PHYS 6110: Mathematical Methods of Theoretical Physics (Dr. Harald W. Griesshammer) in combination with

PHYS 6130: Computational Physics I, Math. Methd's-segment  (Dr. Harald W. Griesshammer)

Lectures Mathematical Methods: Tuesday, Thursday 14:00 to 15:40 in Corcoran 309. All lectures are 100 minutes, equivalent to 4 credit hours.

"Snow Days" (if we need to reschedule lectures, these are possible slots): Fri (preferred) and Mon (if unavoidable), each 08:20 to 10:00 in Cor 309.

Surgery Hours Mathematical Methods: Fridays at 08:20 in Cor 309, if "Snow Day" lecture scheduled, then after lecture, closing by 12:00. Lasts till all questions are answered.
Homework Due: Wednesdays at 16:00hrs. No exceptions. Zero points for assignments not turned in ontime.

Additional office hours by appointment after 3pm in my office. Email what and when to discuss.
email: hgrie <at> gwu.edu

Audience

First-year graduate students.

Goals

Introduction to the mathematical methods of Theoretical Physics with many examples and applications to Physics problems. Focus on skill-building. Not formal but ``heuristic'' proofs. The tools and tricks we discuss form the indispensable back-bone of the graduate curriculum. The lecture
does therefore not provide a comprehensive and systematic study of techniques – given the vastness of the field, that would be futile. Rather, we look for an “alternative narrative”: mathematical perspectives of Physics research.

An incomplete, over-achieving, informal list of Questions to Check Your Progress can be found here. Under no condition is this a survey of material for exams -- neither maximal nor minimal. It might not even be of any use at all.

Prerequisites

Advanced undergraduate Mathematical Methods, Mechanics, Electrodynamics and Quantum Mechanics.
Computing experience on undergraduate level: at least one programming language (Fortran, C).

Co-requisite

For Mathematical Methods: PHYS 6130: Computational Physics I; coordinated with PHYS 6120: Classical Mechanics (Haberzettl)

For Computational Physics: PHYS 6120: Classical Mechanics (Haberzettl)

Exams and Grading

The final grade is a sum of: 

• Exercises/Homework (20% of total): weekly;
• Mid-Term Exam (40% of total): Friday, 26 October time tba in Corcoran 309, 2 hours;
• Final Exam (40% of total): Thursday, 20 December 9:30 to 12:00 in Corcoran 309, 2.5 hours.

In order to pass, you need at least 60% of all points. You will also need at least 50% of the points available in each of the three components
separately. In particular, you need at least 50% of all points in all Problem sheets together (not per sheet!).  An excellent score usually starts at 80% of all points. Exams are closed-book. A sheet with some possibly relevant mathematical formulae will be provided by me in the days before each exam.

Exercises/Homework

Problem sheets are posted Wednesdays on this web-site (see below), due the following Wednesday at 16:00 (no exceptions).
Drop hardcopies in my pigeon-hole in the Physics office or fax to 994-3001, or mail to hgrie <at> gwu.edu. No grace period granted. 
Graded solutions are returned and discussed by the end of the next day, well in time before Surgery hour.
Handwritten solutions must be on 5x5 quadrille ruled paper; electronic solutions must be in .pdf format.  
Use of a "lab-book'' or "journal'' for homework is strongly encouraged.
You may use a symbolic programming language like Mathematica. Submit a paper printout or .pdf file of the code you used, with all results, and with all your documentation or comments. Below, you find my Mathematica code for part of the Stieltjes-Integral. It also shows what I consider to be the bare minimum of documentation. You can supplement your code by a write-up (.pdf or quadrille paper), and vice versa.

Contents (with links to manuscripts -- see Caveat/Warning/Disclaimer)

• Qualitative Methods (1.5 lectures)
• Science (and Not Art) of Approximations (3 lectures)
• Calculus of Variations (1.5 lectures)
• Vector Spaces and Scalar Products (2 lectures)
• Tensor Calculus and Tensor Analysis (3 lectures)
• Functional Analysis: Hilbert Spaces and Operators (3 lectures)
• Partial Differential Equations and Green's Functions (4.5 lectures)
• Complex Analysis (4.5 lectures)
• Groups and their Representations (5 lectures)
• Advanced Topics, including "How To Look Smart In Statistics and Data Analysis" (time permitting)

Syllabus: More Information/Bibliography/Units/Conventions

The only authoritative version of the syllabus contains much more information and is available as as .pdf-file: math-methods18.information.pdf

Bibliography

There is no required reading for this course. You will not be able to find all aspects of the lecture explained well in only one textbook. Moreover, it is an essential part of the learning process to view the same topic from different angles, i.e. using different textbooks. As [Nea p. vii] writes: "It is always useful to get a second viewpoint because it's commonly the second one that makes sense -- in whichever order you read them."

Here is a list of those which I found most useful. If you discover others, tell me.
The Class schedule lists for each lecture recommended readings. 

An asterisk * indicates titles we have on semi-permanent loan in the Graduate Office Corcoran 418, via Course Reserve at Gelman Library. Be social.

Books on which the course is (mostly) based


Further books useful for this course

More texts with many of the topics which are also prerequisites of or covered by this course

Classical texts which are also important as resources/reference


Reference Tables and Collections of Formulae for reference in the University library. You will not need this in the course, you might not even need it ever – but if you do, you will need it desperately.

Lecture Manuscript

A  scanned version of a chapter-by-chapter manuscript can be found by following the links of chapter headings in the Class Schedule and Contents Section. The files are in .djvu-format, which is at present the most condensed way of storing scanned images: 50 scanned pages translate into 1.2 Gbytes of bitmap, or 50 MBytes .pdf or 4.7 MBytes of .djvu. The freeware djvu reader "djvulibre" for all operating systems is available at http://djvu.sourceforge.net/, or as add-on to every decent Linux distribution.

Class Schedule (no exact match, but an outline how we hope to progress)

Date	Topics	Suggested Reading	Exercises
.	Revisit your undergraduate course notes!	Syllabus, Goals	Start-Quiz
28 Aug Tue lecture 1	Syllabus & Philosophy Qualitative Methods (1.5 lectures) Mental Math dimensional analysis, natural system of units	Motivational slides [manu-script Quali 1-12] Mental Math and Mental Physics Amazingly, this fundamental "technique" is not covered in many books. But see Excerpt on Mental Arithmetics from Kurrelmeier/Mais: Electricity and Magnetism and most recently Mahajan: Street-Fighting Mathematics or [Nea 1] plus article in Physics Today Sep 2011 plus Excerpt How To Draw Graphs from Nearing Newell on new SI system (Physics Today July 2014)
30 Aug, Thu lecture 2	scaling relations and mechanical similarity Mental Math: efficient Taylor expansion, estimating roots Illustration: Taylor series (mathematica.nb)  Science (Not Art) of Approximations (3 lectures) Approximations are necessary and good; Taylor expansion of integrands	[manu-script Quali 13-23] [Sni 2 and 12] [V.I. Arnold: Mathematical Methods of Classical Mechanics, Sect. 11] see above [manu-script Approx 1-2, 6-7] also have a look at the online lecture series [Ben]	Problem sheet 1 due 05 Sep film of the "Upshot-Knothole Grable" test at the Nevada Test Site, 25 May 1953
04 Sep, Tue lecture 3	Asymptotic series: saddle-point approximation/method of steepest descend and variable-phase method; Stirling-formula for n! Perturbation Theory: example solving polynomials	[manu-script Approx 7-16]  [AW 5.10] [RHB 25.8] notebook Stieltjes-series excerpt from Appel excerpt from Murdock
06 Sep, Thu lecture 4	Perturbation Theory (cont'd): formulation, application to nonlinear ordinary differential equation (pendulum)	[manu-script Approx 17-25] [Sni 23]	Problem sheet 2 due 12 Sep.
11 Sep, Tue lecture 5	Perturbation Theory (cont'd): application to eigenvalue problems (QM) Calculus of Variations (1.5 lectures) optimisation problems; functionals and their derivatives; linear and local functionals; Euler-Lagrange equation, fixed and variable endpoints	[manu-script Approx 26-29] notebook Pendulum [BF 4.11] [manu-script Var 1-9] full biblio for this section: nice intros: [Sni 25.1-5], [Nea16] functionals: [SG 1.1-2,1.4-5] [BF 2] advanced: [RHB 22.1-5,8], [AW 17.1-8]	Additional Voluntary Problems: Variations
13 Sep, Thu lecture 6	shortest distance between two points; cyclic variables and first integrals examples: brachistochrone, twin paradox; extensions, e.g. several dependent and independent variables	[manu-script Var 10-20] see above	Problem sheet 3 due 19 Sep
14 Sep, Fri 08:20 Special Date & Time	cover contents of lecture 7 below
18 Sep, Tue lecture 7 pre-poned to 14 Sep, Fri 08:20 (Griesshammer on assignment)	should have been: Vector Spaces and Scalar Products (2 lectures) Definitions: vector space, scalar product, dual space, orthogonality, examples; Basis and Vector Components: Einstein's Summation Convention, basis and dual basis, Gram-Schmidt Ortho-Normalisation procedure	[manu-script VSpace 1-11] intro: [Sni 22] review linear algebra: [SG App. A.1-3] [SG 2.1-2] [AW2.6-11] [Nea 6] [BF 1, 3,4.1-3, 5.1-2, 11] [RHB 8] Simmonds: A Brief on Tensor Analysis Suppl. Vector spaces Suppl. Compare Bra-Ket vs. Sum Convention
20 Sep, Thu lecture 8  post-poned to 21 Sep, Fri 08:20 (Griesshammer on assignment)	should have been: Complete Ortho-Normal System/Basis; Abstract operators: definitions, famous operators (e.g. Hermitean, self-adjoint); eigensystems of Hermitean operators, projectors; Rayleigh-Ritz variational principle	[manu-script VSpace 12-19] [SG 2.3, 4.1-3] [Nea 7][AW 17.8] [AW 10.3 and around example 3.1.2][BF 4.3]	Problem sheet 4 due 26 Sep
21 Sep, Fri 08:20 Special Date & Time	cover contents of lecture 8 above
25 Sep, Tue lecture 9	Groups and their Representations (5 lectures) *Script and Contents may be revised* Why groups? Axioms and definitions exemplified by SO(2): Abelian & non-Abelian, finite groups, abstract groups and their concrete realisations: Representations (linear, trivial, faithful, fundamental/defining representations) Nothing On Discrete/Finite Groups Lie (Continuous) Groups: example U(1), SO(2) and the unit circle; defs, Lie groups as manifolds; exp-mapping between tangent space and group	[manu-script Groups 1-11] Suppl. Groups nice: [AW 4.1-5] [SG, 14/15]. [RHB 28, 29.3-4] [BF 10.1-6,8] only treat discrete groups, but nice rep. theory. Useful also: Humphreys: Intro to Lie Algs and Rep Theory; Jones: Groups, Reps and Physics; Georgi: Lie Algebras and Particle Physics; see this link for a fun typo. Zwiebach: Summary on Lie Algebrae	Additional Voluntary Problems: Groups
27 Sep, Thu lecture 10	Lie groups (cont'd): example SO(3); Lie bracket as operation induced on L from G, measuring curvature, geometric and abstract definitions of Lie algebrae, generators, structure constants and complete ortho-normal basis SU(2) and SO(3): Pauli matrices and SU(2), epsilon-tensor, exp-map, rotations and the 3-dimensional sphere	[manu-script Groups 12-19] see above Suppl. Pauli matrices and epsilon-tensor	Problem sheet 5 due 03 Oct
28 Sep, Fri 08:20 Special Date & Time	cover contents of lecture 11 below
02 Oct, Tue lecture 11	should have been: Example SU(2) and SO(3): relation between SU(2) and SO(3) and the 3-dimensional sphere; [Goups 23-25 (Lorentz Group) will be skipped] Some generalisations on relations between algebra and group, classical Lie groups and their algebrae now covered on preceding Friday (see above) instead, cover contents of lecture 12 below	[manu-script Groups 19-22] see above [manu-script Groups 39-40] not relevant for MathMeth exam	.
04 Oct, Thu lecture 12 pre-poned to 28 Sep, Fri 08:20 (Griesshammer on assignment)	should have been: The "Classical Groups": U(n), SU(n), O(n), SO(n) Dash into Representation Theory: How Direct Sums can obscure representations; fundamental and irreducible representations as building elements; Schur's Lemmata; representations in QM	[manu-script Groups 41-45] [manu-script Groups 26-29] neither relevant for MathMeth exam (but for QM!) see above	Problem sheet 6 due 10 Oct
09 Oct, Tue	no lecture (GW Fall Break)
11 Oct, Thu lecture 13  post-poned to 15 Oct, Mon 08:20 (Griesshammer on assignment)	should have been: Dash into Representation Theory (cont'd): looking ahead to QM-I (angular-momentum algebra in Quantum Mechanics); angular-momentum couling by example: isospin invariance of the Strong Nuclear force and the pion-nucleon system (preview of Clebsch-Gordan coefficients, Wigner-Eckart theorem) mathematica notebook here	[manu-script Groups 30-38] not relevant for MathMeth exam (but for QM!) see above [Groups 23-45] not relevant for MathMeth exam	Problem sheet 7 due 17 Oct (last relevant for Midterm) .
15 Oct, Mon 08:20 Special Date & Time	cover contents of lecture 13
16 Oct, Tue lecture 14	Tensor Calculus and Tensor Analysis (3 lectures) metric tensor, orthogonal curvilinear coordinates and the  n-bein, coordinate singularities, scaling factors, line, surface and volume elements; fields and the gradient operator	[manu-script Tensor 1-9] intro: [Sni 5-10 22] [AW 1.6-16,2.6-11] [BF 1, 3,4.1-3] [RHB 8, 10-11] [SG 8.11, 11.2] [Nea 7-8, 12-13] Simmonds: A Brief on Tensor Analysis	Additional Voluntary Problems: Tensor Analysis
18 Oct, Thu lecture 15	transformation of metric and co- and contravariant components under basis changes; tensors and pseudo-tensors, irreducible tensors	[manu-script Tensor 10-19] see above	Problem sheet 8 due 24 Oct
22 Oct, Mon 09:15, Cor 309	Lecturer's Question Time Please email midterm-exam-relevant questions by Sun
23 Oct, Tue lecture 16	differential/nabla operator; Gauss' and Stokes' integration theorems with definition, form of div, rot and Laplace in orthogonal coordinate systems, first definition of Dirac's delta-distribution, Helmholtz's Fundamental Theorem of Vector Calculus	[manu-script Tensor 20-30] see above
25 Oct, Thu lecture 17	Functional Analysis: Hilbert Spaces and Operators (3 lectures) Spaces of Functions and their Bases, Fourier-series, Lesbegue-integral, Hilbert spaces; distributions, extended bases	[manu-script FuAn 1-9] review linear algebra: see [manu-script VSpace] [SG 2.1-2] [BF 5.1-2, 11]	Problem sheet 9 due 31 Oct Additional Voluntary Problems: Functional Analysis.
26 Oct, Fri 09:15, Cor 309	Mid-Term Exam 2:00 hours, closed-book, sheet with mathematical formulae provided.
30 Oct, Tue lecture 18	Dirac's delta-distribution: properties and physical interpretation; Fourier Integral Transform: properties, normalisation, faltung/convolution, transformation of derivatives	[manu-script FuAn 10-21] Suppl. delta-Distibution [Sni 14] [Nea 17.4/5] [SG 2.3] [AW 1.15] [BF 5.3] Suppl. Fourier-Transform [Sni 15] [Nea 15] [SG App. B] [BF 5.6-7] [AW 15.2-7]
01 Nov, Thu lecture 19	conjugate variables and uncertainty relation; Pitfalls of  Operator theory: spectrum/eigensystem of Hermitean operator, Hermitean vs. self-adjoint, spectral decomposition	[manu-script FuAn 22-31] review [manu-script VSpace] CAUTION: many books neglect subtleties! Gieres: [quant-ph/9907069] [SG 2.3, 4.1-3] Porter: Summary on Operator Theory	Problem sheet 10 due 07 Nov
06 Nov, Tue lecture 20	Partial Differential Equations and Green's Functions (4.5 lectures) Linear Partial Differential Equations (PDEs) in Physics: (Poisson, Yukawa, Helmholtz, wave, Schroedinger, heat equations), elementary solutions by Green's functions, a collection of Green's functions, girl on a swing; Formal solution: Green's operator as inverse, Reciprocity Relation	[manu-script Green 1-10] [Sni 18-19] [BF 7.1-5] [SG 5.2,4] [Nea 4.2]	Additional Voluntary Problems: Green's Functions and PDEs
08 Nov, Thu lecture 21	Uniqueness and Boundary Conditions (Potential Theory): Dirichlet and von Neumann boundary condition Collection of Methods to Construct Green's Functions (includes method of image charges) Multipole decomposition of boundary value problems in spherical coordinates: separation of variables,  Illustration: apple and bowling pin multipoles (mathematica.nb, Legendre polynomials) Illustration: Spherical Harmonics as Colours on Surface on Sphere (mathematica.nb) Illustration: Spherical multipole decomposition of  a bowling pin Illustration: Spherical harmonics (mathematica.nb)	[manu-script Green 11-20] [Sni 20] [SG 5.4, 8.2] [BF 5.8] [AW 12] [RHB 18.1-3, 21.1-3,5] [Nea 4.3/7/11/12, 10, 17.6] Suppl. Spherical Harmonics	Problem sheet 11 due 14 Nov.
13 Nov, Tue lecture 22	Legendre's DEq: solution using Frobenius' power-law method, associated Legendre polynomials (=Legendre function of the first kind), spherical harmonics as example to expand in orthogonal functions, elementary and general solution of the spherical Laplace equation	[manu-script Green 21-29] see above
15 Nov, Thu lecture 23	Example: spherical multipole moments of an electrostatic potential in an external field: charge multipoles, generating function and recursion relations for Legendre polynomials, advantages of (spherical) multipoles Review CONS: more complete systems of orthonormal functions, irregular solutions of Sturm-Liouville systems	[manu-script Green 30-38] see above [SG 8.3] [AW 12.10]	Problem sheet 12 due 28 Nov
20 Nov, Tue lecture 24	Orthogonal functions cooking recipe, example wedge and Bessel functions Complex Analysis (4.5 lectures) Complex and analytic functions: Cauchy-Riemann condition	[manu-script Green 39-43] see above [manu-script Complex 1-4] [Sni 16] [Nea 14] [BF 6.1-3] [SG 17.1]	Additional Voluntary Problems: Complex Analysis
22 Nov, Thu	no lecture (Thanksgiving Break)
27 Nov, Tue lecture 25	complex integration, Cauchy-Goursat and Morera Integration Theorems, simply connected domains Calculus of Residues: Laurent series, residues and residue theorem, Cauchy's integration formula, Liouville's theorem on analyticity of bounded functions	[manu-script Complex 6-14] [Sni 17] [BF 6.4/8] [SG 171.2-4, 18.1]	.
29 Nov, Thu lecture 26	Calculus of Residues (cont'd): applications (including Fourier integrals and Heaviside's step function), causality and the retarded Green's function	[manu-script Complex 15-23] see above [BF 6.5-6] [Nea 15.5]	Problem sheet 13 due 05 Dec
04 Dec, Tue lecture 27	Cauchy's principal value prescription, dispersion relations (Kramers-Kronig) Branches of multi-valued functions: logarithm, Riemann sheets and branch cuts Mathematica illustrations: Complex functions and cuts, surface with branch cuts, transition between Riemann surfaces of z^(1/2) and z^(1/3)	[manu-script Complex 24-32] [BF 6.2] [SG 17.6.2, 18.1.2]
06 Dec, Thu lecture 28	Branches (cont'd): combining poles and cuts Analytic continuation: convergence of complex series, examples of analytic continuation, proving Rodrigues' formula Wrap-Up: The End	[manu-script Complex 33-41] [BF 6.7/9] Here a video demonstrating that the sum of positive integers is -1/12 -- since that is usually the point where I loose all credibility in your eyes...	Problem sheet 14 special due 10 Dec 08:00
17 Dec, Mon 15:00 Cor 309 (to be confirmed)	Surgery hour for HW 14
18 Dec, Tue 10:20 Cor 309 (to be confirmed)	Special Surgery Hour and Lecturer's Question Time
20 Dec, Thu 9:30 sharp - 12:00 Cor 309 (to be confirmed)	Final Exam 2:30 hours, closed-book, sheet with mathematical formulae provided.

Computational Physics I Class Schedule (no exact match, but an outline how we hope to progress)

Date	Topics	Suggested Reading	Exercises
31 Aug, Fri starting 15:00 (extra long) Cor 309	Session 1: Introduction derivative algorithms, overflow, underflow, machine precision, crash course on mathematica	Syllabus distributed by email	Problem Sheet 1 due 10 Sep, Mon 10:00 distributed by email
10 Sep, Mon special date starting 08:20 Cor 309	Session 2: Introduction to Fortran (?)	distributed by email	Problem Sheet 2 due 13 Sep, Thu 12:00???? distributed by email
14 Sep, Fri starting 15:00 Cor 309	Session 3: (More) Introduction to Fortran	distributed by email	Problem Sheet 3 due t.b.d. distributed by email
21 Sep, Fri starting 15:00 time t.b.d. Cor 309	Session 4: tba	distributed by email	Problem Sheet 4 due t.b.d. distributed by email
28 Sep, Fri starting 15:00 Cor 309	Session 5: tba	distributed by email	Problem Sheet 5 due t.b.d. distributed by email
5 Oct, Fri starting 15:00 Cor 309	Session 6: Wrap-Up	distributed by email	Problem Sheet 6 due t.b.d. distributed by email
	Project 1	distributed by email	distributed by email
	Project 2	distributed by email	distributed by email