## Sunday, January 29, 2012

### Ben's Physics Problem Archive

I've imported my homework assignments from graduate school into Google Docs, and they are now available in the link to the right.  Their contents are catalogged are in the readme file at the top of the list for easy lookup.

These digitized notes have proven incredibly useful to me, and I provide them here so that others might benefit for them as well.  Please do not copy from them for your homework assignments or exams, as that will only harm you in the long term.

Links to PDFs and my original descriptions of the assignments can be found below.

## Supplemental Problem Archive

 Link Source Description Special Topics in Statistical Physics Exact solution to the Ising model in 1 dimension Special Topics in Statistical Physics Properties of Van Der Waals fluids near critical temperature Biophysics 33-767 Pulling forces on stretched or nearly-stretched polymers and pulling forces on looped polymers. Biophysics 33-767 Viral Budding using the shape equation on a membrane and a spherical colloid particle. Biophysics 33-767 Toroidal vesicles, the Helfrich Hamiltonian, and the shape equation as the linearized term of the Helfrich Hamiltonian. Biophysics 33-767 Calculation of the time autocorrelation function in Fluorescence Correlation Spectroscopy, and the scaled volume fraction of N-aggregates in cylindrical micelles. Biophysics 33-767 The effects of dimensionality of space on a particle's locality under diffusion, a derivation of the Stokes-Einstein relation, and the solubility of oil in water based on the length of the aliphatic chain of the hydrocarbon, in the ideal-gas approximation. Biophysics 33-767 A proof that entropy increases under the diffusion equation in arbitrary dimensions, the time-evolution of a Gaussian distribution under diffusion, and the time-evolution of a square wave under diffusion. Jackson, Ficenec and Teplitz A research outline for Magnetic Monopoles that constituted my notes for the oral qualifying exam. Includes: The Dirac Quantization Condition, Duality Transformations, Dirac Strings, Coulomb's Law for Monopoles, Gauge Transformations, Anomalous Hyperfine Splitting in the Hydrogen Atom, Ionization losses in matter, Nuclear, Atomic and Ferromagnetic binding in matter, and Monopoles in the early universe, the Aharanov-Bohm effect, and quantum mechanical properties of monopoles. Problems: Jackson 6.16, 6.17, 6.18, 6.19 Ficenec and Teplitz Chapter 8 Problems 1-12, 15-22 Original A simple example and intuitive explanation as to why charge on a conductor in equilibrium in free space tends to accumulate on areas with a smaller radius of curvature. Griffiths Quantum Mechanics Problem 4.44 Griffiths 4.44 The expectation value of an unusual operator acting on the singlet state (composed of two spin-half particles). Here you see a brute-force evaluation of the operator that underlies this problem. The procedure at this level is surprisingly simple, but the definitions that underlie it might be intimidatingly complex. In fact, simple substitution of the definitions of the underlying the state and operators ultimately leads to an ugly but straightforward evaluation.

## Semester 4 Problem Archive

 Link Source Description Numerical Analysis Homework 6 Stability Analysis, the Crank-Nicholson Method, Forwards Euler, Central Euler, and Backwards Euler, plus a Mathematica implementation each for the heat equation, the wave equation, and a velocity equation. Kincaid 9.2.2, 9.7.4, 9.7.5 Continuum Mechanics Homework 5 Constitutive classes, ideal fluids, Newtonian fluids, Reiner-Rivlin fluids, independence of observer, the First Representation Theorem for Isotropic Tensor Functions, flows of Reiner-Rivlin fluids, Response functions, and the Piola-Kirchoff stress tensor. Gurtin 16.1, 19.2, 21.1, 22.1, 24.1, 25.2, 27.2, 27.5 Numerical Analysis Computer Project 2 A Mathematica implementation of the Richardson, Jacobi, Gauss-Seidl and the Kaczmarz algorithms, used to study an analog of Simpson's Rule. Continuum Mechanics Homework 4 Angular momentum, rigid motions, inertia tensors, Euler's Equations, Normal stresses, Signorini's Theorem. Gurtin 13.2, 13.3, 14.1, 14.4, 14.5, 14.9, 15.1, 15.3 Continuum Mechanics Homework 3 Identities on simple shears, velocity fields, streamlines, motions under a shift in reference time, Rivlin-Erickson tensors, and potentials. Gurtin 8.1, 8.3, 8.4, 8.5, 9.1, 9.4, 11.2, 11.5 The Rayleigh Instability A simple derivation of the Rayleigh Instability for a fluid cylinder. Numerical Analysis Homework 5 Runge-Kutta formulas, the modified Euler's method, and autonomous systems of first-order equations. Kincaid 8.3.3, 8.3.4, 8.3.5, 8.3.6, 8.6.6 Numerical Analysis Homework 4 Richardson Extrapolation, approximating derivatives, numerical integration rules, Gaussian quadrature, the method of undetermined coefficients, the Newton-Cotes formula, and Euler-Maclaurin formulas. Kincaid 7.1.3, 7.1.6, 7.1.12, 7.1.14, 7.1.15, 7.2.4, 7.2.5, 7.2.8, 7.2.12, 7.2.13, 7.2.20, 7.2.23, 7.3.11, 7.3.15, 7.3.17, 7.3.21, 7.3.25, 7.3.31, 7.4.1, 7.4.2 Continuum Mechanics Homework 2 Component representations, identities on curl and divergence, the divergence theorem, plane strains, isochoric flows, and Korn's inequality. Gurtin 4.3, 4.10, 5.1, 5.2, 6.8, 6.10, 7.1, 7.3, 7.5 Continuum Mechanics Homework 1 Identities on general vector spaces, skew and symmetric tensors, spectra, characteristic spaces, polar decompositions, similarity transforms, differentiation of tensors, differentiation by Jacobians, and orthogonal tensors. Gurtin 1.6, 1.14, 1.15, 2.1, 2.3, 2.6, 3.1, 3.2, 3.6 Numerical Analysis Homework 3 Matrix norms and iterative methods for solving many-variable equations. Kincaid 4.4.1, 4.4.2, 4.4.3, 4.4.8, 4.4.11, 4.4.17, 4.4.18, 4.4.33, 4.4.49, 4.5.3, 4.5.19, 4.6.2, 4.6.12, 4.6.33, 4.6.39 Numerical Analysis Homework 2b The Newton algorithm in complex arithmetic (program, practical example), and the Secant method (program, practical example). Kincaid Computer Projects 3.3.8, 3.3.6 Numerical Analysis Homework 2a Fixed points, Continued Fractions. Kincaid 3.4.3, 3.4.6, 3.4.9, 3.4.12, 3.4.13, 3.4.40, Numerical Analysis Homework 1 Taylor Series, rates of convergence, Newtonian Iteration, Halley's Method. Kincaid 1.1.29, 1.1.32, 1.2.2, 3.1.14, 3.1.16, 3.2.14, 3.2.15, 3.2.16, 3.2.19, 3.2.21,

## Semester 3 Problem Archive

 Link Source Description Quantum Mechanics 3 Final Weakly interacting Bose gases in the Bogoliubov approximation and under the Bogoliubov transformation: both a delta-function interaction potential and a Coulomb interaction are considered. The energies are then found for the resulting Hamiltonian. Next, the BCS theory of superconductivity is used in the context of the Bogoliubov transformation in order to deduce the binding energy of Cooper pairs. Finally, a Hamiltonian for polaritons in semiconductors is found under the Bogoliubov transformation and then polariton operators are constructed, which are then examined and found to show similarities to the component photons and excitations. This exam was never graded nor were solutions given--but I believe my solutions are right. Quantum Mechanics 3 Homework 11 A weakly-interacting Bose gas in the Bogoliubov approximation, just as on the final. Squeezed quantum states, and a general Bogoliubov transformation in a "trap" potential leading up to the Bogoliubov-DeGennes equations.Note: This problem set may contain serious errors, as the feedback given by the professor was poor and I do not agree with many of his criticisms. The Bogoliubov-DeGennes equations' form given here is likely not trustworthy, but the derivation leading up to it is at least plausible. Nuclear and Particle Physics Homework 10 Variables for Dalitz plots are justified, the ratio of the amplitudes of dominant decay modes for short Kaons is determined from experimental data, the number of degrees of freedom in the decay of a particle to N components is examined, and finally a general expression for the decay width of a particle in phase space for 2 and 3 dimensions is integrated to give better-known and much simpler formulas. Quantum Mechanics 3 Homework 10 Coherent eigenstates in the large-particle number limit, Bogoliubov transformations, the Lamb Shift for hydrogenic energy levels and mass renormalization, and the dispersion relation and static dielectric function for a 2D electron gas such as graphene. Note that the professor was rather careless in grading this assignment, so I question the validity of some of my solutions despite being given full credit for them. Quantum Mechanics 3 Homework 9 The plasmon dispersion relation in the long-wavelength limit (I was unable to invert one of the series; can someone tell me how to do this?), zero sound in neutral Fermi gases (e.g., superfluid Helium 3) in the random phase approximation in both the weak-coupling and strong-coupling limits, Landau damping, and the dielectric function describing screening in the random phase approximation. Solids 1 Test 3 An Ising Model (Monte Carlo) simulation of spontaneous magnetization along with examination of the magnetocaloric effect done in Mathematica, and brief explanations of semimetals, ferromagnetism, colossal magnetoresistance, band gaps, hysteresis, the inverse Faraday effect, and the effects of lattice defects on magnetic materials. Quantum Mechanics 3 Homework 8 The equation of motion for the annihilation and density operators for interacting Fermions in the Hartree-Fock (Random Phase) approximation, the pair correlation function for Fermi operators on the free Fermi gas demonstrating the existence of the "Fermi Hole", the first-order energy shift of neutron matter in a neutron Jellium model (please verify that problem 3 is correct; I think it is.), and the mass and first order energy shifts for excitations in the Jellium model. Nuclear and Particle Physics Homework 9 Branching rations for the W and Z bosons including the color factor and production of W bosons by a theoretical quark beam, proton beams, and proton beams considering sea quarks. Quantum Mechanics 3 Homework 7 The first and second-order energy shifts and lifetime induced on an Interaction Hamiltonian by the time translation operator (e.g., gradually turning on the interaction) and verification to the second order of its agreement with the exact value. Calculation of the absorption and emission lifetimes of hydrogenic atoms in a blackbody, proof that these atoms reach their equilibrium distribution exponentially, calculation of these equilibrium values, and calculation of their equilibrium timescale. Nuclear and Particle Physics Homework 8 Spinless electron-muon scattering with mass consideration, massless electron-positron scattering (s- and t-channels) in terms of Mandelstam variables, and the matrix element for electron-positron scattering including spin. Quantum Mechanics 3 Mid-Term The Cosmic Background Radiation from the point of view of the thermodynamics of a photon gas: the Hubble constant is used to find the time dependence of temperature of the early universe viewed as a photon gas. Next, the melting of a crystal is considered using Lindmann's criterion and the melting point is found in the large- and small- Brillouin zone limit. Further, the Casimir effect is then considered, and the Euler-MacLaurin summation formula is used to determine the energy and pressure on a pair of metal plates in the photon vacuum. Finally, the energy splitting and decay time of a neutron as a function of the mass of the pion is considered using perturbation theory. Thanks to Chip and Shiang-Yong for helping me complete this. Solids 1 Test 2 Surface states of a crystal (similar to Ashcroft and Mermin problem 18.2), the quantum mechanics of electron states in a 2D conductor in a strong magnetic field, and the possibility of phononic band gap materials. Is problem 2f right? My test says 'OK' but includes a similar solution nearby. Nuclear and Particle Physics Homework 7 Isospin symmetry in terms of Clebsch-Gordon equations in order to deduce allowed decays of particles, alternative formulations for to charge, spin and isospin quantum numbers, and calculating the magnetic moments of sigma and lambda particles as well as their transition from their symmetrized spin functions. Nuclear and Particle Physics Homework 6 The Yukawa potential in the context of the time-independent Klein-Gordon equation, interaction vertices, weak decay diagrams, and Kaon production by proton-nucleus scattering. Quantum Mechanics 3 Homework 6 Quantum Field Theory: Derivation of the Lorentz Force from the Electromagnetic Hamiltonian, the correlation function for a free gas of bosons and fermions, the matrix elements for one and two photon processes for electrons and photons in an external static charge and the specific case of a free electron gas, and the first-order energy shift of a free electron in the photon gas (it's infinite!) Quantum Mechanics 3 Homework 5 Quantum Field Theory: The optical and acoustic branches of phonons in a crystal lattice with a "site potential" and the physical reason for the apparent shift, justification of the Klein-Gordon scalar field theory from the point of view of a very finely spaced crystal lattice, proof that the electric and magnetic fields are invariant under gauge transformations, the equations of motion for the scalar and vector potentials in the general case, preserving the Coulomb gauge under gauge transformation, and a derivation of the plane wave equations of motion for the scalar and vector potentials. Also, quantization of the electromagnetic field and its related Hamiltonian and a brief analysis of the ramifications of the commutator between electric field and magnetic scalar potential. Nuclear and Particle Physics Homework 5 Measures of momentum transfer and available rest-frame energy for electron-to-muon scattering, pair-creation threshholds for antiproton-proton collisions, inelastic scattering and resonances, and identities on some dot products involving some four-vectors given momentum transfer and the excited state rest mass for inelastic scattering. Quantum Mechanics 3 Homework 4 Quantum Field Theory: Phonons in a one-dimensional linear chain under periodic boundary conditions, its dispersion relation and quantized Hamiltonian, the evaluation of various operators on these Fock states corresponing to these phonons, and also coherent states of phonons. A linear chain of two different types of atoms (alternating) is then explored and its dispersion relation found and evaluated at various limits. Finally, the operator forms in the anharmonic correction to the Hamiltonian for a one-dimensional linear chain of atoms are found. Solids 1 Test 1 Brillouin Lattice planes, using low-angle diffraction experimental data to determine the structure of a powder sample, and a proof that the reciprocal lattice of a simple hexagonal lattice is itself simple hexagonal. Nuclear and Particle Physics Homework 4 Compton scattering of relativistic electrons from a nucleus, derivation of the form-factor for scattering from a homogeneous charge sphere, Rutherford scattering of an electron from a nucleus (actually identical to the hyperbolic orbit problem; namely, the impact parameter and "scattering angle" is found for a hyperbolic orbit), and calculating the differential cross section of Calcium-40 from experimental data. Quantum Mechanics 3 Homework 3 Quantum Field Theory: Coherent states in many-body systems, their overlap with Fock space states, and the average number of particles of a particular momentum in such a coherent state. Bose-Einstein condensates as a coherent state of many-body systems, the momentum operator in the Schrodinger picture. A useful identity on the second-quantized momentum operator allowing one to shift the wavefunction, the result of such a shift acting on the Hamiltonian, and an analysis of its meaning concerning the conservation of total momentum. Finally, Fermi momentum, Fermi energy, and Fermi pressure are found for a free relativistic electron gas and the quantum origin of the Chandrasekhar limit for White Dwarves is revealed. Nuclear and Particle Physics Homework 3 Multiple scattering within a silicon micro-strip detector, pair production near a Lead nucleus, proof that the magnitude of the total four-momentum of a system must be greater than or equal to the mass of the particles, and pair production by a photon near an electron. Quantum Mechanics 3 Homework 2 Quantum Field Theory: The number operator's eigenvalues on a Fock state, proof that the energy term on a Fock state comes from the chi function rather than the kets, particle exchange symmetry, general 3-free particle wave functions for both Bosons and Fermions (directly and with the Slater determinant), the classical Harmonic oscillator and its creation and annihilation quantization, the Heisenberg equations of motion for the quantum field operators, and the action of the density operator on a general Fock state. Nuclear and Particle Physics Homework 2 X-ray emission using Mosely's Law, calcuating the mass of a neutron using data collected by Chadwick in 1932 using a classical elastic scattering approximation, Perkins 1.4 (the flat distribution from the decay of a neutral pion in a relativistic frame and the resluting distribution), and using four-vectors to find the Q-value and momentum of decay products when a Lambda particle decays to a proton and charged pion. Quantum Mechanics 3 Homework 1 Several very nice Quantum Field Theory examples: Commutation relations involving exponentials of arbitrary operators, coherent states of the Quantum Harmonic Oscillator and also a Quantum Harmonic Oscillator under an external force, and Bosonic and Fermionic wave functions for free particles using the Slater Determinant. Nuclear and Particle Physics Homework 1 The binding energy of a Hydrogen atom, the typical orders of magnitude of chemical versus nuclear reactions, a classical model of scattering where two particles collide elastically (considering allowed scattering angles and energies), calculating binding energy from mass excess, and diffraction gratings.

## Qualifying Exam Problem Archive

 Link Source Description 8/18 Study Group Meeting A formula sheet containing formulas you should know for the qualifying exam. We had an inordinate number for the Stat Mech section, so I cut it down to just things necessary for past exams. CMU February 2006 Qualifying Exam General Physics--Radiation Pressure, Drag, Neutrino Mass, the Proton-Proton cycle, dark-line spectra, Zeeman effect, and redshift. CMU February 2006 Qualifying Exam Quantum Mechanics--Ladder operators and commutation operations for a spin-1 particle, as well as results of Stern-Gerlach experiments run on such a particle. CMU February 2006 Qualifying Exam Statistical Mechanics--Energy, Free Energy, and Length of a rubber band in the Canonical ensemble, and the unusual effects of heating this rubber band. CMU February 2006 Qualifying Exam Classical Electrodynamics--A betatron. The betatron condition, magnetic fields necessary to maintain it and small oscillations are explored. My results from part f do not agree. CMU February 2006 Qualifying Exam Classical Mechanics--A sled with a pendulum attached to it sliding down a hill. CMU February 2006 Qualifying Exam Mathematical Methods--The diffusion equation and heat propagation through a uniform box of matter. CMU February 2006 Qualifying Exam Quantum Mechanics--Distinguishable particles in a harmonic potential, raising and lowering operators, and degenerate perturbation theory. CMU February 2006 Qualifying Exam A classical ideal gas in a container, except this gas has a tendency to stick to the walls. Chemical potentials and the ratio of particles on and off the wall are found. CMU August 2005 Qualifying Exam Classical Mechanics--A ball rolling down a circular wedge. CMU August 2005 Qualifying Exam Classical Electrodynamics--Radiation pressure, solar sails, and the limit of Luminosity of a stellar object based on radiation pressure. CMU August 2005 Qualifying Exam Quantum Mechanics--The particle in an infinite well problem, as well as the time development of a Gaussian distribution trapped in the infinite well and a look at the problem of a pair of infinite deltafunction wells. CMU August 2005 Qualifying Exam Statistical Mechanics--Fermi and Bose statistics on a density of states with a band gap, as well as chemical potential and Bose-Einstein condensation. Thanks to RS for bringing my attention to serious errors in this solution. CMU August 2005 Qualifying Exam Mathematical Methods--Green's functions applied to an anharmonic oscillator, treated as a perturbation. I'm not 100% certain that the last part is correct. CMU August 2004 Qualifying Exam Classical Mechanics--Precession of the perihelion position of Earth due to General Relativity, studied from the skeptical point of view that the sun is instead irregularly shaped. CMU August 2004 Qualifying Exam Classical Electrodynamics--Dipole radiation and the differential cross section of a particle being hit by a photon. CMU August 2004 Qualifying Exam Mathematical Methods--Fourier Transforms and Contour Integration CMU August 2004 Qualifying Exam Statistical Mechanics--Energy fluctuation of a classical ideal gas and quantum harmonic oscillators in the canonical ensemble. CMU August 2004 Qualifying Exam Quantum Mechanics--Identical particles in a one-dimensional box. I had a problem with part (e), as I can't do the integral. Could someone give me some advice there? CMU August 2003 Qualifying Exam Classical Mechanics--The Lagrangian of a particle under unusual forces that is constrained to move on the surface of a cylinder. CMU August 2003 Qualifying Exam Classical Electrodynamics--Proof of the wave impediance in a media, boundary conditions of a wave incident on a chargeless, currentless boundary, and the tramsmission and reflection coefficients for normal-incidence.  I am aware of a problem with the coefficients. The correct answer is given, but how my fraction became inverted remains elusive. Please find my error here! CMU August 2003 Qualifying Exam Mathematical Methods--A particle in the box-like problem with a delta-function barrier in the center. This is actually a mass string with another mass tied to the middle. Differential equations and the final form and constraints of the even solutions are given. CMU August 2003 Qualifying Exam Statistical Mechanics--The partition function and energy of a black body, as well as derivation of the "classical" picture. Updated Feb. 14 2007. CMU August 2003 Qualifying Exam Quantum Mechanics--The time-development of a Hamiltonian for a spin-1 system. General differential equations are used (matrix method), and the time development operator is found using the eigensystem of the Hamiltonian. CMU August 2003 Qualifying Exam Quantum Mechanics--The transitions between the 2S and 1S levels of the Hydrogen atom. The Wigner-Eckart theorem is used. Perturbation theory is then set up to be used to estimate the lifetime of the 2S state. We were not able to finish solving the differential equation to get the estimate--can someone look at the work and point out where I have gone wrong? CMU August 2003 Qualifying Exam General Physics--A brief exploration of the possibility of a massive photon from astronomical observations. The relativistic kinetic energy is used along with basic physics. CMU August 2002 Qualifying Exam Classical Mechanics--The frequency of small oscillations of a ball rolling inside of a cylinder under gravitational force only. CMU August 2002 Qualifying Exam Classical Electrodynamics--Photons in linear media, along with their velocities, the boundary conditions, and also a derivation of "skin depth" as a photon attempts to travel through an ohmic conductor. CMU August 2002 Qualifying Exam Mathematical Methods--A toy nuclear reactor, the diffusion equation in integral and differential form and applied in one-dimension with boundary conditions reflecting those of a nuclear reactor. The "critical mass" is found. CMU August 2002 Qualifying Exam Statistical Mechanics--A five-level quantum system in the canonical ensemble. Ratios of the number of particles in different states, the maximum internal energy, and the entropy at very low and very high temperatures are found. Our study group could not find an easy means to expand in part f CMU August 2002 Qualifying Exam Quantum Mechanics--The 21cm line is examined: first, a brief analysis of splitting and shifting of the 1s levels by the fine and hyperfine structure corrections is given. Second, the matrix elements for the hyperfine shifts of the 1s state of Hydrogen are found. This is an excellent, instructive example! CMU August 2002 Qualifying Exam Quantum Mechanics--The particle in the box is revisited under a powerful magnetic field. The density of states of the particle in the box is found, and then the total kinetic energy per electron in the ground state is found as well as the Fermi energy of the particle in the box. CMU August 2001 Qualifying Exam Classical Mechanics--Normal coordinates and the small oscillation frequencies of a double pendulum. CMU August 2001 Qualifying Exam Classical Electrodynamics--The precession of a rotating, charged sphere in a magnetic field, the magnetic moment of a charged sphere, and a capacitor radiating under alternating current including equivalent inductance and capacitance. I believe that there is an error in the dipole radiation portion. CMU August 2001 Qualifying Exam General Physics--The "cutoff" for proton radiation in space explored in light of a gamma-ray interaction with a proton to produce a pion. CMU August 2001 Qualifying Exam Statistical Mechanics--An ideal Fermi gas applied to a white dwarf. Fermi energy, heat capacity, pressure and blackbody radiation are examined. The last parts of this problem are incomplete, starting with the heat capacity estimation. I could not find a good way to estimate the heat capacity, which was necessary for later parts. CMU August 2001 Qualifying Exam Quantum Mechanics--A particle in a spherical well. Orbitals and eigenstates are found, and the problem of two electrons in a spherical well is then explored in terms of both wavefunction and with a perturbed spin-spin interaction. Finally, the ground states are examined in the canonical ensemble and in terms of thermometer construction. CMU August 2001 Qualifying Exam Thermal Physics--The entropy, adiabatic expansion, and heat capacity of a stretched rubber band.

## Semester 2 Problem Archive

 Link Source Description Classical Electrodynamics 1 Final The final from Classical Electrodynamics 1 was the first assignment for the second course. I'm not 100% certain on the fourth problem, so if someone could check that I have it right that would be great. Topics: Superposition Principle, Laplacians, dipole, quadrupole and magnetic moments, non-linear dielectrics, retarded potentials, and a moving particle in Minkowski space. Quantum Mechanics 2 Homework 1 This examines the fundamentals of probability theory in the context of Quantum Mechanical operators. Topics: Phase space, toy quantum models and chain kets, and compatible observables. Classical Electrodynamics 2 Homework 2 Application of Green's Functions to show that the Schrodinger equation is gauge invariant, wave functions in non-current non-charge free space, and Maxwell's equations in 2 dimensions. Quantum Mechanics 2 Homework 2 The Born Rule and Consistent Quantum Theory Topics: Unitary time development, compatible observables, and some paradoxes of Quantum mechanics examined using toy models and the Born rule. StatMechHW1 Request Only Statistical Mechanics Homework 1 Basic reviews and proofs covering elementary Thermodynamics: equations of state and the method of Jacobians. Classical Electrodynamics 2 Homework 3 Alternate forms of Green's functions (Tensor form and an alternate derivation) for the wave equation and retarded potentials of fast-moving charges. Quantum Mechanics 2 Homework 3 A lot of explicit examples involving Consistent Families and the Born Rule. Topics: Consistent quantum histories and scattering with toy models StatMechHW2 Request Only Statistical Mechanics Homework 2 State-counting problems, including Pathria 1.4 and Pathria 1.7. Classical Electrodynamics 2 Homework 4 Jackson 6.1, Jackson 6.18, Jackson 11.17. Quantum Mechanics 2 Homework 4 Partial Traces and toy modelling, as well as the interesting question of measurement and interference patterns from a particle and its own superposition. StatMechHW3 Request Only Statistical Mechanics Homework 3 Examples using the canonical ensemble, Pathria 2.7, dispersion of energy, and the heat capacity of rigid rotators. Classical Electrodynamics 2 Homework 5 The derivation of the Minkowski space Maxwell stress tensor, Green's functions for differential operators using the jump discontinuity method and the eigenfunction decomposition method, and a proof of a property of zeroes of eigenfunctions. Quantum Mechanics 2 Homework 5 These problems examine quantum teleportation, quantum entanglement, quantum coding and cryptography (BB84 protocol), Quantum Mechanics 2 Homework 6 A practice exam: topics covered are consistent histories, the Born rule, partial traces, and quantum teleportation. Classical Electrodynamics 2 Homework 6 A review of linear algebra and eigenvector decompositions, Green's function for a simple harmonic oscillator, Green's function for a quasi-static magnetic field, and Green's function for a two-dimensional electrostatic potential. Jackson 6.3 c. StatMechHW4 Request Only Statistical Mechanics Homework 4 The Bromwich contour, and the canonical ensemble: Pathria 3.26, Pathria 3.29, Pathria 3.32, Pathria 3.17, Pathria 3.19 Classical Electrodynamics 2 Homework 7 Green's functions for a pie-shaped region, inhomogeneous solutions for derivatives of deltafunctions, Jackson 2.7, Jackson 2.19 (my professor and I agree that Jackson has made an error in his solution) Quantum Mechanics 2 Homework 7 Scattering with a practical example, the Born approximation, Cohen-Tannoudji VIII-C-3a, VIII-C-3b, proofs involving operators on Hilbert spaces and their subspaces, and rotation operators and compositions thereof. StatMechHW5 Request Only Statistical Mechanics 2 Homework 5 Pathria 3.23, Pathria 3.12, derivations concerning the partition function in the grand canonical ensemble, Fermi and Bose fluids in the grand canonical model, and the free particle in the box problem in the high-temerature limit using the grand canonical model. Classical Electrodynamics 2 Test 1 Maxwell's Equations in four dimensions, tensor analysis of the Poynting vector, energy density and the Energy-Momentum tensor, and a charged particle between two infinite conducting plates--problem 3 is very similar to Jackson 3.20. Quantum Mechanics 2 Homework 8 Reducible and irreducible representations, spin, orbital angular momentum, addition of angular momentum, Cohen-Tannoudji X-G-1, X-G-3, X-G-4, construction of irreducible tensor operators analogous to spherical harmonics, Bohr frequencies, and simple, explicit examples of the Wigner-Eckart theorem in action and its connection to spherical harmonics. In the final problem, I have been careful to identify every part of the rather hefty notation. Quantum Mechanics 2 Homework 9 A sample test covering the Born approximation, scattering and partial scattering cross-sections and estimating the size of a nucleus from actual measurements of the scattering cross-section, the Wigner-Eckart theorem and the behavior of the rotation group versus the Hamiltonian. Classical Electrodynamics 2 Homework 9 Induced surface charge on dielectrics with the method of images, a derivation of frequency-dependent conductivity in a certain model of a conductor, using Green's functions to construct a solution in another frequency-dependent case, and Jackson 7.12. StatMechHW6 Request Only Statistical Mechanics Homework 6 Pathria 3.30, a interacting harmonic oscillators in the Grand Canonical Ensemble, the expectation value of the mean-square deviation of particle count in the Grand Canonical Ensemble, ideal gases in the GCE, entropy in the GCE, and an ideal gas with adsorption sites within its container. Classical Electrodynamics 2 Homework 10 Time-dependent dielectrics. Jackson 7.21, Jackson 7.22 (with a rather crude result, but still correct), Green's function in proximity of a large dielectric region with a planar boundary, and Green's function for a simple model of an electron in a dielectric medium in the limit of an extremely fast deltafunction pulsed electric field. Quantum Mechanics 2 Homework 10 Perturbation theory on the infinite square well in one and two dimensions and orbital systems and the variational method demonstrated. Cohen-Tannoudji chapter 11 Exercises XI-H-1, XI-H-2, XI-H-5, XI-H-9, and also a very explicit example of the Zeeman effect on the hyperfine structure of atomic deuterium, including very explicit examples of level splitting with Mathematica graphs and code StatMechHW7 Request Only Statistical Mechanics Homework 7 Ideal Bose and Fermi gases with a comparison to the Sackur-Tetrode result, generalization if ideal Bose and Fermi gases to one and two dimensions, a problem on the condensation of ideal bosons to the ground state and the entropy and heat capacity in that region, and numerical verification of why we treat only the zero-energy point in a special way for Bosons. Classical Electrodynamics 2 Homework 11 Derivation of Jackson 11.149, a problem analogous to Jackson 12.6a (derivation of the general boost that will bring a uniform static electric and magnetic fields parallel), and the Lagrangian for a free particle in electric and magnetic fields and proof that its resulting equation of motion is gauge invariant despite it itself not being so. Classical Electrodynamics 2 Homework 12 Jackson 12.6 part b, the relativistic motion of a particle in the degenerate case of uniform but perpendicular electric and magnetic fields, the motion of a particle in a uniform static electric field (analogous to the relativistic rocket), and the relativistic Kepler problem. Thanks to LC for correcting an mistake in this example (5-15-2006). StatMechHW8 Request Only Statistical Mechanics Homework 8 Pathria 8.9, Pathria 8.15, Pathria 5.1, Pathria 5.5, and Curie's Law as it applies to the Ising model. Quantum Mechanics 2 Homework 11 The connection between the interaction and operator forms of the time-independent versus time-dependent perturbation series, Cohen-Tannoudji XIII-E-1, XIII-E-2, XIII-E-7, (spin interactions, the photoelectric effect, and a pulsed harmonic oscillator) electric dipole selection rules and allowed photonic transitions, and the effects of perturbation at short and long times, with a justification of non-exponential decay time. Quantum Mechanics 2 Homework 12 The interaction of polarized light with perturbed energy levels of an electron, Cohen-Tannoudji XIV-D-1 (identical particles), XIV-D-2 (spectral numbers), excited states of Lithium, symmetric and antisymmetric combinations of wave functions, and identical bosons and fermions in a one-dimensional infinite potential well with repulsive perturbation. Quantum Mechanics 2 Practice Final Quantum Computation--considering transitions of Beryllium in the context of polarized laser light, Clebsch-Gordan coefficients, perturbation theory, Consistent Quantum Theory, the Born approximation, and the perturbation due to the Stark effect on the hydrogen atom. Using the Wigner-Eckart Theorem I recall back when I was learning about the Wigner-Eckart theorem, I found the notation rather confusing and available resources rather poor. Here I have written my own small paper on its primary uses and notation. There are many examples here which use it, but this is a summary of its primary uses which I hope will help out anyone who had the same trouble I did, clarifying the notation and what it means.

## Semester 1 Problem Archive

 Link Source Description CMU Placement Exam 2005 The quantum "particle incident on a step function barrier" problem. Mathematical Physics: Analytic Functions and Complex Numbers Simple applications of the Cauchy-Reimann equations, analytic functions, and imaginary numbers. Mathematical Physics: Convergence Convergence and radius of convergence, series, zeroes and poles, the Cauchy-Hadamard formula, and Conformal mapping. Classical Electrodynamics: Delta Function, Coulomb Potential, and Electric Field Advanced integrals in 3-space using Coulomb's Law. Jackson 1.3, and deltafunctions. Probability: The density function of functions of random variables An example demonstrating how this calculation is done. (Cumulative Density Functions) Quantum Mechanics: Uncertainty and the Gaussian Wave Packet Two interesting examples involving the Heisenberg uncertainty principle, and one involving the Gaussian wave packet. Ehrenfest's Theorem. Classical Electrodynamics: Gauss's Law Gauss's Law applied on several scenarios. Jackson 1.4, Jackson 1.5. Mathematical Physics: Branch Cuts Problems mostly concerning avoidance of multiple-valued areas in certain functions analytic over a region. Laurent Series, principal value integrals. Quantum Mechanics: Wave Mechanics A number of examples with simple wave mechanics. Planck energy density, the continuity equation, Cohen-Tannoudji I-2 Mathematical Physics: Contour Integration and Branch Cuts Tests Two sample tests on Contour Integration, complete. Residues, poles, singular points, analyticity, branch cuts. Mathematical Physics: Contour Integration and Branch Cuts A number of problems involving contour integrals solved. ML bounds, Cauchy Principal Value, branch cuts. Classical Electrodynamics: Conductors and Laplacians Conductors from the Laplacian approach, the mean value theorem proved, and a simple Laplacian. Jackson 1.10, Laplace's equation. Classical Electrodynamics: Laplacians and Spherical Harmonics Application of Lapacians to boundary-value problems. Image charges, Jackson 2.2, derivation of Spherical harmonics in two dimensions, and Poisson's equation. Quantum Mechanics: Wave Mechanics Scattering Wave Mechanics and Scattering: this lacks a graph of the solutions to the first problem. It's pretty simple to construct, and a description of how to is there. Cohen-Tannoudji I-7, I-3, transmission and reflection. Quantum Mechanics: Wave Mechanics Scattering Wave Mechanics and Braket notation, creation and destruction operators, time delays, Cohen-Tannoudji V-2 Mathematical Physics: General Linear Algebra Basic linear algebra problems. Normalization, polynomials as a linear space, rotation matrices, similarity transforms, rank. Classical Electrodynamics: Test 1 This reviews earlier EM material. Laplacians, charge distributions, Legendre polynomials. Mathematical Physics: Linear Algebra Methods Linear algebra problems involving matrices more directly. Eigenvectors, eigenvalues, normal matrices, unitary matrices, diagonalization, Hermitian matrices, triangular matrices, positive operators, polar decomposition, operators defined with similarity transforms, simultaneous solutions. Quantum Mechanics: Linear Algebra and Probability Theory. Linear algebra and probability theory problems. Cohen-Tannoudji II-2, II-6, III-1, III-6, III-7, III-8 Quantum Mechanics: Particle in a Box with a Deltafunction Unusual even solutions to the particle in a box problem with a deltafunction. Classical Electrodynamics: Multipole Expansion Problems relating to multipole expansions. Image dipoles, quadrupole moments, Jackson 4.7 Classical Electrodynamics: Multipole and Other Expansions Even more different expansions involving energy and electric field: Taylor, Multipole, and Legendre Polynomial. Tensors, Jackson 4.5, and a minimization-of-energy approach. Mathematical Physics: Basic Probability Distributions, sample spaces etc. Very basic probability. Venn diagrams, sample spaces, probability distributions, marginal distributions, covariance, Binomial and Poisson distributions. Classical Electrodynamics: Magnetostatics Some rather advanced Magnetostatics problems, including Jackson 5.4 and 5.5. Magnetic fields and vector potentials. Quantum Mechanics: The WKB Approximation Mostly the WKB approximation and practical examples. Partition functions, and WKB applied to tunneling. Quantum Mechanics: Angular Momentum Simple problems involving the angular momentum operator and its properties. Cohen-Tannoudji VI-2, VI-4 Mathematical Physics: Probability Sample Test Problems Simple probability, and some Central Limit Theorem problems. Conditional probability, random variables. Mathematical Physics: Markov Chains Advanced probability and Markov Chains. Normal (Gaussian) distributions, radioactive decay, transition matrices, invariant distributions. Classical Electrodynamics Test 2 This reviews magnetostatics and multipole expansions. Dipole and Quadrupole moments, vector potential, energy of systems of dipoles, the Biot-Savart law, magnetic moments. Quantum Mechanics: Rotations of Operators Angular momentum operators under rotation. This has some very interesting problems. Cohen-Tannoudji VI-9, 3D harmonic oscillators Classical Electrodynamics: Photons and Retarded Potentials Interesting problems involving photons, and a very general one involving retarded potentials. I'm not 100% certain on the solution to #3, but it seems plausible. Ohm's law in the language of Maxwell, polarization, polarization of light passing through media, and retarded potentials. Mathematical Physics: Fourier Series and Transformations Problems involving the derivation and applications of Fourier series and transformations. Even and odd extension, and use of Parseval's identity. Griffiths "Introduction to Electrodynamics" Ch.4 I wanted some practice problems for the final, so I did 4.29 to 4.34. Quantum Mechanics: Rotations of Wave Functions Tensor analysis and rotations of wave functions. Lie algebras, spin-orbit coupling, spherical harmonics. Classical Electrodynamics: Retarded Potentials Advanced problems involving retarded potentials, including Jackson 9.1. Some of these problems get ridiculously complex and require a lot of approximation. "EMP" from a power line being turned on suddenly, radiation fields. Griffiths "Introduction to Electrodynamics" Ch.9 Griffiths 9.32, 9.33 and 9.36. I did these to help me study for the final. Mathematical Physics: Differential Equations Matrix methods applied to differential equations, as well as an example of how to use these to find similarity transforms into Jordan form. Convolution, exact differentials, integrating factors, general n-th order differential equations in matrix form, solution basis, Wronskians. Classical Electrodynamics: Special Relativity Special relativity applied to some simple examples, including Jackson 11.6, time dilation, proper time, 4-velocity. Quantum Mechanics: Hydrogen Atom Some special topics including Time Reversal and the Hydrogen Atom. Orbits and the virial theorem. Classical Electrodynamics: 2004 Fall Final For practice, the 2004 Classical Electrodynamics Final. Multipole moments, the method of images, and light reflected from a moving mirror. Differential Equations: Oscillating Spring A friend asked me for help with a problem involving an oscillating spring, and it took me too long to figure it out. The moral? Watch out for whether your solution is overdamp, underdamp, or critically damp! Mathematical Physics: 2003 Final The 2003 final for Mathematical Physics - Power series, vector spaces, general differential equation solutions, random variables. Mathematical Physics: 2004 Final The 2004 final for Mathematical Physics - Fourier transforms, Bessel's equation, indicial equations, Markov chains, matrix forms to differential equations, Cauchy-Reimann conditions, integrating factors, Fourier series, linear independence Mathematical Physics: Differential Equations Advanced topics in Differential Equations including the Frobenius method and Bessel functions. Power series to solve differential equations, singular points of a differential equation, Legendre polynomials, the Bonnet recursion, Wronskians as a tool to predict zeroes of differential equation solutions, modified Bessel functions, eigenfunctions of a differential equation, Fourier-Bessel expansion. Griffiths "Introduction to Electrodynamics" Ch.11 For practice, Griffiths problems 11.24 and 11.19.