Selected Solutions


My solutions to selected excercies in commonly used graduate-level texts are compiled here. Please let me know if there are any errors using the email address to the left. Textbooks used are:

  • Mathematical Methods for Physicists, 7th ed. · Arfken, Weber, and Harris
  • Classical Mechanics, 3rd ed. · Goldstein, Poole, and Safko
  • Principles of Quantum Mechanics, 2nd ed. · Shankar
  • Statistical Mechanics and Thermodynamics · Garrod
  • Classical Electrodynamics, 3rd ed. · Jackson
  • Elementary Particle Physics in a Nutshell · Tully
  • Quantum Field Theory and the Standard Model · Schwartz


  • Arfken Solutions

    Chapter 1:

  • Section 1: 1.5.3, 1.5.4
  • Chapter 7:

  • Section 2: 7.2.5, 7.2.17
  • Section 3: 7.3.4
  • Section 4: 7.4.2
  • Section 5: 7.5.7, 7.5.10, 7.5.11
  • Section 6: 7.6.14, 7.6.22, 7.6.23
  • Section 7: 7.7.4
  • Chapter 10:

  • Section 1: 10.1.3
  • Chapter 11:

  • Section 2: 11.2.3, 11.2.7
  • Section 3: 11.3.3, 11.3.6
  • Section 4: 11.4.7, 11.4.8
  • Section 5: 11.5.7, 11.5.8
  • Section 6: 11.6.11
  • Section 7: 11.7.1, 11.7.4, 11.7.8
  • Section 8: 11.8.10, 11.8.17, 11.8.18b, 11.8.22
  • Section 9: 11.9.3, 11.9.7


  • Garrod Solutions

    Chapters 2,3:

  • Chapter 2: 2.23, 2.26
  • Chapter 3: 3.8, 3.16, 3.25
  • Chapter 4: 4.7, 4.15, 4.16, 4.19, 4.25

    Chapter 5: 5.2, 5.3, 5.9, 5.12, 5.21

    Chapters 6,7,8:

  • Chapter 6: 6.2
  • Chapter 7: 7.4, 7.5, 7.9, 7.15, 7.16, 7.18, 7.19, 7.26, 7.27
  • Chapter 8: 8.4, 8.5


  • Goldstein Solutions

    Chapter 1: 1.13, 1.14, 1.21

    Chapter 2: 2.18

    Chapter 3: 3.10, 3.11, 3.13, 3.21, 3.24, 3.30, 3.33, 3.35,

    Chapter 4: 4.23

    Chapter 5: 5.15, 5.16, 5.18, 5.20

    Chapter 8: 8.3, 8.19, 8.26

    Additional Solutions



    Shankar Solutions

    Chapter 1:

  • Section 8: 1.8.8, 1.8.9, 1.8.10
  • Section 10: 1.10.4
  • Chapter 2:

  • Section 7: 2.7.5
  • Section 8: 2.8.3, 2.8.4, 2.8.5
  • Chapter 5:

  • Section 2: 5.2.2, 5.2.4
  • Section 3: 5.3.1
  • Chapter 7:

  • Section 3: 7.3.6
  • Section 5: 7.5.4
  • Chapter 12:

  • Section 2: 12.2.4
  • Section 3: 12.3.6, 12.3.7, 12.3.8
  • Section 5: 12.5.3, 12.5.12, 12.5.13
  • Section 6: 12.6.8, 12.6.9, 12.6.10
  • Chapter 15:

  • Section 2: 15.2.4
  • Chapter 17:

  • Section 3: 15.3.4


  • Jackson Solutions

  • Chapter 2: 2.2, 2.7, 2.12, 2.13
  • Chapter 4: 4.1, 4.2, 4.7


  • Other Electromagnetism Solutions

  • Solution Set 1: General Forms of Gauss' and Stokes' Theorems, Surface Charge Densities of Parallel Plate Capacitor, Quantum Capacitance, Force due to Self-Capacitance, Capacitance Matrix
  • Solution Set 2:Generalized Gauss' Divergence Theorem, Electric Field Boundary Conditions, Self Capacitance of Ellipsoid of Revolution, Force Between Point Charge and Grounded Sphere, Ensemble of Point Charges, Green's Function Technique - Conducting Sphere
  • Solution Set 5: Green's Function in Free Space, Angled Capacitor, Dielectric Spherical Capacitor, Step-Function Dielectric, Dielectric Parallel Plate Capacitor
  • Solution Set 6: Interaction of Polarizable Molecules, Polarizability of Electron in Harmonic Potential, Dielectric Spherical Shell, Dielectric Infinite Cylinder
  • Solution Set 7: Coaxial Conductors, Magnetic Field inside Cylindrical Cavity, Axially Symmetric Magnetic Field, Infinite Sheet in Uniform Magnetic Field, Rotating Charged Sphere
  • Solution Set 8: Toroidal Electromagnet with Gap, Electromagnetic Momentum, Type I Superconductor, Self-Inductance of a Long Solenoid, Mutual Inductance of Coaxial, Displaced Current Loops, Inductance of Coaxial Cable
  • Solution Set 9: Transfer Function, Control Circuit, Polarization of Electromagnetic Waves, Superposition of Plane Waves
  • Solution Set 10: Complex Polarization Amplitude, Complex Wave Vector, Relative Strength of Electromagnetic Wave Components in the Vacuum, Complex Components of Refractive Index for a Metal, Electron Density of Interstellar Medium, Electromagnetic Waves in a Plasma
  • Solution Set 11: Complex Dielectric Constant: Drude Approximation, Sokhotski-Plemelj Theorem, Kramers-Kronig Relations, Position-Dependent Permittivity, Surface Magnetic Field Wave at a Dielectric Boundary
  • Solution Set 12: Dielectric Response of Quantum Electron Gas, Rectangular Waveguide, Circular Waveguide, Transverse Electromagnetic (TEM) Modes in Coaxial Waveguide, Lumped Element Model
  • Solution Set 13: Fields of Moving Point Charge, Point Charge with Constant Velocity, Larmor Formula, Linear Motion of Charged Particle, Linear Motion of Charged Particle, Energy of Proton from van de Graaff Accelerator
  • Solution Set 14: Hertz Vector - Partial Field Expansion, Alternative Solutions to the Wave Equation, Vector Solutions to the Wave Equation, Covariant Tensor - Change of Coordinate Basis, Contraction of Tensor, Metric Tensors
  • Solution Set 15: The Metric Tensor, Covariant Derivative of Scalar, Covariant Derivative of a Covector, Euler - Lagrange Equations, Special Relativity, The Potential Tensor, Transverse Electromagnetic Waves


  • Quantum Mechanics Solutions

  • Solution Set 1: Hydrogenic atoms, Electron bound to dielectric plane, Electron in annular potential
  • Solution Set 2: Interference of neutron beams, Electron in uniform magnetic field, Coherent States
  • Solution Set 3: Anharmonic oscillator, Potential of classical radius of proton, Hydrogen atoms in electric field
  • Solution Set 4: Stark effect for hydrogen n=3 levels, Spins in inhomogeneous magnetic field, Spin of electron, Hydrogen in magnetic field, Total angular momentum of electron, Hydrogen fine structure - Integrals
  • Solution Set 5: Commutator of electric and magnetic fields, Free radiation field, Momentum of electromagnetic waves, Electric field correlation function
  • Solution Set 6: Quantization of free radiation field, Selection rules for transitions of hydrogenic atoms, Transitions of hydrogenic atoms, Lifetime of excited hyperfine state
  • Solution Set 7: Electron in short range potential, Two electron atoms: Coulomb repulsion, Excited states of helium
  • Solution Set 8: Charged particle scattering from neutral hydrogen, Quantum hard sphere scattering, The deuteron


  • Quantum Field Theory and Particle Physics Solutions

  • QFT-I Solution 1: Dimensional transmutation, Dimensional analysis in natural units, Spinless relativistic particle and causality violation
  • QFT-I Solution 2: Relativity exercises, Two-dimensional quantum harmonic oscillator, Spinless boson, Divergences in QFT, Conserved currents of infintiesimal Lorentz transforms
  • QFT-I Solution 3: Lorentz transformation properties, Schwartz 10.2, Continuity euations and the Dirac equation, Dirac spinors, Schwartz 10.5
  • QFT-I Solution 4: Schwartz 11.1, Schwartz 11.4, Schwartz 11.8
  • QFT-I Solution 5: Invariance of Dirac Lagrangian, Feynman propagator as Green's function, Schwartz 6.1
  • QFT-I Solution 6: Scalar Yukawa theory
  • QFT-I Solution 7: Amplitudes for spinor Yukawa processes, Complex scalar field
  • QFT-II Solution 1: Classical electromagnetism, Scalar QED, Electron spin & Lorentz boost, Schwartz 9.1
  • QFT-II Solution 2: Bhabha scattering, QED with massive photon (dark photon), QED with radiation in final state
  • QFT-II Solution 3: Renormalization of scalar QED, Photon-photon scattering (Euler-Heisenberg Lagangian)
  • Moller scattering
  • Tully Solutions: Chapter 2 - Problems 1, 4, 5, 8, 11, 17, 18