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Course, academic year 2025/2026   
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Quantum Theory of Resonances - NBCM134
Title: Kvantová teorie rezonancí
Form of teaching: lecture
Guaranteed by: Department of Chemical Physics and Optics (32-KCHFO)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019
Duration in semesters: 1
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0, Ex [HT]
Capacity: unlimited
Maximum number of enrolled students: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Repeated enrollment: 2 / 2 / 2 / 2
Guarantor: doc. Mgr. Jaroslav Zamastil, Ph.D.
RNDr. Milan Šindelka, Ph.D.
Teacher(s): RNDr. Milan Šindelka, Ph.D.
Annotation -
This course is intended for students who have completed introductory courses in quantum mechanics and quantum chemistry and would like to pursue scientific work in the field of atomic, molecular, and optical physics. The students will gain a deep understanding of the basic formalism of non-Hermitian quantum mechanics, become familiar with the basic concepts in the theory of non-adiabatic quantum dynamics, and expand their insight into quantum chemistry regarding the development of Gaussian basis sets.
Last update: Kapsa Vojtěch, RNDr., CSc. (25.06.2025)
Aim of the course -

The course is designed as part of preparation for scientific work in the field of atomic, molecular and optical physics by introducing students into advanced chapters of quantum mechanics.

Last update: Kapsa Vojtěch, RNDr., CSc. (25.06.2025)
Literature -

[1] J. R. Taylor: Scattering Theory (The Quantum Theory of Nonrelativistic Collisions), Dover Publications, 2000.

[2] P. Roman: Advanced Quantum Theory, Addison-Wesley, 1965.

[3] N. Moiseyev: Non-Hermitian Quantum Mechanics, Cambridge, 2011.

Last update: Kapsa Vojtěch, RNDr., CSc. (19.02.2018)
Teaching methods -

lectures

Last update: Kapsa Vojtěch, RNDr., CSc. (28.02.2018)
Course assessment methods and requirements for successful completion, grading scheme -

The students can choose either an oral exam (corresponding to the syllabus and to the contents of the presented lectures) or a written (open book) exam based upon a mini-project related to scattering theory.

Last update: Kapsa Vojtěch, RNDr., CSc. (28.02.2018)
Syllabus -
  1. Non-Hermitian metastable states (taught by Kaprálová)
    • explanation of complex energy and divergent character of wavefunctions of dissipative eigenstates of Hamiltonian (resonances) based on quasiclassical theory
    • one-dimensional Hamiltonian in Laplace-Fourier representation, complex scaling transformation of Hamiltonian
    • physical interpretation of complex continuum obtained by solving complex scaled Hamiltonians based on quasiclassical theory
    • numerical pitfalls in application of the complex scaling method - detailed analysis of the problem using Wigner representation
    • generalization of complex scaling transformation - “smooth exterior complex scaling”
    • stabilization diagram through real-defined scaling of coordinate (avoided crossings, branching points, extrapolation of energy into complex plane using Padé approximants)
    • use of artificial Hamiltonian perturbation (complex absorption potential, artificial addition of charge force)
  2. Non-adiabatic quantum dynamics for a parameterized two-state Hamiltonian (taught by Kaprálová)
    • diabatic vs. adiabatic two-state Hamiltonian parameterized by one variable (avoided crossing, mixing angle and non-adiabatic coupling, adiabatic theorem); Hamiltonian parameterized by two variables (exceptional points and conical intersection, geometric phase)
    • differential equations of motion for diabatic representation (derivation of the Landau-Zener formula for the probability of a jump to the opposite state when passing through an avoided crossing)
    • adiabatic perturbation theory (transition points, complex-time method, Stückelberg oscillations, solutions for the Landau-Zener and Dykhne-Davis-Pechukas models)
    • chirally asymmetric exchange of states in the case of contour encircling of an exceptional point
  3. Long-range Gaussian electronic basis sets (taught by Šmydke)
    • Gaussian basis sets in ab initio calculations
    • Rydberg quantum states and their significance
    • discussion of the Hartree-Fock method
    • Long-range Gaussian basis sets
Last update: Kapsa Vojtěch, RNDr., CSc. (25.06.2025)
 
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