66913 - Physical Chemistry 1

Learning outcomes

The students learn: to master their mathematical knowledge in order to apply it to problems in Physical Chemistry; the fundamentals of molecular symmetry and of quantum mechanics for following studies in atomic and molecular structure;to apply the methods of quantum mechanicsfor studying the electronic properties, especially the energy levels,of atoms and simple molecules.

Course contents

• Module 1

1. Vector spaces and linear transformations: Vector spaces, Matrix algebra, Matrices and Linear transformations, Determinants, Invertible matrices, Orthogonal matrices; Complex matrices, The eigenvalue problem, Similarity trasformations and diagonalization, Hermitian matrices. Function spaces.
2. Molecular symmetry and group theory: Symmetry operations and elements; The algebra of simmetry operators; Groups: definition, the multiplication table, some properties and definitions; Point groups; Symmetry operations as linear transformations in the ordinary 3D space; Matrix representations of symmetry groups; Base functions to build matrix representations; Equivalent representations; Reducible and irreducible representations; Great orthogonality theorem; Character tables.
3.  Symmetry and quantum mechanics: The postulates of quantum mechanics: states, operators and observables; The Schroedinger equation; The meaning of the wavefunction; Time evolution; The matrix formulation; The symmetry of the Hamiltonian; Symmetry and degeneration; Integrals and selection rules.
4.  Differential equations: Separable differential equations of first order, Linear first-order differential equations, Second-order homogeneous linear equations with constant coefficients, Examples: the classical harmonic oscillator and the particle in a one-dimensional box and in a ring, Second-order inhomogeneous linear equations. Separation of variables, Examples: the particle in a rectangular box and in a circular box.

• Module 2 and 3

1. Heisenberg Uncertainty Principle
2. Wave Function of One or More Free Particles
3.  Particle in a Box
4.  Harmonic Oscillator
5.  Rigid Rotor
6. Hydrogen Atom and Hydrogen-like Atoms
7. Variational Principle
8. Perturbation Theory
9. Introduction to Multi-electron Atoms
10. Laboratory work: Numerical solutions of quantum mechanical problems 

Readings/Bibliography

• Lecture notes on Group theory and simmetry, L. DORE, Editografica 2024.
• The Chemistry Math Book, E. STEINER, Oxford, 2008, 2nd ed.
• Molecular Quantum Mechanics, P.W. ATKINS and R.S. FRIEDMAN, Oxford, 2010.

Suggested readings:

• Quantum Mechanics. The Theoretical Minimum, L. SUSSKIND and A. FRIEDMAN, Penguin Books, 2015.
• Quantum Mechanics, G. AULETTA, M. FORTUNATO and G. PARISI, Cambridge University Press, 2009.

Teaching methods

The course is organized in three learning modules: Mathematical Methods for Chemistry (5 credits)  and Atomic and Molecular Structure (4 credits) with its additional module of laboratory work (1 credit). Classes of the first module are given in the first semester; classes of the second and third  modules are given in the second semester.

Classes are organized as lectures in the classroom,  in-class exercises and, for the third module only, laboratory exercises.

All students must attend Module 1, 2 on Health and Safety, online.

Assessment methods

Learning assessment is evaluated only by means of the final examination. This aims at verifying the student's knowledge and skills by means of one test for each semester.

For the first module there is first a written examination with exercises, which lasts 3 hours.  A minimum grade of 16/30 is required for the admission to the oral exam, where, after a discussion of the written test, two questions concerning the course contents are asked to the student.

For the second and third modules, there will be a written test where the student will have to solve one numerical problem, followed by a viva examination relative to the course contents and the laboratory activity.

The final grade is the arithmetic mean of the grades obtained for each learning module.

Teaching tools

Video projector, notebook, blackboard.

Office hours

See the website of Luca Dore

See the website of Assimo Maris

See the website of Luca Bizzocchi