UČNI NAČRT PREDMETA / COURSE SYLLABUS Predmet: Kvantna mehanika Course title: Quantum mechanics Študijski program in stopnja Study programme and level Univerzitetni študijski program 1.stopnje Fizika First cycle academic study program Physics Študijska smer Study field Letnik Acade mic year Semester Semester vse 3 prvi all 3 first Vrsta predmeta / Course type Univerzitetna koda predmeta / University course code: obvezni predmet/compulsory course??? Predavan ja Lectures Seminar Seminar Vaje Tutorial Klinične vaje work Druge oblike študija Samost. delo Individ. work 45 30 135 7 ECTS Nosilec predmeta / Lecturer: Prof. dr. Anton Ramšak, doc. dr. Tomaž Podobnik Jeziki / Languages : Predavanja / Slovensko/Slovene Lectures: Vaje / Tutorial: Slovensko/Slovene Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti: Vpis v letnik. Opravljen izpit iz vaj kot pogoj za pristop k ustnemu izpitu. Prerequisits: Enrollement status. Final examination pending on succesfully completed numerical exercises. 140
Vsebina: Schroedingerjeva enačba: Lastnosti Schroedingerjeve enačbe in valovne funkcije za en delec. Vloga simetrij: parnost, obrat časa, rotacije, translacije. Operatorji: Vpeljava operatorjev in pričakovane vrednosti za verjetnostno gostoto, tok verjetnosti, hitrost, gibalno količino, pospešek. Klasična limita. Formalizem kvantne mehanike: Postulati kvantne mehanike. Matrični zapis operatorjev. Povezava med valovno funkcijo in Diracovim formalizmom: reprezentacija -p in -r. Operator časovnega razvoja. Operator in lastne funkcije tirne vrtilne količine. Primeri: Uporaba formalizma kvantne mehanike na posebnih primerih: valovni paket, harmonični oscilator in koherentno stanje, krogelno simetrični problemi, elektron v magnetnem polju. Primerjava s klasično mehaniko. Spin: Obravnava spina in tirne vrtilne količine. Paulijeve matrike. Nelokalnost kvantne mehanike. Teorija motenj: Prvi in drugi red za nedegeneriran spekter. Prvi red za Content (Syllabus outline): Schroedinger equation: Properties of the Schroedinger equation and the wave function for one particle. Symmetries: parity, time reversal, rotations, translations. Operators: Introduction of operators and expectation values for the probability density, current, velocity, momentum, acceleration. Classical limit. Formalism of quantum mechanics: Postulates of quantum mechanics. Matrix formulation. Connection between the wave function and the Dirac formalism: p- and r- representations. Time evolution operator. Operator and eigenfunctions of orbital angular momentum. Examples: Application of the formalism in special cases: wave packet, harmonic oscillator and coherent state, spherical symmetric problems, electron in magnetic field. Relation to classical mechanics. Spin: Formalism for spin and angular momentum. Pauli matrices. Nonlocality of quantum mechanics. Perturbation theory: The first and the second order for non-degenerate spectrum. The first order for 141
degeneriran spekter. Časovno odvisna motnja. Fermijevo zlato pravilo. Primeri. Sipanje delcev in sipalna matrika. degenerate spectrum. Time dependent perturbation. Fermi golden rule. Examples. Scattering of particles and the scattering matrix. Temeljni literatura in viri / Readings: 1. F. Schwabl, Quantum Mechanics. Springer-Lehrbuch, Heidelberg, 2002. 2. J. J. Sakurai, Modern Quantum Mechanics. Addison Wesley Longman, 1994. 3. E. Merzbacher, Quantum Mechanics. John Wiley & Sons, New York, 1970. 4. L.D. Landau, E.M. Lifshitz, Quantum Mechanics. Pergamon Press Ltd., London, 1958. Cilji in kompetence: Cilji: Razumevanje podobnosti in razlik med klasično in kvantno mehaniko. Zna obravnavati količine, ki so v naravi kvantizirane. Zna formulirati in rešiti enoelektronske probleme s tem v zvezi. Kompetence: Teoretično razumevanje. Sposobnost modeliranja in reševanja fizikalnih problemov. Globlje poznavanje teorije kvantne mehanike. Sposobnost iskanja po strokovni literaturi. Objectives and competences: Objectives: Understanding of similarities between classical and quantum mechanics. Understanding of quantized quantities in nature. Ability to formulate and to solve related single electron problems. Acquired competence: Theoretical understanding. Modeling and solving the models of physical systems. In depth knowledge of the quantum mechanics. Acquired capacity to do independent literature search. 142
Predvideni študijski rezultati: Znanje in razumevanje Kandidat zna uporabiti osnovne principe kvantne verjetnosti. Uporaba Kandidat zna pojasniti črtasti spekter izsevane svetlobe, valovne lastnosti delcev, spin. Refleksija Abstraktno modeliranje fizikalnih sistemov. Prenosljive spretnosti - niso vezane le na en predmet Podlaga za razumevanje mikroskopskih pojavov: fizike kondenzirane snovi, atomov, molekul, jeder, in osnovnih delcev. Intended learning outcomes: Knowledge and understanding Knowledge of the principles of quantum probability. Competences Learn the line spectrum of light, wave properties of particles, spin of the electron. Reflection Abstract modeling of physical systems. Portable competences not connected with a single subject The basic knowledge for the understanding of microscopic effects: condensed matter physics, atoms, molecules, nucleai and elementary particles. Metode poučevanja in učenja: Predavanja, individualne konzultacije, računske vaje, domače naloge. Learning and teaching methods: Lectures, numerical exercices, homeworks and consultations. Načini ocenjevanja: Delež (v %) / Assessment: 143
2 pisna kolokvija iz vaj, ustni izpit. Ocene 1-5 (negativno), 6-10 (pozitivno) (po Statutu UL). Weight (in %) 50% 50% 2 tests on numerical exercises or a written examination, oral examination. Grading: 1-5 (negative), 6-10 (positive). Reference nosilca / Lecturer's references: 1. A non-adiabatically driven electron in a quantum wire with spin-orbit interaction, T. Čadež, J.H. Jefferson, and A. Ramšak, New J. Phys. 15, 013029 (2013). 2. Geometric analysis of entangled qubit pairs, A. Ramšak, New J. Phys. 13, 103037 (2011). 3. Geometrical view of quantum entanglement, A. Ramšak, Europhys. Lett. 96, 40004 (2011). 4. Spin qubits in double quantum dots - entanglement versus the Kondo effect, A. Ramšak, J. Mravlje, R. Žitko, and J. Bonča, Phys. Rev. B 74, 241305(R) (2006). 5. Entanglement of two delocalized electrons, A. Ramšak, I. Sega, and J.H. Jefferson, Phys. Rev. A 74, 010304(R) (2006). 144