CH-453 / 3 credits

Teacher: Vanicek Jiri

Language: English

## Summary

The course covers several exact, approximate, and numerical methods to solve the time-dependent molecular Schrödinger equation, and applications including calculations of molecular electronic spectra. More advanced topics include introduction to the semiclassical methods and Feynman path integral.

## Content

1. Review of classical molecular dynamics.
Langrangian and Hamiltonian formalisms, phase space.
Classical molecular dynamics and thermodynamics in phase space.
2. Exact real-time quantum dynamics.
Time-dependent Schrödinger's equation. Born-Oppenheimer approximation and potential energy surfaces.
Time-correlation functions.
Methods of quantum propagation of wave functions.
Split operator method and the fast Fourier transform.
3. Approximate methods for quantum dynamics.
Sudden approximation.
Adiabatic approximation.
Time-dependant perturbation theory.
Fermi's Golden Rule.
Time-dependent Hartree method.
4. Semiclassical dynamics.
Old quantum theory and the WKB approximation.
Wigner function.
Van Vleck propagator.
Semiclassical initial value representation.
5. Quantum thermodynamics.
Feyman path integral approach
- interpreted as imaginary-time dynamics
- interpreted as classical thermodynamics of a polymer chain.
Path integral Monte Carlo method.
Path integral molecular dynamics.

## Learning Outcomes

By the end of the course, the student must be able to:

• Solve the time-dependent Schrödinger equation with a basis method.
• Derive and apply the sudden and adiabatic approximations.
• Derive the time-dependent perturbation theory and Fermi's Golden Rule.
• Apply the time-dependent perturbation theory and Fermi's Golden Rule to molecular transitions induced by electromagnetic field.
• Expound the connections between the Newtonian, Lagrangian, and Hamiltonian approaches to classical mechanics.
• Expound how electronic spectra can be computed via the autocorrelation functions.
• Apply the Fourier and split-operator methods to solve the time-dependent Schrödinger equation numerically.
• Expound the connection between quantum dynamics and quantum thermodynamics and how it can be used to compute molecular quantum thermodynamic properties with the Feynman path integral.

## Assessment methods

Grade: 25% exercises during the semester; 75% oral exam

## Supervision

 Office hours Yes Assistants Yes

## In the programs

• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Molecular quantum dynamics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Molecular quantum dynamics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Molecular quantum dynamics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Molecular quantum dynamics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Molecular quantum dynamics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Molecular quantum dynamics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Exam form: Oral (summer session)
• Subject examined: Molecular quantum dynamics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Molecular quantum dynamics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional

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