To students: if you find yourself with no idea how to begin a problem, send me a query and I will consider putting a hint on this page. There is little educational value in banging one's head against the wall for hours in frustration, and I can well remember doing that myself.
To faculty: a solutions manual with all the problems solved is available upon request from me. The way to get it is to email me.
Warning: There are problems with known defects in the second printing, 3.4, 5.9, 10.2—7, 11.1, 13.6, 17.9, and 25.5. Please do not work on or assign these problems without taking into account the corrections. I apologize for difficulties they have caused to students, and encourage anyone who finds a difficulty with other problems to send me their observations as soon as possible.
Return to book home page. Send comments to marder@chaos.ph.utexas.edu
Jump to Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Easy
Easy
Medium. The structures described by this problem are realized in nature by helical molecules, and by charged particles trapped in beams.
Medium. This problem shows that five-fold axes cannot exist, explaining some of the excitement that surrounded the discovery of the quasicrystals discussed in chapter 5.
Medium. The chapter discusses potential ground states of interacting particles, and this problem provides a specific example of what that means. Parts (a) and (c) need to be done numerically, but part (b) is best done analytically.
Easy.
Easy.
Easy.
Easy.
Easy.
Medium. This problem continues to address the question of how crystals choose their structure, now by asking how well various structures fill space. Warning: Distance between particles must be 2, not 1.
Medium. This problem is similar to 1.5, but needs to be carried out in three dimensions rather than in two. It requires some numerical work.
Easy.
Medium. This problem concerns the definition of Miller indices employed by crystallographers.
Easy.
Easy.
Easy. The Bragg picture of scattering is a standard topic, and a famous achievement, but this is the only reference to it in this text.
Medium-hard. Probably the fastest way to check the possible crystal structures is to write computer programs, so some numerical work required. Use Equations (3.38) and (3.39). The absence of spots is as important in determining which lattice is possible as is their presence. One has to pay attention to all the cutoffs mentioned in the problem. Warning: the figure is missing 3 spots which correspond to Bragg angle of 18.4 degrees. Download corrected figure here (8K, pdf, 24K PostScript). I have seen one copy of the book where a random blob of ink emerged from the printing process in the middle of the figure, making this exercise an even more accurate example of the challenges of performing experiments than I intended.
Medium. To do the second half of the problem, it is necessary to form an analogy between the response of a solid to light, and a damped harmonic oscillator, without worrying too much about what the analogy means. Section 20.2.2 provides an explanation of where one might get the idea for this analogy.
Medium-easy. The calculation is only a few lines long.
Medium. Many texts describe all the terms in 3.51 in an intuitive way. I have never been able to feel confident that they really can be derived in such a fashion, so this problem describes a formal path.
Medium. Requires numerical work. My C code for this problem is less than 50 lines long.
Medium. The angle q in Eq. (4.1) is really just a distraction and can be argued away. Note that a superlattice must contain at least two points away from the origin that are not collinear. Their existence can be used as part of the solution.
Easy. I had wanted to pose a problem proving the analog of Eq. (4.3) for a bcc lattice. The proofs I found myself were so involved that it did not seem worthwhile. There is probably a simple proof; I would appreciate learning it.
Easy. The simplest calculations of surface energy involve bond-breaking arguments. Counting the numbers of bonds that break in forming new surfaces is easy if one has a picture to work with, and much more difficult otherwise.
Medium. This problem introduces the Wulff construction, shows how the equilibrium shape of a crystal is related to the surface free energy, and shows why crystals have facets. Some computation required.
Medium. Diffusion-limited aggregation became a bit of an industry after Witten and Sander's original Letter. There was a great deal of analytical as well as numerical work. The review article by Erzan et. al. can be used to discover some relatively recent references. This problem leads the student through the simplest version of the algorithm, and requires some computaion. The most familiar pictures involved roughly spherical aggregates, rather than ones in a channel.
Medium. This problem goes through the brief calculation that underlies the popularity of the Verlet algorithm in molecular dynamics calculations.
Medium.
Easy.
Medium-hard. The issues behind the question in part (f) are quite subtle. Requires numerical work. The computational results should apparently contradict statements by Landau and Lifshitz. Landau is not wrong, and neither is the computer.
Medium. The image of soap cells comes from an apparatus with two plates separated by about ¼ inch into which one blows soap bubbles. I learned about the topic from James Glazier, Joel Stavans, and Steve Gross.
Medium, but the problem is not posed entirely correctly. Energy must be a periodic function of angles. Since it is not, one must assume that bk>>1, in which case angles approaching p can be neglected, and integrals over angles extended to infinity.
Easy.
Medium. The final part needs to be done numerically.
Medium-hard. Requires numerical work. This algorithm for constructing quasicrystals is much more efficient than inflation schemes. Warning: the sum you need to form in part (c) is \sum_j n_j \hat e_j.
Fairly easy.
Fairly easy. Assume that the electrons live in a square box of size L with periodic boundary conditions, as in Eq. 6.7.
Easy.
Easy.
Medium. This problem illustrates the way that some conventional approximations must be re-examined when sample dimensions approach atomic dimensions.
Medium. This problem is mathematical, and has little physical content, but gives good practice in working with contour integrals.
Medium. This problem is again largely mathematical; the result is remarkable, but I must admit that it is not as useful as it is elegant.
Medium. This problem provides practice in plotting functions, and may help build some intuition about the shape of the Fermi function. Should be done on a computer.
Meadium easy. Good mathematical result to know.
Easy.
Fairly easy. Small error here; the first part of this problem is the same as problem 7.1, and 7.1 provides more hints. The results are different at maxima and minima from the results at saddle points. Write out the energy as a quadratic form in the vicinity of the extremum and perform the integral in each case.
Medium-hard. Requires a fair amount of numerical work. My FORTRAN program is about 100 lines long.
Medium. This is a standard analytical approach to the Kronig-Penney model. I did the algebra in a symbolic manipulation program. In part (d), also set hbar=1. Requires some plotting.
Fairly easy.
Hard. In addition to making sense of the regular representation, one must do a fair amount of work within a symbolic algebra program. I had been curious as to how group representations for point groups could be obtained from a definite formal procedure, and this problem provides an answer.
Easy.
Easy. Standard exercise having to do with the nearly-free electron approximation.
Easy. Note that in figure 8.3 the Fermi surface is discontinuous in (A), but is continuous and smooth in (B) and (C)
Easy.
Easy. It is useful to perform these constructions by hand at least once.
Easy.
Medium. This problem provides a first introduction to tight-binding models. In addition, it provides an excellent illustration of Bloch's theorem in a context where most students do not immediately recognize that it applies. Requires some numerical work.
Hard. This problem concerns the analytical properties of Wannier functions, and introduces Berry's phase. Although there are many hints, the problem is long and rather subtle.
Fairly easy.
Fairly easy. This result is rather important, and is referred to quite frequently. Warning: In part (a) you must include a term that accounts for the interaction energy of the positive background charge with itself. This constant offset has no effect upon the Hartree-Fock equations.
Medium. Screening is an extremely important physical idea, and appears in a large number of contexts.
Medium. Establishes the mathematical preliminaries needed to find the solutions of the Thomas-Fermi equation.
Medium--hard. Finds the solution of the Thomas-Fermi equation numerically, using the results of Problem 9.4. Requires numerical work.
Medium hard. This problem works through the details of establishing a variational lower bound for the energy of the hydrogen atom. In his Reviews of Modern Physics article Lieb has a tighter lower bound than this, but this one is more similar to his bound for many-electron systems. For the final part of the problem, I had in mind that students would verify the calculations taking place between (9.85) and (9.93).
Medium. The calculations associated with APW, LMTO, and other schemes are all somewhat painful.
Medium. The results of this problem will be used later in order to find the resistance of liquid aluminum, and to interpret optical data. Warning: the values listed on p. 257 should be U_0=-31.30eV, d=0.350, and Rc=0.943
--7. Hard (or at least Long). These five problems provide a series of exercises that lead students through the creation of a very primitive band structure code. I tried several other approaches to this problem over the years, but this one worked the best. If assigned over the course of five weeks, or as an end-of-semester project, these problems are not too burdensome. The hints should be sufficient to catch most errors before they get out of hand. Requires lots of numerical work. Warning: the value listed on p. 257 should be U_0=-31.30eV, d=0.350, and Rc=0.943 Once these changes are made, all the numerical values on pages 256 and 257 change, although they are correct given the value of -36.31 eV that has been suggested. The final graph on page 258 changes as well, although the alterations are difficult to make out by eye. If you prefer to carry out the problems with corrected values of the pseudopotential, and do not have the fourth printing where this has been fixed, obtain corrected pages here (pdf, 28K) Warning: in parts (e) and after, all energies are in Rydbergs, not eV.
Crystal fcc bcc hcp
A_6 14.4539 12.2537 14.4549
A_{12} 12.1319 9.1142 12.1323
A_6^2/2A_{12} 8.6102 8.2373 8.6111
Easy. The first part is no more than recalling an earlier result from the text. The second two parts rely on simple geometrical features of Brillouin zones.
Medium. This is a good exercise in quantum-mechanical perturbation theory.
Medium. The exercise is largely mathematical, but the result can be used to answer physically interesting questions.
Medium. With the hints, this problem is not especially hard. It shows how to perform Ewald summation for systems with many particles in each unit cell.
Medium easy. The problem provides a good illustration of how to obtain interesting physical results from a system by employing simple but sensible approximations.
Easy.
Easy.
Easy.
Easy.
Medium easy. This problem provides the only coverage of how wave speeds vary with direction in a crystal.
Medium. Surface waves are important in many contexts ranging from earthquakes to nondestructive testing through acoustic signals.
Easy.
Medium. The possibility of flipping liquid crystals with external fields lies behind liquid crystal displays, which however use electrical rather than magnetic fields.
Easy. Some numerical work required.
Medium. Use short-range cutoff on log function when log(0) appears.
Medium hard. This problem demonstrates one of the basic facts of fracture mechanics. Note that in part (b) there is a shift of coordinates to bring the right-hand crack tip to the origin.
Medium. This problem derives properties of stress fields near sharp tips.
Medium hard. This problem demonstrates the phenomenon of lattice trapping of cracks.
Medium. An exercise in conformal mapping.
Medium easy. Shows difference between fluctuations in two and three dimensions.
Medium. This problem shows how viscosity arises from kinetic theory.
Hard. This problem follows the classic calculation of Stokes for flow around a sphere. There are additional subtleties that the problem does not mention. The energy of this flow is divergent, and the formulas for the flow field are incorrect at large distances from the sphere.
Medium. This problem describes vortices observed far from their cores.
Medium. This problem describes vortices by providing a particular model of the vortex core.
Easy.
Medium easy.
Medium hard. This classic calculation of Bogoliubov shows how interactions between bosons change the excitation spectrum of the dilute Bose gas. The problem is the first in the text to make substantial use of second quantization, and also introduces the Bogoliubov transformations that will be employed later in the study of superconductivity. Take u and v to be real.
Medium-easy. First of two problems showing why Bloch oscillations are not naturally observed in metals.
Medium-easy. Second problem showing the effect of damping upon Bloch oscillations. Requires numerical work.
Medium. Derives the effective mass theorem.
Medium. Like Problem 8.6 works with one-dimensional tight-binding models.
Somewhat hard. The calculations needed for this problem are fairly lengthy, although they are not conceptually tremendously difficult.
Medium-hard. Somewhat easier than the previous problem, but still rather involved. To find the equation for k, it helps to recall that Lagrangian dynamics are unchanged if one subtracts from the Lagrangian a total time derivative.
Medium. Simpler formal approach to semiclassical dynamics than the wave packets used in this chapter; this approach is not careful enough to produce the anomalous velocity.
Medium-easy.
Medium. Similar to Problem 16.7, but now including magnetic fields, which make this problem somewhat more difficult. Do NOT try to include anything having to do with anomalous velocities in this problem. The formalism is not really up to handling it, so far as I know.
Easy.
Easy.
Medium-easy, given the discussion in the text in the vicinity of Eqs. 17.39 and 17.40.
Easy.
Medium-easy. Note to instructors: solutions manuals distributed before fall 2003 actually obtain c_\mu rather than c_V. To correct the final answer, multiply by 3/2.
Medium. The problem motivates the definition of figures of merit for thermoelectric materials.
Medium. Derives Thomson effect.
Medium-easy. The Hall effect is derived from the Drude model.
Medium-hard. The Hall effect is derived within the context of the Boltzmann equation. Warning: Add a term of the form D \partial f/\partial \mu to the right hand side of equation (17.190), and neglect a term proportional to \partial^2 f/\partial \mu^2 during the calculation. The latter term changes the shape of the electron distribution function; keeping it would, I believe, involve going beyond the relaxation time approximation.
Medium. This exercise in Fermi liquid theory is similar to a calculation performed in the course of the Sommerfeld expansion in Chapter 6.
Medium-hard. The calculations needed to verify this result from Fermi liquid theory are somewhat elaborate.
Medium-hard. This problem builds on the result of Problem 10.2, and additional conclusions summarized in Problem 10.3, using a simple pseudopotential to estimate the resistance of liquid aluminum. The calculation is originally due to Ashcroft. Requires numerical work.
Medium-hard. This problem extends Problem 8.6 towards incommensurate potentials, and explores the scaling properties of localized states. Requires numerical work.
Medium-hard. This problem introduces transfer matrices, and shows how they can be used to construct solutions of Schrödinger's equation. Requires numerical work. The final part is quite subtle, but requires no calculation.
Medium-hard. This problem tests whether certain quantities introduced in the chapter are recognized when they appear in a new context, and also illustrates the difficulties involved in plotting quantities that seem easy to define. Numerical work required.
-7. Hard. These three problems work through details of the scaling theory of localization. They depend upon a result of Pichard and André (Eq. 18.135) that is not proved in the text. The result is also not easy to deduce from any single paper. The problems provide an excellent illustration of the power of scaling theories. I tried seven other numerical approaches to the localization problem, but the scaling arguments defeated all of them by a wide margin. These problems can be assigned over the course of 3-4 weeks, or as an end-of-semester project.
Medium-hard. Analytical exploration of weak localization.
Moderately easy.
Moderately easy. Involves nothing but drawing pictures and talking about them.
Medium.
Medium-easy. Deals with the mathematics of the ideal diode equation.
Medium. A fair amount of algebra is required.
Medium. Numerical work required. This problem shows how to use Fourier transforms to compute the analytic functions that appear in the Kramers-Kronig relations.
Medium-easy.
Easy. Looking back at the chapter on elasticity provides hints on what to do.
Medium. Numerical work required. This problem provides a very useful illustration of two complementary descriptions of functions, one where the function has tightly spaced poles, and the other where the function acquires a smooth imaginary part.
Medium.
Medium. This result will be useful in Section 27.3.1 to set up the microscopic theory of superconductivity. The sum over k states also contains (implicitly) a spin sum. The integral has to be broken up into three regions to avoid places where the denominator vanishes
Easy. This result was presented as Eq. 10.13, and used in Problem 10.2.
Medium-easy. Involves analyzing experimental data.
Medium-hard. These calculations show how to calculate Einstein A and B coefficients for hydrogen. There is a subtlety related to A21(\omega) and A21. The first of these gives the transition rate in a very tiny frequency interval around \omega, where tiny means small compared with the the width of the lineshape F12. The second is the full transition rate given by integrating over the lineshape. The first is proportional to the lineshape, and in the second, the lineshape is replaced by 1.
Hard. This problem introduces the Gunn effect, and involves a fairly lengthy set of calculations.
Hard. This problem finds properties of the Frenkel exciton. It is very difficult to carry out unless one relies upon the formalism of second quantization.
Medium. This problem follows upon the previous one and finds the dispersion relation for the Frenkel excitons.
Easy.
Medium. The calculations here are quite similar to those performed for cyclotron resonance, but for Faraday rotation the approximations described here are most appropriate.
Medium. Shows the relation between time-independent and time-dependent perturbation theory in the case of polarons.
Medium. The final result is simple, but the integrals are a little complicated.
Medium. Urbach tails have been difficult to explain, and all this problem will do is lead the student to obtain an answer that seems reasonable but does not correspond in detail to experiment.
Easy.
Medium. This problem makes use of the nearly-free electron approximation, and shows the connection between optical data and pseudopotential parameters such as those used in Problems 10.2 and 18.1.
Medium-hard. This problem is similar to 22.2, and to the computations for cyclotron resonance. Note that (23.77) equals zero, and after (23.79) \epsilon_{yy} should be \epsilon_{xy}.
Easy.
Easy.
Medium. Requires some numerical work.
Easy.
Medium-easy.
Medium-easy.
Medium. Deals with mean field theory for antiferromagnets.
Medium. Analytical solution of the 1-d (Lenz-)Ising model with the high-temperature expansion.
Medium. Mean field theory for superlattices in a slightly elaborate situation. Final part requires some numerical work.
Medium.
Medium-easy. Not hard, but physically interesting.
Easy. Mainly involves talking about a picture.
Moderately easy.
Medium. These calculations justify the Peierls substitution, and set the stage for the Hofstadter butterfly.
Easy. Reviews energy levels of an electron in a magnetic field.
Medium-hard; at least, somewhat lengthy. This calculation is due to Darwin, and addresses worries one might have about Landau's calculation of energy levels in a magnetic field. Although interactions of electrons with boundaries are crucial to the classical argument that magnetic fields do not affect thermodynamic quantities, Landau's calculation appears to treat boundaries carelessly. The calculation treats them more carefully and obtains the same result. Warning: p. 741, Equation (25.157) six final terms on lhs should be multiplied by R. p. 742, Equation (25.160) third term from left should not have factor of 1/\nu.
Easy. Involves talking about a picture.
Medium. Careful demonstration that the ferromagnetic state is the ground state of the ferromagnet.
Medium. Combines arguments from the Stoner model with Fermi liquid theory.
Easy.
Moderately easy. The last part of the problem means to ask why the corner elements of the matrix are zero.
Medium. Requires a fair amount of calculation.
Hard. This calculation reproduces a classic result of Anderson. It makes use of local densities of states, which were defined in Chapter 18.
Moderately easy.
Hard. The derivation of the t-J model from the Hubbard model is, I believe, originally due to Schrieffer and Wolf who used canonical transformations. This derivation is somewhat simpler, and uses degenerate perturbation theory.
Medium-easy.
Medium.
Medium. Parts (a), (b), and (c) lead to the result 27.253, which however is not quite right. Must replace $\pi$ by $2\pi$ in (27.253); it is normal in the literature on Josephson junctions to use $\Phi_0=hc/2e$. In the Hall effect literature, $\Phi_0=hc/e$. I have adopted the Hall effect convention, which means that formulas for the Josephson effect acquire unfamiliar factors of 2.
Easy.
Medium. A reasonable way to get used to working with the BCS formalism.
Medium-hard. The computations involve second quantization and are fairly elaborate.
Medium-hard. Once again, the computations involve second quantization and are fairly elaborate. The state |G> is defined in Eq. 27.118.
Medium. This problem settles picky questions about gauge invariance within the BCS theory.
Medium. This problem works through some nasty integrals that arise in demonstrating the Meissner effect.