# Preliminary Examination Information

### Physics Preliminary Exam

**The 2024 Preliminary Exam is scheduled for May 30–31, 2024.**

***** Baylor Physics Ph.D. Students can find additional logistical details and FAQs here: ****Baylor Prelim Info Presentation**** *****

The Physics Preliminary Exam for admission to candidacy for the Ph.D. will be given over the course of two days. The exam consists of four parts: Part I (Classical Mechanics), Part II (Quantum Mechanics), Part III (Electricity and Magnetism), and Part IV (Statistical Mechanics).

Physics Ph.D. students who will have completed their first-year of study in May and have not already passed all four parts of the exam must take the exam. Students working toward the terminal Masters degree may take the exam in place of an oral exam.

Each part of the exam will consist of 5 problems. All 5 problems will be scored.

The Physics Department will provide the following for use during the exam: a simple scientific calculator (TI-30XA), Schaum’s *Mathematical Handbook of Formulas and Tables* (Spiegel *et al.*), and *The Cambridge Handbook of Physics Formulas* (Woan).

The topics covered by the four parts of the Preliminary Exam are listed below.

Old exam problems are available on the web in a PDF file. The exams (from 2007 to 2023) may be downloaded from the file located at

Note that the exam format has changed since 2007 (for example, there were previously 6 problems per part). Even older versions (from 1967 till 2006) are available at

Since these documents are long, please conserve paper by printing specific pages only as you use them.

Best regards,

The Preliminary Exam Committee

### Part I. Classical Mechanics

The Classical Mechanics (CM) part of the 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) part of the 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 Griffiths, *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 One 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) part of the Preliminary Exam will test basic concepts of electricity and magnetism 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 transformations; 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) part 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 Huang, *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-Thermodynamics**

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 derivatives; 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 transport 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 sums 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; Bose-Einstein condensation; heat engines and refrigerators; efficiency of a cycle; Carnot engines and maximal efficiency allowed by the 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 Butkov, *Mathematical Physics*, Arfken and Weber, *Mathematical Methods for Physicists*, or Wyld, *Mathematical Methods for Physics*. The topics that may be covered in the exam correspond to the material presented in PHY 5360 Mathematical Physics (the first semester of graduate Mathematical 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 derivatives 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