Quantum Mechanics

(in Polish) kierunkowe
(in Polish) specjalnościowe
obligatory courses

Algebra 0900-FG1-1AL
Analysis I 0900-FS1-1AM1
Analysis II 0900-FS1-1AM2
Elements of Classical Electrodynamics 0900-FS1-2EEK
Elements of Quantum Mechanics 0900-FS1-3EMK
Elements of Theoretical Mechanics 0900-FS1-2EMT

Algebra 0900-FG1-1AL
Analysis I 0900-FS1-1AM1
Analysis II 0900-FS1-1AM2
Elements of Classical Electrodynamics 0900-FS1-2EEK
Elements of Quantum Mechanics 0900-FS1-3EMK
Elements of Theoretical Mechanics 0900-FS1-2EMT

The main goal of the subject is to apply quantum mechanics to explain those physical phenomena, which otherwise remain outside the scope of classical physics. Specially one needs to learn and use Dirac's notation for describing expressions, which are very difficult to handle in their full explicit mathematical form. The relevance of various symmetries of the analyzed systems are discussed. The perturbation methods show the reality of practical quantum mechanical computations for physical systems.

(in Polish) w sali

Quantum Mechanics is the one semester course of the subject. It includes 30 hours of the lecture and 30 hours of the discussion session.

The content is following:

1. Dirac's formalism and mathematical notation.

2. Time independent perturbation methods for non-degenerate spectrum of unperturbed Hamiltonian operator.

3. Time independent perturbation methods for degenerate spectrum of unperturbed Hamiltonian operator.

4. Fine structure for the hydrogen atom spectrum.

5. Variational method for estimating the ground state energy.

6. Binding energy for the hydrogen ion molecule. ground energy for Helium atom.

7. Time independent perturbation method for two level systems.

8. Transitions between two levels of energy of electron in the hydrogen atom.

9. Relativistic equations in quantum mechanics: Klein-Gordon, Weyl, Dirac.

10. Pauli's equation for electron in external electromagnetic field.

Quantum Mechanics is the one semestral course of the subject. It includes 30 hours of the lecture and 30 hours of the discussion session (2 hours of the lecture and 2 hours of the discussion session per a week).

Educational profile: general academic.

Type of the studies: full-time.

Block (unit): theoretical physics, mandatory subject.

Field of knowledge and discipline of science: physical science, quantum mechanics.

Year of the studies, semester: 1st year, 1st semester, graduate studies.

Introductory conditions: course of analysis, course of algebra, course of classical mechanics, elements of classical electrodynamics, elements of

quantum mechnics

Didactic methods: lecture, solving the problems, homework, discussions, consultations, unassisted studying.

ECTS points: 9.

Balance sheet of the student's work: lecture (45 hours), discussion session (45 hours), homework (90 hours), discussions (5 hours), consultations (15 hours), unassisted studying (90 hours).

Quantitative indicators: lecture (2 ECTS points), discussion session (2 ECTS points), homework (2 ECTS points), discussions (0,5 ECTS points), consultations (0,5 ECTS points), unassisted studying (2 ECTS points).

The content is following:

1. Dirac's formalism with bra and ket states, brackets for the expectation values for linear operators. It is compared with the mathematical notation

for the functional vector spaces.

2. The time independent perturbation method, the energy corrections in the first and second orders of perturbation, the wave function correction in the

order of perturbation is valid only if the unperturbed Hamiltonian operator has non-degenerate spectrum.

3. The time independent perturbation method for degenerated spectrum of unperturbed Hamiltonian, breaking of degeneracy by the perturbation Hamiltonian.

Criteria of choosing "good combinations" of wave functions with equal unperturbed energies.

4. The fine structure of the hydrogen atom spectrum as the 1-st order correction. The perturbation Hamiltonians: relativistic corrections

to the kinetic energy, spin-orbit interaction for electron in the Coulomb spherical potential.

5. Theorem for ground-state energy. Theorem for 1-st excited state energy. Choice of trial wave function.

Variational parameters. Minimization of the expectation values for the Hamiltonian operators.

6. Binding energy for the hydrogen ion molecule. The symmetric and anti-symmetric trial wave function from the hydrogen atomic wave functions.

Ground state energy for helium atom with two electrons. Effective electric charge of helium atom nucleus.

7. The time dependent perturbation calculations for the two level system. Time dependence for stationary wave function with many

modes with different energies, the time independent probability density. The time dependent perturbation Hamiltonians.

Wave function as the modified stationary wave function - constant coefficients transformed into functions of time. Ordinary differential equations

derived from the time dependent Schroedinger equation. Solution of equations in 0-th, 1-st and 2-nd order of perturbation.

8. Transitions between states with different energies in the hydrogen atom. The matrix elements for the electric dipole moment.

The sinusoidal perturbation. The principle behind the laser. Non coherent perturbation, averaging over polarizations and propagation direction

for electromagnetic wave. Selection rules for matrix elements, allowed and non-allowed transitions, meta-stable excited states for the hydrogen

atom.

9. The dispersion relation in the special relativity. Relativistic equation in quantum mechanics for scalar particle - Klein-Gordon equation.

states with positive and negative energies, troubles with probability density for negative energy. Factorisation of dispersion relation

for massless particles, the Weyl equation, the Weyl spinors with left and right helicities. The mass term coupling of the Weyl spinors,

Dirac's bispinors. Chiral (Weyl) representation, standard (Dirac) representation.

10. The minimal coupling principle for the Dirac equation. Non-relativistic limit for the positive energy wave function, the Pauli equation

for electron in external electromagnetic field.

1) L. I. Schiff: "Quantum mechanics"

2) J.J. Sakurai, J.J.Napolitano: "Modern quantum mechanics".

3) D.J. Griffiths; "Introduction to quantum mechanics.

4) S. Weinberg, "Lectures on quantum mechanics".

5) I. Białynicki-Birula, M. Cieplak, J. Kamiński: "Theory of Quanta"

A student:

1. Knows role of quantitative models and abstract descriptions of physical object and physical phenomena in the area of fundamental parts of physics.

2. Knows restrictions of applicability for chosen physical theories, models of objects and description of physical phenomena.

3. Understands formal structure of basic physical theories, is able to apply appropriate mathematical tools for quantitative description of physical phenomena from chosen parts of physics.

4. Has knowledge of quantum mechanics foundations, of formalism and probabilistic interpretation of this theory, knows theoretical description and mathematical tools for analysis of chosen quantum systems.

5. Can understandingly and judgmentally examine professional literature and Internet sources, with regard to studied problems of quantum mechanics.

6. Understands structure of physics, treated as a branch of science, acquires cognisance of connections between its domains and theories, knows examples of false physical hypothesis and false physical theories.

7. Is capable to use known mathematical tools for defining and solving chosen problems of theoretical and experimental physics.

8. Can present theoretical formulation of quantum mechanics and is able to perform theoretical analysis of chosen quantum systems, using relevant mathematical tools.

9. Knows limitations of his knowledge and understands necessity of further education, of upgrading personal, professional and social competencies.

10. Is able to search individually information in literature and Internet sources,

Labels:

K_W22, K_U20.

Students take part in lectures broaden of computer simulations, illustrating transmitted contents. They are stimulated for asking the questions and for discussion.

Written and oral examinations undergo after the end of the course of Quantum Mechanics. They verify acquirement of knowledge.

Students get the series of questions, exercises and problems for individual and unassisted solving. Content of the series of questions is correlated with the lecture. During the course, students present solutions of given problems. Lecturer is advised to pay close attention to understanding used concepts and clarity of presentations. He stimulates students group for asking the questions and discussions. Lecturer tries to create sense of responsibility for team inside the students group and he encourages the group to join work.

Assessment of student learning is based on the grade, which includes:

1. Ability to solve the problems from define parts of quantum mechanics.

2. Ability to present the solutions.

3. Ability to discuss subjects and problems of the course.

4. Ability to use the literature and Internet sources.

5. Ability to collaborate inside the team.

6. Creative approach to solved problems.

Permanent grading by lecturer.

Final grade is expressed by the number established in the study regulation, which includes evaluation of the knowledge, abilities and competencies of the student.