Course Webpage for PHYS-500A (Fall 2025) at SIU--Carbondale

Course Webpage for
PHYS 500A: Mathematical Methods (Fall 2025)
Department of Physics, Southern Illinois University--Carbondale
Last updated: Dec 12, 2025. (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: MWF 11 AM-12:30 PM; MW 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: 2025F [2023F, 2021F, 2020F]
  - Informal Notes: This is not a substitute for a textbook.
- 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: [2024F, 2023F, 2022F, 2021F, 2020F, 2019F, 2018F]
- Miscellaneous
  - Grades will be posted on SIU MyCourses (password protected).

COURSE CONTENT:

Vector calculus:
1. Vector algebra
  - Index notation,
    Scalar product, Kronecker \(\delta\)-symbol,
    Vector product, Levi-Civita symbol,
  - Vector identities involving non-commuting (quantum) operators
    Vector identities involving gradient operator \({\bf\nabla}\)
2. Vector field
  - Gradient operator, divergence, and curl
    Divergence theorem, Curl theorem
  - Dyadic notation
    Basis independent methods for gradient operations
    Identities: \( {\bf\nabla} \, {\bf r} ={\bf 1} \) and \( {\bf\nabla} \, r =\hat{\bf r} \)
3. \(\delta\)-function
  - Gauss's law, Ampere's law
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
2. Transformation matrix
3. 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
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
Keywords: complex number, analytic function

Examples:
- Roots of a cubic equation, Cardano formula
- Analytic continuation of Beta function and Zeta function
Midterm Exam 01 (September 26, Friday)
Preview of Midterm Exam 01
Quantum mechanics
1. Probabilities for measurements and selections in sequential acts
2. Dynamics in terms of commutation relations
Resources:
Homework No. 06
Keywords: commutation relations

Examples:
- Stern Gerlach experiment
- Quantum Zeno effect
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.
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. 08
Keywords: Forced harmonic oscillator

Examples:
- Method of images in electrostatics
- Simple, damped, forced, harmonic oscillator
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. 09
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
Midterm Exam 02 and Solutions (October 31, Friday)
Preview of Midterm Exam 02
Spherical harmonics
1. Generating function for spherical harmonics
2. Differential equation for spherical harmonics
3. Orthogonality relation for spherical harmonics
4. Completeness relation for spherical harmonics
5. Addition theorem for spherical harmonics
Resources:
Homework No. 10
Keywords: spherical harmonics, multipole expansion
Cylinder harmonics
1. Generating function for Bessel functions
2. Differential equation for Bessel function
3. Orthogonality relation for Bessel functions
4. Completeness relation for Bessel functions
5. Addition theorem for Bessel functions
6. Modified Bessel functions
7. Free Green function
Resources:
Homework No. 11
Keywords: Bessel functions, Modified Bessel functions
Final Exam (December 12, Friday, 8-10 AM)

More information:

Evaluation:

Homework 30%

Midterm Exams (2x20%) 40%

Classroom Quizzes 10%
Final Exam 20%

Exam schedule:

Midterm Exam 01 --- September 26 (Friday)

Midterm Exam 02 --- October 31 (Friday)

Final Exam --- December 12 (Friday, 8:00-10:00 AM)

Last updated on Dec 12, 2025 (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 26 (Friday)
Midterm Exam 02	---	October 31 (Friday)
Final Exam	---	December 12 (Friday, 8:00-10:00 AM)