The “Theoretical Chemistry” course intends to teach the fundamentals of quantum mechanics, with a carefully balanced choice of basic topics, ranging from the mathematical tools required to correctly formulate and solve the easiest problems, to the postulates of quantum mechanics and their application to problems of medium difficulty.
The course will provide an introduction to a selection of topics of quantum mechanics, which are treated in a way that will enable the student to tackle in the future more advanced topics, both in physical chemistry of molecules and of the solid state.
The course intends at the same time to complement (in the choice of topics) and to deepen the knowledge acquired in the physical chemistry courses, and will provide the students the tools and basic skills to engage in the study of solid state matter. Every topic of the course has its correct and rigorous mathematical and physical description, and often a comparison is made between classical and quantum mechanics.
The classes will be taught following the classical method (i.e. using the blackboard). The use of the computer and the projector will be limited to the visualization of scripts (downloaded from the ““Wolfram demonstration project” website) which will help to further clarify some of the problems that will be solved during the classes. The general teaching technique of the course is to introduce and explain the theoretical concepts and then to invest much of the time in applying these principles to solve a real physical problem. The course follows extensively some of the chapters of the suggested book and the associated solutions manual. Whenever new concepts are introduced that are not present in these books, the teacher provides he students with additional documentation.
1. A brief historical introduction to quantum mechanics
2. Complex numbers
General definitions, addition, product, powers and roots of complex numbers.
3. Wave equation of classical mechanics
Definitions, boundary conditions, general solution of the wave equation, normal modes of vibration, standing waves as superposition of travelling waves, orthogonality of standing waves.
4. Probability and statistics
First and second moment of a distribution, standard deviation, continuous distributions, Gaussian distribution, problems.
5. The Schrödinger equation and the particle in a box
One dimensional wave equation, introduction to the operators, eigenvalue problems, solutions for the particle in a box, correspondence principle, calculation of average values, practical examples, uncertainty principle.
6. Postulates and general principles of quantum mechanics
Classical dynamic variables and comparison with classical mechanics, well-behaved functions, Postulate 1, behavior of the wave equation in a discontinuous potential, Postulates 2-4, examples, commutator of two operators and its relation with the uncertainty principle, exercises with commutators, Hermitian operators, Dirac notation, orthogonality of eigenfunctions of Hermitian operators, Gram-Schmidt orthogonalization, Fourier series, probability of obtaining a given value of a physical observable, time dependent Schrödinger equation and its plausibility, consequences of measuring the position of a particle.
7. Systems with piecewise constant potentials
Potential step, reflection and transmission coefficients, evanescent wave, potential barrier, tunneling, comparison between classical and quantum probability, resonances, potential well, calculation of bound states in a potential well, behavior of a free particle inside and outside a potential well, comparison with the ionic potential, Kronig-Penney model for a particle in a periodic potential, its matrix formulation and general solution, allowed energy bands and their physical interpretation.
8. Approximate methods
Variational methods, trial functions that depend linearly on the variational parameters, trial functions that do not depend linearly on the variational parameters, introduction to perturbation theory, first and second order perturbation theory, examples.
9. Atomi multi-elettronici
Atomic units, helium atom and related Hartree-Fock equations, antisymmetry of the electron wave functions, Slater determinants, Hartree-Fock-Rothaan method and related results for atoms, correlation energy, atomic term symbols.
Quantum Chemistry (Inglese), by Donald A. McQuarrie, seconda edizione (2007)
Problems and Solutions to Accompany Donald A. McQuarrie's Quantum Chemistry (Inglese) (2007), di Helen O. Leung e Mark D. Marshall
Additional documentation provided by the teacher
LIBERATO MANNA (President)
ADRIANA SACCONE (President Substitute)
from october 19th 2020 (according to the schedule published on http://www.chimica.unige.it/didattica/Home_SC and/or https://corsi.unige.it/9018)
The exam is based on an oral examination
The exam will assess the level of comprehension of the concepts learned during the course and the ability of the student to apply those concepts to solve elementary quantum mechanical problems
