Phys 631 Mathematical Methods of Theoretical Physics Fall 2018

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Hilary Peters
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1 Phys 631 Mathematical Methods of Theoretical Physics Fall 2018 Course information (updated November 10th) Instructor: Joaquín E. Drut. drut at .unc.edu. Office: Phillips 296 Where and When: Class: Phillips Mo-We-Fr, 10:10am-:00am Mo-We: Lectures Fr : Lectures / midterm exams (see below) Office hours: Wednesdays after lecture. Website: Course Sakai site Bibliography: - Required: Arfken, Weber & Harris, Mathematical Methods for Physicists, Seventh Edition. [] - Recommended: - Brown & Churchill, Complex variables and Applications, Eighth Edition. [BC] - Greiner & Müller, Quantum Mechanics: Symmetries, Second Edition. [GM] - Friedman, Principles and Techniques of Applied Mathematics. - Garrity & Pedersen, All the Mathematics You Missed: But Need to Know for Graduate School. Exams: Midterm 1: Friday, September 28th, 10:00am, Phillips 222. Midterm 2: Friday, November 16th, 10:00am, Phillips 222. Final: Friday, December 14th, 8am, Phillips 222. Other remarks: - The holidays affecting this course are: - Monday, September 3rd (Labor day). - Friday, October 12th (University day). - Spring break: October 17th 5pm - October 22nd 8am. - Thanksgiving break: November 21st (Wed) - November 26th 8am. - First lecture: Wednesday, August 22nd. - Last lecture: Wednesday, December 5th.

2 Introduction The goal of this course is to cover the mathematical methods you will need for quantum mechanics, electromagnetism, solid-state physics, quantum field theory, nuclear and particle physics, and beyond. To that end, we will discuss four main topics: linear algebra, group representation theory (with an emphasis on Lie groups), second-order partial differential equations, complex analysis, as well as related topics such as Fourier transforms, Green s functions, and special functions. plans are useless, but planning is indispensable - D. Eisenhower Contents Module 0: Mathematics review: Vector analysis, theorems of vector calculus, Dirac delta. Taylor series. Binomial expansions. Module 1: Linear algebra. Vector spaces, determinants, matrices, eigenvalue problems. Bases, components, basis change, Jacobian matrix and determinant. The discrete Fourier series as a unitary transformation. Matrix groups. Group representation theory. Module 2: Ordinary differential equations. Sturm-Liouville theory. Second-order partial differential equations. Classification. Laplace & Poisson equations. Wave equation. Heat flow / diffusion equation. Separation of variables. Fourier and Laplace transforms. Green s functions. Orthogonal polynomials. Special functions. Module 3: Complex analysis. Complex numbers and elementary functions. Analytic functions and series. Contour integrals, residues, and poles. Cauchy s theorem. Applications. Grading A linear scale between will be used. There will be no grading on the curve. HP: ; P: ; LP: ; F: Activity Percentage of grade Notes Homework sets 35% Complete 75% or more of each HW set to get a full credit for that set. Midterm 1 (September 28th) 20% Midterm 2 (November 16th) 20% Final (December 14th) 25% Prerequisites You will need to have Mathematica installed on your computer. UNC can provide you with a license.

3 Attendance policy Class attendance is mandatory for everyone. If you know in advance that you will miss a class and you have a legitimate reason, you need to contact me prior to the class hour by . If you miss because of an acute onset of illness, and you are still alive, you need to contact me as soon as possible. Lectures No cell phones, open laptop computers (except for classroom exercises) or other electronic devices are allowed during class. You are not allowed to make audio or video recordings in class without instructor permission. Homework Homework assignments will be posted online and the due date will be indicated at the top of the first page. Your solutions should be uploaded to Sakai. The homework will be graded and returned to you as fast as possible. I expect the solutions to be the results of your individual work. This means that copying someone else's solutions is a serious violation of the Honor Code. Course calendar and tentative schedule of lectures. Meeting # Date Lecture HW available / due Notes 1 Week 1: Aug 22 Course logistics. Review: series HW00: Review Due: Aug 31st Ch.1 & 3 2 Aug 24 Series of functions. Uniform convergence. Dirac delta. Ch.1 & 3 3 Week 2: Aug 27 Properties of Delta. Power series. Integration tricks. Ch.1 & 3 4 Aug 29 Linear algebra: Vector spaces. Linear independence. Subspaces. Bases. Dimension. Inner product. Hilbert spaces. 5 Aug 31 Hilbert spaces of functions. Orthogonal bases. Gram- Schmidt process. Operators. HW01: Linear algebra I Due: Sep 15th (shifted to Sep 19th due to Hurricane Florence) Ch.3 & 5 Ch.3 & 5 Week 3: Sep 3rd LABOR DAY. No class. 6 Sep 5 Operators. Matrices. Identity. Inverse. Adjoint. Hermitian, unitary, and orthogonal matrices/operators. Ch.3 & 5

4 7 Sep 7 Isomorphisms. Inverses. Determinants. Ch.2 & 6 8 Week 4: Sep 10 Determinants. Eigenvalue problems. Similarity transformations. Ch.2 & 6 9 Sep 12 Hermitian eigenvalue problems. Simultaneous diagonalization. Spectral decomposition. 10 Sep 14 Completeness relations. Expectation values. Normal matrices. Linear systems. LU decomposition. Class cancelled due to Hurricane Florence Class cancelled due to Hurricane Florence Ch.2 & 6 Week 5: Sep 17 Group theory. Definition & examples. Group representations. Class cancelled due to Hurricane Florence. HW02: Linear algebra II Due: Sep 28th 12 Sep 19 Group theory. Definition & examples. Discrete groups New Florenceinduced deadline for HW Sep 21 Group Representations. 14 Week 6: Sep 24 Group Rep. Theory 15 Sep 26 Group Rep. Theory Q&A on Tu & Thu before exam 16 Sep 28 MIDTERM EXAM 1 Everything up to and including Week 5 HW03: Group theory Due: Oct 12th HW02 is part of midterm 17 Week 7: Oct 1 Group Rep. theory: SO(n) and SU(n). 18 Oct 3 Complex analysis: Differentiation, analytic functions, contour integrals. 19 Oct 5 Complex analysis: Cauchy integral theorem and integral formula. Liouville s theorem. 20 Week 8: Oct 8 Complex analysis: Taylor and Laurent expansions. Examples. Singularity classification.

5 21 Oct 10 Complex analysis: Singularities. Residue theorem. Calculation of residues. Oct 12 UNIVERSITY DAY. No class. HW04: Complex analysis I Due: Oct 26th 22 Week 9: Oct 15 Complex analysis: Using residues to calculate integrals. 23 Oct 17 Complex analysis: Using residues to calculate integrals. Conformal mapping. 24 Week 10: Oct 22 FALL BREAK: Oct 5pm Oct 8am Complex analysis: Conformal mapping. 25 Oct 24 Complex analysis: Hilbert transform, Cauchy principal value. 26 Oct 26 Complex analysis: Dispersion relations, evaluating infinite sums. HW05: Complex analysis II Due: Nov 9th 27 Week : Oct 29 ODEs : First order, linear; second order. Example: Bessel equation. 28 Oct 31 ODEs : Second order. Series solutions. Bessel functions. 29 Nov 2 ODEs : Second order. Second solution. Fuchs theorem. 30 Week 12: Nov 5 ODEs : Sturm-Liouville theory. Self-adjoint differential operators. 31 Nov 7 PDEs : Examples. Separation of variables for the diffusion equation. Generalized Fourier series. 32 Nov 9 PDEs : Spherical coordinates and Legendre polynomials. Rodrigues formula. Other examples: Hermite and Laguerre polynomials. HW06: PDEs Due: Nov 26th 33 Week 13: Nov 12 PDEs

6 34 Nov 14 PDEs Q&A on Tu & Thu before exam 35 Nov 16 MIDTERM EXAM 2 Everything from Week 6, up to and including Week 12 HW06 is NOT part of midterm 36 Week 14: Nov 19 Exam review. THANKSGIVING BREAK: Nov 21 Nov 8am 37 Week 15: Nov 26 PDEs HW07: Due: Dec 5th 38 Nov 28 PDEs 39 Nov 30 PDEs 40 Week 16: Dec 3 PDEs 41 Dec 5 LDOC 42 Dec 14th 8am-am FINAL EXAM: Everything Disclaimer The instructor reserves the right to make changes to the syllabus, including due dates and test dates (excluding the officially scheduled final examination), when unforeseen circumstances occur. These changes will be announced as early as possible so that students can adjust their schedules.

San Jose State University Department of Mechanical and Aerospace Engineering ME 230, Advanced Mechanical Engineering Analysis, Fall 2015

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