### Part I. Classical Mechanics

The Classical Mechanics (CM) Preliminary Exam will test basic concepts of classical mechanics and related applications to physical problems. The exam will cover both (i) material presented in PHY 5320 (the first semester of graduate CM at Baylor) and (ii) material generally presented at the undergraduate level. The topics that may be covered in the exam are given in the list below.

The graduate level at which these topics will be covered is on par with Goldstein, Poole & Safko, *Classical Mechanics*. The level of the undergraduate material in the exam is representative of that found in typical undergraduate textbooks such as Thornton & Marion, *Classical Dynamics.*

__Elementary Principles__

Newton's laws; equations of motion; conservation laws & symmetries; work and energy; center of mass; elastic and inelastic collisions

__Lagrangian and Hamiltonian Mechanics__

Hamilton's principle; d'Alembert's principle, principle of virtual work; Lagrangian equations of motion; Lagrangian with constraints; generalized coordinates and momenta; undetermined multipliers; canonical equations;

__Central Force Motion__

Equivalent one-body problem; classification of orbits, Virial theorem; Kepler's laws; planetary motion; scattering

__Dynamics of Rigid Bodies__

Moment of inertia; inertia tensor; orthogonal transformations; eigenvalues and eigenvectors; Euler angles; Euler equations of motion; rotating coordinate systems; Coriolis effect

__Oscillations__

Simple harmonic oscillators; damped oscillators; driven oscillators; coupled oscillators and normal modes

__Hamilton Equations of Motion__

Hamilton Equations of motion; cyclic coordinates and conservation theorems; principle of least action

__Special Relativity__

Postulates of special relativity; Lorentz transformations; length contraction and time dilation

### Part II. Quantum Mechanics

The Quantum Mechanics (QM) Preliminary Exam will test basic concepts of quantum mechanics and related applications to physical problems. The exam will cover both (i) material presented in PHY 5370-5371 (the full year of graduate QM at Baylor) and (ii) material generally presented at the undergraduate level. The topics that may be covered in the exam are given in the list below.

The graduate level at which these topics will be covered is on par with Sakurai, *Modern Quantum Mechanics*, Chapters 1-7. Good references for applications are the Complements sections in Cohen-Tannoudji et al., *Quantum Mechanics*, Volumes I and II. The level of the undergraduate material in the exam is representative of that found in typical undergraduate textbooks such as Griffith, *Introduction to Quantum Mechanics* (which is used in PHY 3372-3373); Gasiorowicz, *Quantum Physics*; or Shankar, *Principles of Quantum Mechanics.*

__Fundamental Concepts and Formalism__

Wave-particle duality; de Broglie and Compton wavelengths; Dirac notation of bras and kets; state vectors; matrix representations; wave functions in position and momentum space; physical observables and hermitian operators; symmetry translations and (anti)-unitary operators; Hilbert space; commutation relations and uncertainties; Heisenberg uncertainty relations; complete sets of commuting operators; expectation values; probabilities; eigenstates and eigenvalues; pure and mixed states; Schroedinger and Heisenberg pictures

__Time Independent Schroedinger Equation in 1-Dimension__

Stationary states; free particles; infinite square well; harmonic oscillators; creation & annihilation operators; delta-function potential; finite square well; bound states versus scattering states

__Quantum Mechanics in Three Dimensions__

Three dimensional Schroedinger Equation; Schroedinger Equation in spherical coordinates; angular momentum (including addition of); orbital angular momentum; spin; hydrogen atom; spin-1/2 systems

__Identical Particles__

Non-interacting particles; Boltzmann statistics and distributions; bosons; Bose statistics and distributions; fermions; Fermi statistics and distributions; exchange forces; Young tableaux, atoms & the periodic table; solids; band structure

__Time Independent Perturbation Theory__

Nondegenerate perturbation theory; degenerate perturbation theory; 2nd order approximations; fine structure of hydrogen; Zeeman effect; hyperfine splitting

__Variational Principle__

Upper bounds to observable values; ground state and ground state energy; helium; hydrogen molecule ion

__WKB Approximations__

Classical region; tunneling; quantization conditions; bound state decay

__Time Dependent Perturbation Theory__

2nd order approximations; two-level systems; Fermi's golden rule; radiation emission and absorption; spontaneous emission; selection rules

__Adiabatic Approximation__

adiabatic theorem; dynamic phase; geometric phase; Berry's phase; Aharanov-Bohm effect

__Scattering__

(Differential) cross sections; Born approximation; partial wave analysis; optical theorem; Eikonal approximation; phase shifts; Coulomb and Rutherford scattering; plane waves and spherical waves; identical particles & symmetry considerations; low energy scattering; bound states

### Part III. Electricity and Magnetism

The Electricity & Magnetism (E&M) Preliminary Exam will test basic concepts of electromagnetics and related applications to physical problems. The exam will cover both (i) material presented in PHY 5330 (the first semester of graduate E&M at Baylor) and (ii) material generally presented at the undergraduate level. The topics that may be covered in the exam are given in the list below.

At the graduate level, problems will be based at the level of Jackson, *Classical Electrodynamics*, Chapters 1-6. There may be undergraduate-level problems on any of the topics listed below. The undergraduate material on the exam is representative of that found in textbooks such as Griffith, *Introduction to Electrodynamics*, Marion & Heald, *Classical Electromagnetic Radiation*, and Schwartz, *Principles of Electrodynamics*. Also, topics like partial wave techniques and scattering are covered in many quantum mechanics textbooks.

Most undergraduate textbooks use the MKS (SI) system while graduate-level texts use the Gaussian system. You are expected to know the difference between the two systems; however, you are free to use the formulas in either system of units.

__Electrostatics__

Coulomb's law; Gauss' law; scalar potential; image methods; boundary conditions; Green functions; boundary value problems; multipole expansion; dielectric media; capacitance; eigenfunction expansions

__Magnetostatics__

Boundary conditions; Ampere's law; Biot-Savart law; vector potentials; magnetic materials; mutual and self-inductance

__Time-varying Fields__

Macroscopic Maxwell equations; current and energy/momentum conservation; gauge tranformations; retarded and advanced Green functions; Faraday's law

__Electromagnetic Waves__

Wave equations; EM waves in dielectrics; Kramers-Kronig relations; EM waves in conductors; reflection and refraction at interfaces; waves guides; resonant cavities; energy loss in wave guides and resonant cavities; radiation from sources; partial wave techniques; multipole fields; scattering and optical theorem

__Relativistic Formulations__

Lorentz transformations; field equations; conservation laws

### Part IV. Statistical Mechanics

The Statistical Mechanics (SM) Preliminary Exam will test basic concepts of statistical mechanics and thermal physics. The exam will cover both (i) material presented in PHY 5340 (graduate Statistical Mechanics at Baylor) and (ii) material generally presented at the undergraduate level. The topics that may be covered in the exam are given in the list below.

The graduate level at which these topics will be covered is on par with Pathria, *Statistical Mechanics* and Juang, * Statistical Mechanics.* The level of the undergraduate material in the exam is representative of that found in typical undergraduate textbooks such as Kittel and Kroemer, * Thermal Physics,* Reif, *Fundamentals of Statistical and Thermal Physics;* Morse, *Thermal Physics,* Schroeder, *An Introduction to Thermal Physics* (which was used in PHY 4340); or Bowley and Sanchez, *Introductory Statistical Mechanics.*

**Concepts-Theromodynamics**

__Equilibrium and the Three Laws of Thermodynamics__

Thermal equilibrium and the definition of temperature and pressure; equilibrium and the definition of chemical potential; spontaneous heat transfer from hot to cold systems; non-thermal transfer of energy; exact differentials; first and second laws of thermodynamics; entropy maximum principle; equations of state; internal energy; Helmholtz and Gibbs free energies; enthalpy; free energy as available work; minimum principles for Helmholtz and Gibbs free energies; manipulation of partial derivaties; Maxwell relations; quasi-static processes; reversible and irreversible processes; heat capacities; basic calorimetry

First order and critical phase transitions

Latent heat; Maxwell construction for first-order transition

Kinetic Theory

Molecular flux; Maxwell distribution of velocities; pressure and internal energy derived from kinetic theory; kinetic theory: mean free path and tranport in dilute gases; transport coefficients (viscosity, thermal conductivity, diffusion)

**Concepts-Statistical Mechanics**

Statistical definition of entropy; thermal equilibrium and the definition of temperature; statistical ensembles (microcanonical, canonical, grand canonical); densities of single-particle states; functions as sms over quantum states; Boltzmann factor; Bose, Fermi, and Boltzmann statistics; partition functions; particles and the Gibbs factor; equipartition theorem; random walks and mean values; fields

**Applications**

Classical ideal gas; blackbody radiation in a cavity; Debye theory of the heat capacity of a solid; van der Waals equation of state; paramagnetism in a classical spin system; paramagnetism in a degenerate Fermi gas; BE condensation; heat engines and refrigerators; efficiency of a cycle; Carnot engines and maximal efficiency allowed by second law

### Mathematical Physics

Mathematical methods will be applied within problems among the CM, QM, EM, and SM sections. The Mathematical Physics will test basic concepts of mathematical physics at the level of

*Mathematical Physics* Butkov,

*Mathematical Methods for Physicists* Arfken and Weber, or

*Mathematical Methods for Physics* (Wyld). The topics that may be covered in the exam correspond to the material presented in PHY 5360 Math Physics (the first semester of graduate Math Physics at Baylor) and are given in the list below.

Vectors, Matrices, and Coordinates

Vector algebra; scalar product, triple products; coordinate transformations, rotation matrices; scalar and vector fields; gradient, divergence, and curl; vector differentiation and integration; Gauss' Theorem, Stokes' Theorem; curvilinear coordinates

Functions of a Complex Variable

Complex numbers: basic algebra and geometry; De Moivre Formula, Euler's Formula; complex functions, branches, and Riemann surfaces; analytic functions, Cauchy-Riemann conditions; Cauchy integral theorem, Cauchy integral formula; complex sequences and series, Taylor and Laurent series; singularities and the residue theorem

Fourier Series

Definition and examples of Fourier series; complex form of Fourier series; convergence of Fourier series; Parseval's Theorem

Integral Transforms

Fourier transforms; properties and examples; Fourier Integral Theorem (Inversion Theorem); Laplace transforms; properties and examples; Laplace transforms of derivaties and integrals; Mellin inversion integral; convolution theorem

Linear Differential Equations of Second Order

The Wronskian; general solution of the homogeneous equation; nonhomogeneous equations, variation of constants; power series solutions; Frobenius method

Partial Differential Equations

Laplace's equation and Poisson's equation; wave equation and the diffusion equation; method of separation of variables; integral transform methods; method of eigenfunction expansions

Green Functions

The Green function method; Green function for the Sturm-Liouville operator; eigenfunction expansions for Green functions