## 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