ... = anything. Hint: You're looking for the coefficients A and B in ... where the signs are determined by ... (b) Check this general result against three or four entries in Table 4.8. (References: Table 4.8 & Equations) ... ... [4.136] ... ... ... Get solution
52. Find the matrix representing Sxfor a particle of spin 3/2 (using, as always, the basis of eigenstates of Sz). Solve the characteristics equation to determine the eigenvalues of Sx. Get solution
53. Work out the spin matrices for arbitrary spin s, generalizing spin 1/2 (Equations 4.145 and 4.147), spin 1 (Problem 4.31), and spin 3/2 (Problem 4.52).Answer:...where... Get solution
54. Work out the normalization factor for the spherical harmonics, as follows. From section 4.12 we know that ... The problem is to determine the factor ... (which I quoted, but did not derive, in Equation 4.32). use Equations 4.120, 4.121, and 4.130 to obtain a recursion relation giving ... in terms of ...solve it by induction on m to get ... up to an overall constant, ... Finally, use the result of Problem 4.22 to fix the constant. You may find the following formula for the derivative of an associated Legendrefunction useful: ... ... ... ... ... Get solution
55. The electron in a hydrogen atom occupies the combined spin and position state...(a) If you measured the orbital angular momentum squared (L2), what values might you get, and what is the probability of each?(b) Same for the z component of orbital angular momentum (Lz).(c) Same for the spin angular momentum squared (S2).(d) Same for the z component of spin angular momentum (Sz)Let J ≡ L + S be the total angular momentum.(e) If you measured J2, what values might you get, and what is the probability of each?(f) Same for Jz.(g) If you measured the position of the particle, what is the probability density for finding it at r, θ, ф?(h) If you measured both the z component of the spin and the distance from the origin (note that these are compatible observables), what is the probability density for finding the particle with spin up and at radius r? Get solution
56. (a) For a function f(ф) that can be expanded in a Taylor series, show that...(where φ is an arbitrary angle). For this reason, ... is called the generator of rotations about the z-axis Hint: Use Equation 4.129, and refer to Problem 3.39.More generally, ... is the generator of rotations about the direction ..., in the sense that exp(...) effects a rotation through angle φ (in the right-hand sense) about the axis .... In the case of spin, the generator of rotations is .... In particular, for spin 1/2... [4.200]tells us how spinors rotate.(b) Construct the (2 × 2) matrix representing rotation by 180° about the x-axis, and show that it converts “spin up” (χ+) into "spin down" (χ‒), as you would expect.(c) Construct the matrix representing rotation by 90° about the y-axis, and check what it does to χ+(d) Construct the matrix representing rotation by 360° about the z-axis. If the answer is not quite what you expected, discuss its implications.(e) Show that... [4.201] Get solution
57. The fundamental commutation relations for angular momentum (Equation 4.99) allow for half-integer (as well as integer) eigenvalues. But for orbital angular momentum only the integer values occur. There must be some extra constraint in the specific form L = r x p that excludes half-integer values.` Let a be some convenient constant with the dimensions of length (the Bohr radius, say, if we're talking about hydrogen), and define the operators ... ... ... Get solution
58. Deduce the condition for minimum uncertainty in ... (that is, equality in the expression... for a particle of spin ½ in the generic state (Equation 4.139). Answer: With no loss of generality we can pick a to be real; then the condition for minimum uncertainty is that b is either pure real or else pure imaginary. ... Get solution
59. ... ... ... Get solution
60. [Refer to Problem 4.59 for background.] Suppose...where B0 and K are constants.(a) Find the fields E and B.(b) Find the allowed energies, for a particle of mass m and charge q, in these fields. Answer:... [4.209]where to... and ... Comment: If K = 0 this is the quantum analog to cyclotron motion; ω1 is the classical cyclotron frequency, and it's a free particle in the z direction. The allowed energies, ... are called Landau Levels.46 Get solution
61. [Refer to problem 4.59 for background.] In classical electrodynamics the potentials A and ... are not uniquely determined; the physical quantities are the fields, E and B. ... Refer to problem 4.59: ... Get solution
