Chapter #6 Solutions - Introduction to Quantum Mechanics - David J. Griffiths - 2nd Edition

1. Suppose we put a delta-function bump in the center of the infinite square well: ... where α is a constant. (a) Find the first-order correction to the allowed energies. Explain why the energies are not perturbed for even n. (b) Find the first three nonzero terms in the expansion (Equation 6.13) of the correction to the ground state, .... ... Get solution

2. For the harmonic oscillator ... the allowed energies are ... where ... in is the classical frequency. Now suppose the spring constant increases slightly: ... (Perhaps we cool the spring, so it becomes less flexible.) (a) Find the exact new energies (trivial, in this case). Expand your formula as a power series in ?, up to second order. (b) Now calculate the first-order perturbation– in the energy, using Equation 6.9. What is H' here? Compare your result with part (a). Hint: It is not necessary—in fact, it is not permitted —to calculate a single integral in doing this problem. (Reference: Equation) ... Get solution

3. Two identical bosons are placed in an infinite square well (Equation 2.19). They interact weakly with one another, via the potential ... (where V0 is a constant with the dimensions of energy, and a is the width of the well). (a) First, ignoring the interaction between the particles, find the ground state and the first excited state—both the wave functions and the associated energies. (b) Use first-order perturbation theory to estimate the effect of the particle-particle interaction on the energies of the ground state and the first excited state. ... Get solution

4. (a) Find the second-order correction to the energies ...for odd n. (b) Calculate the second-order correction to the ground state energy ... for the potential in Problem 6.2. Check that your result is consistent with the exact solution. (Reference: Problems 6.1) Suppose we put a delta-function bump in the center of the infinite square well: ... where α is a constant. (a) Find the first-order correction to the allowed energies. Explain why the energies are not perturbed for even n. (b) Find the first three nonzero terms in the expansion (Equation 6.13) of the correction to the ground state, ... (Reference: Equation) ... (Reference: Problems 6.2) For the harmonic oscillator ... the allowed energies are ... where ... in is the classical frequency. Now suppose the spring constant increases slightly: ... (Perhaps we cool the spring, so it becomes less flexible.) (a) Find the exact new energies (trivial, in this case). Expand your formula as a power series in ?, up to second order. (b) Now calculate the first-order perturbation– in the energy, using Equation 6.9. What is H' here? Compare your result with part (a). Hint: It is not necessary—in fact, it is not permitted —to calculate a single integral in doing this problem. (Reference: Equation) ... Get solution

5. Consider a charged particle in the one-dimensional harmonic oscillator potential. Suppose we turn on a weak electric field (E), so that the potential energy is shifted by an amount H' = —qEx.(a) Show that there is no first-order change in the energy levels, and calculate the second-order correction. Hint: See Problem 3.33.(b) The Schrodinger equation can be solved directly in this case, by a change of variables: x′ = x — (qE/mω2). Find the exact energies, and show that they are consistent with the perturbation theory approximation. Get solution

6. Let the two “good” unperturbed states be ... ... ... Get solution

7. Consider a particle of mass m that is free to move in a one-dimensional region of length L that closes on itself (for instance, a bead that slides frictionlessly on a circular wire of circumference L, as in Problem 2.46). (a) Show that the stationary states can be written in the form ... ... Notice that – with the exception of ground state (n =0) – these are all doubly degenerate. (b) Now suppose we introduce the perturbation ... (Reference: Problem 2.46 & Equations 6.9, 6.27) Problem 2.46 Imagine a bead of mass m that slides frictionlessly around a circular wire ring of circumference L. (This is just like a free particle, except that ψ (x + L) = ψ (x).) Find the stationary states (with appropriate normalization) and the corresponding allowed energies. Note that there are two independent solutions for each energy ... —corresponding to clockwise and counter-clockwise circulation; call them ... How do you account for this degeneracy, in view of the theorem in Problem 2.45 (why does the theorem fail, in this case)? ... ... Get solution

8. Problem 6.8 Suppose we perturb the infinite cubical well (Equation 6.30) by putting a delta function "bump" at the point ...: ... Find the first-order corrections to the energy of the ground state and the (triply degenerate) first excited states. ... Get solution

9. Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian, in matrix form, is...where V0 is a constant, and e is some small number (e 1).(a) Write down the eigenvectors and eigenvalues of the unperturbed Hamiltonian= 0).(b) Solve for the exact eigenvalues of H. Expand each of them as a power series in e, up to second order.(c) Use first- and second-order nondegenerate perturbation theory to find the approximate eigenvalue for the state that grows out of the nondegenerate eigenvector of H0. Compare the exact result, from (a). Get solution

10. In the text I asserted that the first-order corrections to an ≪-fold degenerate energy are the eigenvalues of the W matrix, and I justified this claim as the "natural" generalization of the case n = 2. Prove it, by reproducing the steps in Section 6.2.1, starting with...(generalizing Equation 6.17), and ending by showing that the analog to Equation 6.22 can be interpreted as the eigenvalue equation for the matrix W. Get solution

11. (a)Express the Bohr energies in terms of the fine structure constant and the rest energy (mc2) of the electron.(b) Calculate the fine structure constant from first principles (i.e., without recourse to the empirical values of ... Comment: The fine structure con­stant is undoubtedly the most fundamental pure (dimensionless) number in all of physics. It relates the basic constants of electromagnetism (the charge of the electron), relativity (the speed of light), and quantum mechanics (Planck's constant). If you can solve part (b), you have the most certain Nobel Prize in history waiting for you. But I wouldn't recommend spending a lot of time on it right now; many smart people have tried, and all (so far) have failed. Get solution

12. Use the virial theorem (Problem 4.40) to prove Equation 6.55. ... Get solution

13. In Problem 4.43 you calculated the expectation value of ...in the state .... Check your answer for the special cases s = 0 (trivial), s =-1 (Equation 6.55), s = -2 (Equation 6.56), and s = -3 (Equation 6.64). Coalmen% on the case s = -7. ... Get solution

14. Find the (lowest-order) relativistic correction to the energy levels of the one-dimensional harmonic oscillator. Hint: Use the technique in Example 2. Get solution

15. Show that ...is not, for hydrogen states with l = 0. Hint: For such states ... so ... Using integration by parts, show that ... Check that the boundary term vanishes for ..., which goes like ... ... Get solution

16. Evaluate the following commutators: (a) [L • S, L], (b) [L • S S] (c) [L • S, J], (d) [L • S, L2], (e) [L • S, S2], (f) [L • S, J2]. Hint: L and S satisfy the fundamental commutation relations for angular momentum (Equations 4.99 and 4.134), but they commute with each other. Get solution

17. Derive the fine structure formula (Equation 6.66) from the relativis- tic correction (Equation 6.57) and the spin-orbit coupling (Equation 6.65). Hint. Note that j = I ± 1/2; treat the plus sign and the minus sign separately, and you'll find that you get the same final answer either way. Get solution

18. The most prominent feature of the hydrogen spectrum in the visible region is the red Balmer line, coming from the transition n — 3 to n = 2. First of all, determine the wavelength and frequency of this line according to the Bohr theory. Fine structure splits this line into several closely spaced lines; the question is: How many, and what is their spacing ? Hint: First determine how many sublevels the n = 2 level splits into, and find E1fs for each of these, in eV. Then do the same for n = 3. Draw an energy level diagram showing all possible transitions from n = 3 to n = 2. The energy released (in the form of a photon) is (E3 — E2) + A E, the first part being common to all of them, and the ∆ E (due to fine structure) varying from one transition to the next. Find ∆ E (in eV) for each transition. Finally, convert to photon frequency, and determine the spacing between adjacent spectral lines (in Hz)—not the frequency interval between each line and the unperturbed line (which is, of course, unobservable), but the frequency interval between each line and the next one. Your final answer should take the form: "The red Balmer line splits into (???) lines. In order of increasing frequency, they come from the transitions (1) j = (???) to j = (???), (2) j = (???) to j = (???), .... The frequency spacing between line (1) and line (2) is (???) Hz, the spacing between line (2) and line (3) is (???) Hz…….. Get solution

19. The exact fine-structure formula for hydrogen (obtained from the Dirac equation without recourse to perturbation theory) is 16 ... ... ... Get solution

20. Use Equation 6.59 to estimate the internal field in hydrogen, and characterize quantitatively a "strong" and "weak" Zeeman field. ... Get solution

21. Consider the (eight) n = 2 states, |2ljmj>. Find the energy, of each state, under weak-field Zeeman splitting, and construct a diagram like Figure 6.11 to show how the energies evolve as Bext increases. Label each line clearly, and indicate its slope. Get solution

22. Starting with Equation 6.80, and using Equations 6.57, 6.61, 6.64 and 6.81, derive Equation 6.82. ... Get solution

23. Consider the (eight) n — 2 states, |2l ml ms〉. Find the energy of each state, under strong-field Zeeman splitting. Express each answer as the sum of three terms: the Bohr energy, the fine-structure (proportional to α2), and the Zeeman contribution (proportional toμBbext).If you ignore fine structure altogether, how many distinct levels are there, and what are their degeneracies? Get solution

24. If l = 0, then j = s, ... for weak and strong fields. Determine ... (from Equation 6.72) and the fine structure energies (Equation 6.67), and write down the general result for the I = 0 Zeeman effect—regardless of the strength of the field. Show that the strong-field formula (Equation 6.82) reproduces this result, provided that we interpret the indeterminate term in square brackets as 1. ... Get solution

25. Work out the matrix elements of H'z and H'is, and construct the W-matr given in the text, for n = 2. Get solution

26. Analyze the Zeeman effect for the n = 3 states of hydrogen, in the weak, strong, and intermediate field regimes. Construct a table of energies (analo­gous to Table 6.2), plot them as functions of the external field (as in Figure 6.12), and check that the intermediate-field results reduce properly in the two limiting cases. Get solution

27. Let a and b be two constant vectors. Show that...(the integration is over the usual range: O...for states with l = 0. Hint: r = sin θ cosϕ i + sin θ sin ϕ j + cos θk. Get solution

28. By appropriate modification of the hydrogen formula, determine the hyperfine splitting in the ground state of (a) muonic hydrogen (in which a muon—same charge and g-factor as the electron, but 207 times the mass—substi­tutes for the electron), (b) positronium (in which a positron—same mass and g-factor as the electron, but opposite charge—substitutes for the proton), and (c) muonium (in which an anti-muon—same mass and g-factor as a muon, but opposite charge—substitutes for the proton). Hint: Don't forget to use the reduced mass (Problem 5.1) in calculating the "Bohr radius" of these exotic "atoms." Inci­dentally, the answer you get for positronium (4.82 x 10-4 eV) is quite far from the experimental value (8.41 x 10-4 eV); the large discrepancy is due to pair anni­hilation (e+ + e- → y + y), which contributes an extra (3/4)AE, and does not occur (of course) in ordinary hydrogen, muonic hydrogen, or muonium. Get solution

29. Estimate the collection to the ground state energy of hydrogen due to the finite size of the nucleus. Treat the proton as a uniformly charged spherical shell of radius b, so the potential energy of an electron inside the shell is constant: —e2/(4neob)-, this isn't very realistic, but it is the simplest model, and it will give us the right order of magnitude. Expand your result in powers of the small parameter (b/a), where a is the Bohr radius, and keep only the leading term, so your final answer takes the form...Your business is to determine the constant A and the power n. Finally, put in b ~ 10"15 m (roughly the radius of the proton) and work out the actual number. How does it compare with fine structure and hyperfine structure? Get solution

30. Consider the isotropic three-dimensional harmonic oscillator (Problem 4.38). Discuss the effect (in first order) of the perturbation...(for some constant X)(a) on the ground state;(b) (b) the (triply degenerate) first excited state. Hint: Use the answers to Problems 2.12 and 3.33 Get solution

31. Van der Waals interaction. Consider two atoms a distance R apart. Because they are electrically neutral you might suppose there would be no force between them, but if they are polarizable there is in fact .a weak attraction. To model this system, picture each atom as an electron (mass in, charge -e) attached by a spring (spring constant k) to the nucleus (charge +e), as in Figure 6.14. We'll assume the nuclei are heavy, and essentially motionless. The Hamiltonian for the unperturbed system is ... ... Get solution

32. Suppose the Hamiltonian If, for a particular quantum system, is a function of some parameter ... be the eigenvalues and ... ... ... Get solution

33. The Feynman-Hellmann theorem (Problem 6.32) can be used to determine the expectation values of ... for hydrogen.23 The effective Hamiltonian for the radial wave functions is (Equation 4.53) ... ... ... ... Get solution

34. Prove Kramers' relation... Get solution

35. (a) Plug s = 0, s = 1, s = 2, and s = 3 into Kramers' relation (Equation 6.104) to obtain formulas for .... Note that you could continue indefinitely, to find any positive power. (b) In the other direction, however, you hit a snag. Put in s = -1, and show that all you get is a relation between .... (c) But if you can get ..., and check your answer against Equation 6.64. ... Get solution

36. When an atom is placed in a uniform external electric field ..., the energy levels are shifted—a phenomenon known as the Stark effect (it is the electrical analog to the Zeeman effect). In this problem we analyze the Stark effect for the n = 1 and n = 2 states of hydrogen. Let (the field point in the z direction, so the potential energy of the electron is ... Treat this as a perturbation on the Bohr Hamiltonian (Equation 6.42). (Spin is irrelevant to this problem, so ignore it, and neglect the fine structure.) (a) Show that the ground state energy is not affected by this perturbation, in first order. (b) The first excited state is 4-fold degenerate: .... Using degenerate perturbation theory, determine the first-order corrections to the energy. Into-how many levels does E2 split? (c) What are the "good" wave functions for part (b)? Find the expectation value of the electric dipole moment ... Get solution

37. Consider the Stark effect (Problem 6.36) for the n = 3 states of hydrogen. There are initially nine degenerate states, ψ31m(neglecting spin, as before), and we turn on an electric field in the z direction.(a) Construct the 9 × 9 matrix representing the perturbing Hamiltonian. Partial answer: ⟨300|z|310⟩ = −3..., ⟨310|z|320⟩ = −3..., ⟨31±l|z|32±l⟩ = −(9/2)a.(b) Find the eigenvalues, and their degeneracies. Get solution

38. Calculate the wavelength, in centimeters, of the photon emitted under a hyperfine transition in the ground state (n = 1) of deuterium. Deuterium is "heavy" hydrogen, with an extra neutron in the nucleus; the proton and neutron bind together to form a deuteron, with spin 1 and magnetic moment...the deuteron g-factor is 1.71. Get solution

39. In a crystal, the electric field of neighboring ions perturbs the energy levels of an atom. As a crude model, imagine that a hydrogen atom is surrounded by three pairs of point charges, as shown in Figure 6.15. (Spin is irrelevant to this problem, so ignore it.)(a) Assuming that r ≪ d1, r ≪ d2, and r ≪ d3, show thatH′ = V0 + 3(β1x2 + β2y2 + β3z2) – (β1 + β2 + β3)r2,where...(b) Find the lowest-order correction to the ground state energy.(c) Calculate the first-order corrections to the energy of the first excited states (n = 2). Into how many levels does this four-fold degenerate system split, (i) in the case of cubic symmetry, β1 = β2 = β3; (iii) in the case of tetragonal symmetry, β1 = β2 ≠ β3, (iii) in the general case of orthorhombic symmetry (all three different)?... Get solution

40. Sometimes it is possible to solve Equation 6.10 directly, without having to expand ...in terms of the unperturbed wave functions (Equation 6.11). Here are two particularly nice examples. (a) Stark effect in the ground state of hydrogen. (i) Find the first-order correction to the ground state of hydrogen in the presence of a uniform external electric field ... (the Stark effect—see Problem 6.36)) Hint: Try a solution of the form ... your problem is to find the constants A, B, and C that solve Equation 6.10. (ii) Use Equation 6.14 to determine the second-order correction to the ground state energy the first-order correction is zero, as you found in Problem 6.36(a))) Answer: ... (b) If the proton had an electric dipole moment p, the potential energy of the electron in hydrogen would be perturbed in the amount H, ep cos 0 4re eor2 (i) Solve Equation 6.10 for the first-order correction to the ground state wave function. ... ... Get solution