Course Webpage for PHYS-500A (Fall 2024) at SIU--Carbondale

Course Webpage for
PHYS 500A: Mathematical Methods (Fall 2024)
Department of Physics, Southern Illinois University--Carbondale
Last updated: Dec 13, 2024. (This page is no more maintained.)

General information:
- Instructor: K. V. Shajesh (Office: Neckers 491, Tel.: +1(618)453-5212)
- Class meetings: MF 9:35-10:50 AM, in Neckers 404.
- Office hours for instructor: MTRF 11 AM-12 PM; MF 3-4 PM.
- Syllabus for the course. The Provost Office has put together a Syllabus Attachment to assist the student. It lists and provides web links for many elements that are important for a student.
Resources:
- References:
  - Methods of Theoretical Physics, Part 1 and 2, by Philip M. Morse and Herman Feshbach.
  - Mathematical Methods for Physicists, 6th Edition, by George B. Arfken and Hans J. Weber.
- Lectures
  - Lecture recordings: [2023F, 2021F, 2020F]
- Quizzes and Homework
  - Quizzes: Due in class in the beginning of the next lecture.
  - Homework: Due in Mail Box before 4:30 PM on due date.
- Exams
  - Old Exams are available in Past course webpages: [2023F, 2022F, 2021F, 2020F, 2019F, 2018F]
- Miscellaneous
  - Grades will be posted on SIU MyCourses (password protected).

COURSE CONTENT:

Vector calculus:
1. Vector algebra, dot product, cross product
  - Index notation
  - Dyadic notation
  - Vector algebra for non-commuting (quantum) operators
2. Vector field
  - Gradient operator
  - Divergence theorem
  - Curl theorem
  - Basis independent methods for evaluating gradient operations
3. Delta function
Resources:
Homework No. 01
Homework No. 02
Keywords: Vector calculus, vector field, delta function

Examples:
- Gauss' law in electrostatics
Curvilinear coordinates:
1. Basis vectors, reciprocal basis vectors
  - Orthogonality, metric
  - Completeness relation
  - Transformation matrix
2. Connection, curvature
Resources:
Homework No. 03
Keywords: curvilinear coordinates, connection, curvature

Examples:
- Reciprocal basis in crystallography
Complex analysis:
1. Addition, subtraction, multiplication, subtraction, exponentiation
  - Polar representation (Argand diagram)
  - Roots of unity
  - Roots of a cubic equation, Cardano formula
2. Analytic function
  - Cauchy-Riemann conditions
  - Conjugate functions
3. Cauchy's theorem
  - Contour integrals with poles
  - Contour integrals with branch points
  - Analytic continuation of Gamma function
Resources:
Homework No. 04
Homework No. 05
Homework No. 06
Keywords: complex number, analytic function

Examples:
- Explore Cardano formula
- Analytic continuation of Beta function and Zeta function
Midterm Exam No. 01 (September 27, Friday)
Fourier eigenfunctions:
1. Discrete Fourier series
  - Completeness and orthogonality
  - Differential equation
  - Green's function
2. Continuous Fourier integals
  - Completeness and orthogonality
  - Differential equation
  - Green's function
3. Half-range Fourier series
  - Completeness and orthogonality
  - Differential equation
  - Green's function
Resources:
Homework No. 07
Keywords: Fourier series, Fourier transformation

Examples:
- Fourier transform of Heaviside step function.
Harmonic oscillator
1. Simple harmonic oscillator
2. Damped harmonic oscillator
3. Forced harmonic oscillator
Resources:
Homework No. 08
Keywords: Forced harmonic oscillator

Examples:
- Introduction to Green's function.
Midterm Exam No. 02 (November 1, Friday)
Green's function
1. Inhomogeneous differential equation, particular solution
2. Wronskian
3. A particular solution in terms of Green's function
4. Reciprocity
5. Continuity conditions for Green's function
6. Retarded, advanced, Feynman, and Wheeler, Green's function
Resources:
Homework No. 09
Keywords: Forced harmonic oscillator

Examples:
- Method of images in electrostatics
Legendre polynomials
1. Generating function for Legendre polynomials
2. Recurrence relation for Legendre polynomials
3. Differential equation for Legendre polynomials
4. Rodrigues formula for Legendre polynomials
5. Orthogonality relation for Legendre polynomials
6. Completeness relation for Legendre polynomials
Resources:
Homework No. 10
Keywords: Legendre polynomials, multipole expansion

Examples:
- Electric potential of 2l-pole
- Electric potential due to charged sphere
- Electric potential due to charged disc
- Electric potential due to charged ring
Spherical harmonics
1. Inversion symmetry of Laplacian
2. Solid harmonics versus surface (or spherical) harmonics
3. Generating function for spherical harmonics: Null vectors
4. Differential equation for spherical harmonics
5. Orthogonality conditions for spherical harmonics
Resources:
Homework No. 11
Homework No. 12
Keywords: Spherical harmonics

Examples:
- Addition theorem
Final Exam (December 13, Friday, 8:00-10:00 AM)

More information:

Evaluation:

Homework 30%

Midterm Exams (2x20%) 40%

Classroom Quizzes 10%
Final Exam 20%

Exam schedule:

Midterm Exam 01 --- September 27 (Friday, in class)

Midterm Exam 02 --- November 1 (Friday, in class)

Final Exam --- December 13 (Friday, 8:00-10:00 AM)

Last updated on Dec 13, 2024 (and is no more maintained) by K. V. Shajesh (kvshajesh@gmail.com).

Homework	30%
Midterm Exams (2x20%)	40%
Classroom Quizzes	10%
Final Exam	20%

Midterm Exam 01	---	September 27 (Friday, in class)
Midterm Exam 02	---	November 1 (Friday, in class)
Final Exam	---	December 13 (Friday, 8:00-10:00 AM)