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.
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
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
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
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
Upper bounds to observable values; ground state and ground state energy; helium; hydrogen molecule ion
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 theorem; dynamic phase; geometric phase; Berry's phase; Aharanov-Bohm effect
(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.
Coulomb's law; Gauss' law; scalar potential; image methods; boundary conditions; Green functions; boundary value problems; multipole expansion; dielectric media; capacitance; eigenfunction expansions
Boundary conditions; Ampere's law; Biot-Savart law; vector potentials; magnetic materials; mutual and self-inductance
Macroscopic Maxwell equations; current and energy/momentum conservation; gauge tranformations; retarded and advanced Green functions; Faraday's law
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
Lorentz transformations; field equations; conservation laws
Part IV. Statistical Mechanics & Mathematical Physics
The Statistical Mechanics section of Part IV of the 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.
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
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)
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
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
The Mathematical Physics section of Part IV of the Preliminary Exam 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
Definition and examples of Fourier series; complex form of Fourier series; convergence of Fourier series; Parseval's Theorem
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
The Green function method; Green function for the Sturm-Liouville operator; eigenfunction expansions for Green functions