Elements of Quantum Mechanics

(in Polish) ogólne kierunkowe
(in Polish) kształcenia ogólnego
(in Polish) kierunkowe
obligatory courses

Algebra 0900-FG1-1AL
Analysis I 0900-FS1-1AM1
Analysis II 0900-FS1-1AM2
Elements of Theoretical Mechanics 0900-FS1-2EMT

Algebra 0900-FG1-1AL
Analysis I 0900-FS1-1AM1
Analysis II 0900-FS1-1AM2
Elements of Theoretical Mechanics 0900-FS1-2EMT

The main goal of the subject is to acquaint the students with theoretical foundations of quantum mechanics. The differences between classical and quantum approach are especially emphasized.

Due to well-known problems in quantum mechanics assimilation, the phenomena leading to introduction of quantum mechanics for description the micro scale physical world, are widely discussed.

(in Polish) w sali

Elements of Quantum Mechanics are the one semestral course of the subject. It includes 45 hours of the lecture and 45 hours of the discussion session.

The content is following:

1. Experimental foundations of quantum mechanics.

2 The measurement in quantum mechanics.

3. Operators in quantum mechanics. Schrodinger equation.

4. The wave-function.

5. Eigen-functions and eigenvalues in quantum mechanics.

6. Principle of the uncertainty.

7. Bounded states in quantum mechanics.

8. Scattering problems in quantum mechanics.

9. Harmonic oscillator.

10. Hydrogen atom.

11. Orbital momentum operators.

12. Spin 1/2.

13. Statistics for identiocal particles.

14. Pauli's exclusion principle and the periodic table of elements.

15. Chemical bond in the hydrogen molecule.

Elements of Quantum Mechanics are the one semestral course of the subject. It includes 45 hours of the lecture and 45 hours of the discussion session (3 hours of the lecture and 3 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: 3rd year, 5th semester, undergraduate studies.

Introductory conditions: course of analysis, course of algebra, course of classical mechanics.

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) Physical foundations of quantum mechanics – discovery of new physical phenomena, unexplained by classical theories: radioactivity, Roentgen's rays, ideal black-body radiation, atomic spectra. Planck's hypothesis. De Broglie's concept of waves of matter. Bohr – Sommerfeld quantization rules. Bohr's correspondence principle. Experiments showing necessity of inventing new theory: experiments with diffraction, Stern - Gerlach experiment, Davisson – Germer experiment. Wave-particle duality.

2) Law of radioactive decay, decay constants. Decription of ideal black-body, Wien's law, Stefan – Boltzmann law. Bohr - Sommerfeld quantization rules for some physical systems: atom of hydrogen, particle in uniform gravitational field.

3) Role of measurement in quantum mechanics. Measurement of position and momentum, measurement of lifetime and energy. Heinsenberg's uncertainty principle, complementarity principle. Space and time evolution of wave packet. Gaussian wave packet. Wavefunction. Heuristic introduction of Schrödinger's equation.

4) Probability of finding classical particle in definite point of space. Heisenberg's microscope. Examples of wavefunctions.

5) Schrödinger's equation with potentials. Statistical interpretation of wavefunction. Normalization of wavefunction. Probability density. Continuity equation, closing probability current and probability density in quantum mechanics. Operators in quantum mechanics: position operator, momentum operator. Eigenvalues of operators, eigenfunctions of operators (in general).

6) Examples of normalizing wavefunctions. Calculations of probability density. Calculations of probability current. Determining mean values and eigenvalues for differential and matrix operators. Momentum operator, its eigenvalues and eigenfunctions.

7) Hamiltonian and total energy operator in time-dependent Schrödinger's equation. Separation of time in Schrödinger's equation, stationary state Schrödinger's equation, kinetic energy operator. Boundary conditions for Schrödinger's equation. Continuity of wavefunction and its derivative in case of Schrödinger's equation. Eigenvalues and eigenfunctions of operators in quantum mechanics. Hermitian operators. Ehrenfest's theorem.

8) Calculations of mean values of total and kinetic energy operators. Calculations of mean values in quantum mechanics with help of Ehrenfest's theorem. Examples of hermitian operators and their properties.

9) Linearity of Schrödinger's equation. Orthogonality of eigenfunctions, orthonormal basis of eigenfunctions. Wavefunction expansion into eigennfunctions. Quantum reduction (collapse) of wavefunction. Eigenfunctions of total energy operator and momentum operator.

10) Wavefunction expansion into eigenfunctions and making superpositions of wavefunctions – some examples. Probability of finding one quantum state into another.

11) General formulations of Heisenberg's uncertainty principle: uncertainty principle for canonically conjugate variables, operator formulation of uncertainty principle. Solution of Schrödinger's equation for free particle.

12) Variance and standard deviation in quantum mechanics. Heisenberg's uncertainty principle applied to some quantum system – examples. Function minimalizing uncertainty principle - Gaussian wavefuncktion (wawe packet). Analysis of free particle wavefunction.

13) Bound states in quantum mechanics. Solution of Schrödinger's equation for finite and infinite potential well; wavefunctions, discerte levels of energy. One-, two- and three-dimensional cases. Scattering states in quantum mechanics. Scattering in one dimension: transmission and reflection coefficients. Tunnel effect.

14) One-dimensional scattering of quantum particle on finite potential step. One-dimensional scattering of quantum particle on finite and infinite potential well and finite potential barrier.

15) Angular momentum operator: commutative algebra, eigenvalues, eigenfunctions. Spherical harmonics. Hydrogen atom in quantum mechanics: separation of variables for Schrödinger's equation with Coulomb potential in spherical coordinates, asymptotic behavior of radial part of wavefunction, energy levels, Laguerre polynomials, complete wavefunction, degree of degeneration.

16) Harmonic oscillator in quantum mechanics: asymptotic behaviour of solution, energy levels, Hermite polynomials, wavefunctions, classical limit. Creation and annihilation operators for harmonic oscillator in quantum mechanics.

Discussion session embraces the same scope of the program as the lecture does and comprises its illustration.

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

2) L. D. Landau, E. M. Lifshitz: "Quantum Mechanics, Non-Relativistic Theory"

3) R. Liboff: "Introductory Quantum Mechanics"

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

5) R. P. Feynman, R. B. Leighton, M. Sands: "Lectures on Physics", vol. 3: "Quantum Mechanics"

6) Brojan, J. Mostowski, K. Wódkiewicz: "Problems in Quantum Mechanics" (in Polish)

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 informations in literature and Internet sources, including explorations in foreign languages.

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 Elements 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.