## Physics 230 — Mathematical Methods of Physics

### Objectives and Goals

This course is designed to introduce first-year graduate student to mathematical concepts and tools needed for research, and more advanced math courses. The subject exposes the students to the level of mathematical rigor required for doctoral research. It helps students acquire the mathematical methods and tools for other graduate course (particularly E&M, QM and SM), necessary research while earning their Ph.D.'s, and understanding journals and papers (e.g. PRLs) necessary for their study. This course also introduces the students to the mathematical tool, Mathematica.

The course provides exposure to the following topics:

• Coordinates [coordinate transformations, Jacobian, cylindrical and spherical coordinates, grad, div, curl, Laplacian]
• Partial Differential Equations [PDE's of physics, separation of variables, Laplace and Helmholtz's equations, spherical harmonics]
• Ordinary Differential Equations [general DEs, linear DEs, Wronskians, particular solutions, Green's functions, series expansions, Legendre polynomials]
• Eigenfunction methods [Hermitian operators, Sturm-Liouville theorem, orthogonal series, delta functions, closure, Green's functions]
• Infinite Series [summation of series, convergence of power series, asymptotic series]
• Complex Variables [Cauchy-Riemann relations, elementary functions, conformal transformations, Cauchy's theorem and integral formula, Taylor and Laurent series, poles, zeroes, branch points, residue theorem]
• Integration [contour integration, principal values, Gaussian integrals, Gamma functions, error functions, saddle point integration]
• Integral Transforms [Fourier series, non-periodic functions, Fourier transforms, convolution theorem, Parseval's theorem, power spectra, Laplace transforms]
• Probability [permutations and combinations, probability distributions, random walks, binomial, Poisson, Gaussian, transformation of variables, moment generating functions, central limit theorem]
• Mathematica [solving equations, integrals, power series, vectors and matrices, FFT, graphics, visualization]

### Methods and Approach

Format:
This course is taught through 75-minute lectures (2 per week), a one-hour computer lab (one per week), and weekly homework sets. Lectures generally involve blackboard presentations and demonstrations by the professor. Computer lab and recitations involve practice with Mathematica and its applications.
Homework sets generally consist of 8-10 problems designed to take approximately 10 hours of concentrated effort to complete. Students are encouraged to discuss the homework with their peers, but first they have to made a reasonable effort to find the main idea on their own. They are required to write solutions independently and understand them.
Students are responsible for all material covered in the lectures and in the homework problems (the Mathematica notebook). Some concepts and applications that are essential for future courses are covered only in the homework.
Text:
• Required: Donald A. McQuarrie Mathematical Methods for Scientists and Engineers
• Recommendation: M.R Spiegel and J. Liu Mathematical Handbook of Formulas and Tables
• Other: Bibliography posted on the website ~palmer/Phy230/bibliograph.php
• Tests: 2 one-hour tests (10% each). The dates to be announced first day of class,
• Final Exam: 24-hour take-home during finals week (20%).
• Homework Assignments: 10 problems sets, 6% each (60%). Due dates to be announced first day of class.
Prerequisites:
Undergraduate courses in intermediate calculus (such as MTH 103 or equivalent) are required. Students must have some familiarity with partial differentiation, multiple integrals, differential vector calculus (grad, div, curl, Laplacian), integral vector calculus (divergence and Stokes theorems), matrices and determinants, simultaneous linear equations, and complex numbers.

### Sample Syllabus

• Lecture 1: Integrals and Special Functions
• Lecture 2: Special Functions; Steepest Descent; Infinite Series
• Lecture 3: Infinite Series; Power Series; Summing Series
• Lecture 4: Asymptotic Expansions; ODEs
• Lecture 5: General ODEs
• Lecture 6: Linear ODEs
• Lecture 7: Series Solutions of ODEs; Legendre Polynomials
• Lecture 8: Series Solutions of ODEs; Bessel Function
• Lecture 9: Qualitative Methods for ODEs
• Lecture 10: Boundary conditions; Eigenfunctions; Orthogonal Functions
• Lecture 11: Orthogonal Series
• Lecture 12: Generating Function; Hermitean Operators; Sturm-Liouville Theory
• Lecture 13: Sturm-Liouville Theory; Delta Functions
• Lecture 14: Green's Functions - 1D
• Lecture 15: Green's Functions - 3D; Orthogonal Coordinates
• Lecture 16: Orthogonal Coordinates; PDEs of Physics
• Lecture 17: PDEs: General Techniques; Separation of Variables
• Lecture 18: Laplace's eqn: Cylindrical Coordinates (Bessel)
• Lecture 19: Laplace's eqn: Spherical Coordinates (Spherical Harmonics); Helmholtz' eqn
• Lecture 20: Fourier Series and Transforms
• Lecture 21: Convolution, Correlation, and Power Spectrum Density
• Lecture 22: Laplace Transforms
• Lecture 23: Complex Variables
• Lecture 24: Complex Variables
• Lecture 25: Contour Integration
• Lecture 26: Contour Integration; Conformal Mapping
• Lecture 27: Basic Probability; Permutations and Combinations; Random Variables
• Lecture 28: Probability Distributions; Stochastic Processes
• Lecture 29: Characteristic Functions; Central Limit Theorem