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Lecturer(s)
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Procházka Vít, doc. Mgr. Ph.D.
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Vyšín Ivo, RNDr. CSc.
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Richterek Lukáš, Mgr. Ph.D.
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Říha Jan, Mgr. Ph.D.
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Course content
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1. Historical introduction. Old quantum theory of light, corpuscular-wave dualism, radiation. Planck's law, the photoelectric effect, the Compton effect. Bohr's theory of the structure of atoms . De Broglie waves, diffraction of electrons. 2. The concept of wave function, its physical meaning. Properties of wave functions. Representation of physical quantities, linear Hermitian operators, operator equations. Mean values ??of physical quantities. Operators of specific physical variables, commutation relations, the uncertainty principle. 3. Schrödinger equation, stationary and non-stationary states. Green's function. Limit transitions to classical mechanics. Rate of change of physical quantities, by the time derivative operator. Ehrenfest theorem. Parity condition. 4. Applications. Solutions for rectangular potentials, one-dimensional, three-dimensional potential well, the method of separation of variables. A potential barrier, the tunnel effect, cold emissions, a radioactive decay. One-dimensional and three-dimensional quantum linear harmonic oscillator (LHO). The particles in a spherically symmetric potential field. A model of the hydrogen atom, orbitals. Mechanical and magnetic orbital angular momentum of the electron. 5. Approximate methods of solving problems in quantum physics. The perturbation theory, variational methods. The stationary perturbation theory, non-degenerate and degenerate states, the non-stationary perturbation theory, the Fermi rule. Direct and general variational methods. 6. Free particles, Green's function of a free particle. 7. Representation theory. Wave functions and operators as vectors and matrices in a Hilbert space. Dirac notation. Coordinate, impulse and energy representations. Schroedinger's and Heisenberg's attitudes. The density matrix, pure and mixed states. 8. The spin of particles. its experimental discovery. Pauli spin matrices, the Hamiltonian of a particle in an electromagnetic field. The Pauli equation, basics of relativistic quantum mechanics. the Klein-Gordon-Fock equation, the Dirac equation. 9. Basic concepts of statistical physics. The phase space, the Liuoville theorem. Microcanonical , canonical and grand canonical ensembles. The statistical definition of entropy. 10. Statistical properties of the sum and the statistical integral, calculations of thermodynamic quantities, applications for some systems (Maxwell's distribution of velocities, monoatomic and diatomic ideal gases, the concept of paramagnetism). 11. Statistical distribution. fermions and bosons, Bose Einstein condensation. 12. A model of a photon gas and blackbody radiation.
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Learning activities and teaching methods
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Monologic Lecture(Interpretation, Training), Work with Text (with Book, Textbook), Activating (Simulations, Games, Dramatization)
- Attendace
- 24 hours per semester
- Homework for Teaching
- 10 hours per semester
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Learning outcomes
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The course Fundamentals of Modern Physics consists of lectures and problem-solving exercises. The aim of the course is to provide a basic understanding of the principles, mathematical tools, and interpretations of quantum and statistical physics that are essential for understanding and explaining problems in contemporary physics, and to develop students' ability to apply this knowledge when solving both numerical and conceptual problems. Attention will be given to the connections between classical and quantum physics, the contribution of quantum physics to the description of physical phenomena, and the potential applications of quantum and statistical physics in other natural science disciplines, such as chemistry or biology. At the same time, the course guides future physics teachers toward the didactic transformation of difficult topics in modern physics, toward critical engagement with sources, models, analogies, and visualizations, and toward reflection on the possibilities for their use in school physics.
The course is primarily focused on acquiring knowledge and skills in quantum and statistical physics; at the same time, it develops selected KRAAU competencies. The student: a) Understands and explains the historical and physical foundations of quantum physics, describes the fundamental experimental phenomena leading to the abandonment of the classical description of the microscopic world, and uses the appropriate technical terminology. KRAAU: 1.1.1; 1.1.2; 1.1.3. b) Use the basic mathematical tools of quantum physics, particularly wave functions, operators, eigenvalues, expected values, commutation relations, and the Schrödinger equation, and interpret the physical significance of the results. KRAAU: 1.1.1; 1.1.3; 1.1.6. c) Solves typical problems in quantum mechanics, such as a potential well, a potential barrier, the tunnelling effect, a quantum harmonic oscillator, and the hydrogen atom, and evaluates the validity of the approximations and models used. KRAAU: 1.1.1; 1.1.6; 1.2.2. d) Explains the basic principles of spin, representations in quantum mechanics, the density matrix, and the fundamental concepts of relativistic quantum mechanics at a level appropriate for a bachelor 's-level course. KRAAU: 1.1.1; 1.1.7; 1.2.3. e) Understands the basic concepts of statistical physics, uses the microcanonical, canonical, and grand canonical ensembles, explains the statistical definition of entropy, and applies the statistical sum to simple physical systems. KRAAU: 1.1.1; 1.1.3; 1.2.3. f) Distinguishes between classical and quantum statistical descriptions, explains the difference between fermions and bosons, applies the Fermi-Dirac and Bose-Einstein distributions, and describes the physical significance of a photon gas and blackbody radiation. KRAAU: 1.1.1; 1.1.7; 1.2.3. g) Critically evaluates specialized, textbook, and popular science sources on modern physics; recognizes simplifications or misleading analogies; and assesses the suitability of models, visualizations, and digital simulations for teaching. KRAAU: 1.1.3; 1.1.4; 1.2.2; 1.2.5; 6.2.1; 6.2.3. h) Identifies typical difficulties and misconceptions among students related to quantum physicssuch as duality, the wave function, uncertainty, measurement, the tunnel effect, spin, the statistical behaviour of microparticles, and blackbody radiationand proposes appropriate didactic strategies to address them. KRAAU: 1.2.1; 1.2.2; 1.2.6; 2.2.1; 4.2.1. i) Designs a model teaching situation, activity, demonstration, analogy, visualization, or digital simulation for a selected topic in modern physics, sets a learning objective, and proposes a method for assessing students' understanding. KRAAU: 1.2.1; 1.2.2; 1.2.5; 2.1.1; 4.1.1; 4.1.3. j) Reflects on their own understanding of difficult concepts in modern physics and assesses how this understanding would influence their future teaching of physics in high school or in a popular science context. KRAAU: 2.5.1; 4.3.2; 6.1.1.
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Prerequisites
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Recommended prerequisites: knowledge of general physics at the level of introductory college courses, particularly mechanics, electricity and magnetism, optics, waves, and thermodynamics; mathematical background including differential and integral calculus, differential equations, complex numbers, vectors, matrices, eigenvalues, and eigenfunctions. Students are expected to be able to solve basic physics problems, work with mathematical models, and interpret the physical significance of the results.
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Assessment methods and criteria
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Student performance
Assessment of learning outcomes takes place in person. Throughout the semester, students' performance is assessed based on their ability to solve numerical, conceptual, and interpretive problems in quantum and statistical physics; their work with mathematical models, physical interpretations, graphs, operators, wave functions, and statistical data sets; and their participation in discussions on the historical, epistemological, and applied aspects of modern physics. Requirements include: a) regular attendance at seminars (at least 50%) and active participation in solving problems, b) ongoing work on numerical and conceptual problems in quantum physics, statistical physics, and their applications, c) the ability to explain the physical significance of the mathematical relationships, models, and approximations used, rather than merely substituting values into formulas, d) critical engagement with scholarly, textbook, or popular science sources on modern physics, including distinguishing precise physical formulations from simplifications, analogies, or potentially misleading interpretations, g) a passing grade in the laboratory exercises as a prerequisite for admission to the final exam, h) a final exam assessing understanding of fundamental principles and mathematical tools, as well as the ability to apply theory to specific physical situations; this includes a brief self-reflection on one's own understanding of a selected challenging concept in modern physics and a reflection on ways to teach it effectively. The course includes formative work with the Competency Framework for Teacher Education Graduates, particularly in the areas of subject-specific competence, the didactic transformation of challenging physics content, critical engagement with sources, the use of digital tools, assessment of understanding, and reflection on one's own professional development.
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Recommended literature
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Beiser, A. (1975). Úvod do moderní fyziky. Praha.
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Bořkovec M. a kol. Kompetenční rámec absolventa a absolventky učitelství. Praha. 2023.
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Čulík F., Noga M. (1982). Úvod do štatistickej fyziky a termodynamiky. Bratislava.
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Davydov, A. S. (1978). Kvantová mechanika. Praha.
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Feynman, R. P. (2013). Přednášky z fyziky 1-3. Praha.
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Gold, H., Tobochnik, J. (2010). Statistical and Thermal Physics: With Computer Applications.
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Greiner, W., Neise, L., Stöcker, H.:. Thermodynamics and Statistical Mechanics.
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Greiner, W. (1989). Quantum Mechanics I.
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Opatrný, T., Richterek, L., Vyšín, I., Říha, J. (2013). Základy moderní fyziky. Olomouc.
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Skála L. (2005). Úvod do kvantové mechaniky. Praha.
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Thornton S. T., Rex A. (2013). Modern Physics For Scientists and Engineers.
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