COURSE GOALS: Course goals are to acquire theoretical and practical knowledge in the theory of ordinary and partial differential equations.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
1. KNOWLEDGE AND UNDERSTANDING
1.1. demonstrate a thorough knowledge and understanding of the fundamental laws of classical and modern physics;
1.2. demonstrate a thorough knowledge and understanding of the most important physics theories (logical and mathematical structure, experimental support, described physical phenomena);
1.3. demonstrate knowledge and understanding of basic experimental methods, instruments and methods of experimental data processing in physics;
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1. identify and describe important aspects of a particular physical phenomenon or problem;
2.2. recognize and follow the logic of arguments, evaluate the adequacy of arguments and construct well supported arguments;
2.3. use mathematical methods to solve standard physics problems;
3. MAKING JUDGMENTS
3.1. develop a critical scientific attitude towards research in general, and in particular by learning to critically evaluate arguments, assumptions, abstract concepts and data;
4. COMMUNICATION SKILLS
4.1. communicate effectively with pupils and colleagues;
4.2. present complex ideas clearly and concisely;
4.4. use the written and oral English language communication skills that are essential for pursuing a career in physics and education;
5. LEARNING SKILLS
5.1. search for and use professional literature as well as any other sources of relevant information;
5.2. remain informed of new developments and methods in physics and education;
5.3. develop a personal sense of responsibility for their professional advancement and development;
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon passing the course on Mathematical Methods of Physics II, the student will be able to:
* define and analyse basic notions in the theory of ordinary and partial differential equations
* solve homogeneous linear differential equations with constant coefficients and construct particular solutions using the method of variation of parameters
* formulate the Legendre differential equations, derive Legendre polynomials as solution of Legendre differential equation and understand basic properties of Legendre polynomials
* derive formulas for the associated Legendre differential equations and for spherical harmonics
* formulate the Laplace equation in Euclidian, spherical and cylindric coordinate system
* formulate the Wave equation in Euclidian, spherical and cylindric coordinate system
* formulate the Bessel differential equation, derive formulas for the Bessel functions and understand basic properties of the Bessel functions
* formulate the Schrodinger equation for the hydrogen atom and derive the formulas for the Laguerre polynomials as the radial part of the solution of the Schrödinger equation.
1. Ordinary differential equations
2. Linear differential equations. Linear differential equations of the first order
3. Existence and uniqueness theorems for the Cauchy problem for the homogeneous linear equation of n-th order
4. Linear independence of functions and the Wronskian
5. Linear differential equation with constant coefficients
6. Nonhomogeneous differential equations. The method of undetermined coefficients. The method of Variation of Parameters.
7. Solving differential equations by power series.
8. Second order linear differential equation with regular singularities
9. Legendre polynomials and Legendre differential equation. A Generating Function for Legendre Polynomials.
10. The associated Legendre functions. Spherical harmonics
11. Laplace's equation. The method of separation of variables
12. Wave equation
13. Bessel functions and Bessel differential equation
14. Schrodinger equation. Laguerre polynomials
REQUIREMENTS FOR STUDENTS:
Students are expected to regularly attend lectures, exercises and do homework. Furthermore, students are required to pass two colloquiums during the semester, and to achieve at least 40% of the total number of points on them.
GRADING AND ASSESSING THE WORK OF STUDENTS:
Grading and assessing the work of students during the semesters:
* Two written exams
* Home works
Grading at the end of semester:
* final oral exam
Contributions to the final grade:
* 10% of the grade is carried by the results of the home works and on presence
* 60% of the grade is carried by the results of the two written exams
* the oral exam carries 30% of the grade.