Application deadline

Type of course

Admission requirements

Course content

At the beginning of last century, Physics was a well established science. Its main foundations were two kind of fundamental entities:

• particles were the basic constituents of matter, obeyed the Newton¿s equations of motion and were the subject of Mechanics.
• electromagnetic waves were the interpretative tools for light-related phenomena in a broad sense.

Chemistry on the other hand was still in its infancy: even the existence of atoms (Dalton¿s theory) was still debated!

The advent of quantum mechanics (QM) shaked physics at its foundation by showing that a particle and a wave could be two manifestation of the very same physical object and it was merely a matter of which one was more apparent and/or useful in a given context. Additionally, it provided a crucial link between Physics and Chemistry: atoms and molecules (whose existence was finally accepted by the scientific community at large) are so small that the particle-wave duality cannot be ignored to explain their behavior and their fundamental properties.

The goal of the present course is to present the foundation of QM in a rigorous albeit simple way in order to show how it is nowadays employed in the modeling of atoms and molecules.

The course will start by introducing the axiomatic foundations of QM. Its postulates will be presented by showing their implication for the description and interpretation of physical phenomena at atomic and molecular level.

The postulates will then be employed on simple models such as the particle in a box and the harmonic oscillator. The latter, which has relevant implications for chemistry (e.g. molecular vibrations and  description of photons) will be covered in detail, by making use of the ladder-operator formalism. We will later consider rotational motion, which is the basis to understand the shape of atomic orbitals, the electron spin, the structure of atomic spectra. The last ¿exact¿ model we will cover is the hydrogen atom by deriving its wavefunctions, thus showing the meaning of all four quantum numbers assigned to electrons in atoms: we will see how this description allows to interpret the spectrum of the hydrogen atom (almost) fully: the missing details will be touched upon but their full explanation requires relativity which is covered in another course (KJE-3104).

Exactly solvable models provide a good starting point to show the features of many-electron atoms and molecules. However real systems are far more complex and in order to treat them properly a range of tools needs to be employed: group theory will allow to extract as much information as possible from a system by the investigation of its symmetry properties; for systems that cannot be solved exactly, one can make use of perturbation theory, which assumes that the deviations from the exact system are small and hence yield small deviation from the ideal behavior; in its time-dependent version, perturbation theory provides the framework to interpret atomic and molecular spectra; the Born-Oppenheimer approximation allows to separate nuclear and electronic motion and Hartree-Fock theory can be employed to obtain the simplest yet quite accurate description of the electronic structure of an atom or a molecule.

Objectives of the course

Knowledge:

The candidate

• will illustrate the postulate of quantum mechanics (wave functions and the state of a system, operators and observables, measurements and probability, the Schrödinger¿s equation and time evolution)
• will explain group theory (symmetry elements, definition of a group, group representations, orthogonality theorems, characters)
• will present time-independent and time-dependent perturbation theories.
• will show how the Born-Oppenheimer approximation leads to the separation of electronic and nuclear motions.
• will recognize the limitations of the B-O approximation and show when it is applicable and when it does not hold.
• will show how the variation principle can be used to find optimal solutions for electronic structure problems.

Skills:

The candidate

• will derive the wavefunction of exactly solvable models such as the particle in the box, the harmonic oscillator (using ladder operators), the hydrogen atom.
• will apply group theory to make predictions about structure and properties of atoms and molecules on the sole basis of their intrinsic symmetry.
• will employ time-independent perturbation theory to interpret features of atomic spectra and explain anharmonic deviation in vibrational spectra of molecules.
• will employ time-dependent perturbation theory to explain transition between states and resonance effects.
• will show how molecular orbitals are obtained based on the time-independent Schrödinger equation of molecular systems.
• will derive the Hartree-Fock equations from the variation principle.

General competence:

The candidate...

• will make use of quantum mechanics to interpret physical phenomena at the microscopic scale.
• will recognize the presence of symmetry in atoms and molecules in order to take advantage of it.
• will identify small components of a Hamiltonian operator and set up the framework for their perturbative treatment.
• will employ time-dependent perturbation theory to explain the band structure of atomic and molecular spectra.
• will show how molecular vibrations are connected to the Born-Oppenheimer approximation through the molecular electronic structure.
• will connect molecular properties and spectra to the necessary quantum chemical modeling required to compute them.

Language of instruction and examination

Teaching methods

Assessment

The evaluation consists in an oral exam at the end of the course (duration approximately 1h). The final grade (A-F scale) is an overall evaluation of the oral exam.

The students are expected to attend lectures and seminars. During the lectures the theoretical framework will be presented. Seminars will be used to show how to use the acquired theoretical knowledge on concrete problems.

Candidates that receive the grade F can repeat the oral exam the following semester.

Recommended reading/syllabus

Atkins and  Friedman, Molecular Quantum Mechanics, Oxford University Press

Details of the course are given through Fronter, the learning portal of the university. Only registered students for the course will have access to Fronter.