Elements of Quantum Mechanics 390-FS1-3EMK
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) Black-body radiation – classical description, ultraviolet catastrophe, Planck’s quantum theory, Wien’s law, Stefan-Boltzmann law.
2) Photoelectric effect – classical description, Einstein’s quantum theory. Compton effect. X-rays.
3) Diffraction experiments – wave nature of matter. Wave-particle duality. de Broglie matter waves.
4) Measurements of atomic spectra. Old quantum theory. Bohr-Sommerfeld quantization rules applied to various physical systems (hydrogen atom, particle in a constant, uniform gravitational field).
5) The role of measurement in quantum mechanics. Measurement of position and momentum; measurement of time and energy. Heisenberg uncertainty principle, complementarity principle. Propagation of wave packets in time and space. Gaussian wave packet. Wave function. Heuristic introduction to the free Schrödinger equation.
6) 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).
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) 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.
9) Expansion of wave functions in terms of eigenfunctions. Constructing superpositions of wave functions – examples. Calculating the probability of finding a state described by one wave function within a state described by another wave function.
10) 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.
11) 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.
12) 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. Law of radioactive decay, decay constants.
13) 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.
14) 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.
15) 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.
Mode
Prerequisites (description)
Course coordinators
Term 2024: | Term 2025: | Term 2026: | Term 2023: |
Type of course
General: (in Polish) kierunkowe (in Polish) kształcenia ogólnego obligatory courses (in Polish) ogólne kierunkowe | Term 2026: obligatory courses (in Polish) ogólne kierunkowe (in Polish) kierunkowe (in Polish) kształcenia ogólnego |
Requirements
Prerequisites
Number of hours of remote classes
Learning outcomes
The graduate knows and understands:
1. to an advanced degree, the concepts, principles, and theories specific to physics and astronomy within the scope of the curriculum; KP6_WG1
2. and is able to explain descriptions of physical regularities, phenomena, and processes using the language of mathematics; in particular, is able to independently derive fundamental theorems and laws; KP6_WG3
The graduate is able to:
1. analyze problems in the fields of physical sciences and astronomy and find solutions based on known theorems and methods; KP6_UW1
2. perform quantitative analyses and formulate qualitative conclusions based on them; KP6_UW2
3. use the tools of advanced mathematics and the mathematical methods of physics to describe and model fundamental physical phenomena and processes; is able to independently derive theorems and equations describing fundamental phenomena and laws of nature, and to conduct proofs of these theorems and laws; KP6_UK2
4. learn independently. KP6_UU1
The graduate is prepared to:
1. critically evaluate their own knowledge and the information they receive; KP6_KK1
2. consult scientific and popular-science literature to deepen and broaden their knowledge, while being aware of the risks associated with obtaining information from unverified sources, including the Internet; KP6_KO2
Assessment criteria
Students take part in lectures broaden of computer simulations, illustrating transmitted contents. They are stimulated for asking the questions and for discussion.
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.
Bibliography
1) R. Eisberg, R. Resnick "Quantum physics"
2) L. I. Schiff: "Quantum mechanics"
3) L. D. Landau, E. M. Lifshitz: "Quantum Mechanics, Non-Relativistic Theory"
4) R. Liboff: "Introductory Quantum Mechanics"
5) I. Białynicki-Birula, M. Cieplak, J. Kamiński: "Theory of Quanta"
6) R. P. Feynman, R. B. Leighton, M. Sands: "Lectures on Physics", vol. 3: "Quantum Mechanics"
7) Brojan, J. Mostowski, K. Wódkiewicz: "Problems in Quantum Mechanics" (in Polish)
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: