# THEORETICAL CHEMISTRY

OVERVIEW

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.

## AIMS AND CONTENT

LEARNING OUTCOMES

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.

AIMS AND LEARNING OUTCOMES

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.

Teaching methods

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.

SYLLABUS/CONTENT

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, vibration of a rectangular membrane

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, particle in a box model applied to pi electrons in linear conjugated hydrocarbons, uncertainty principle

6. Vectors

Definitions, operations with vectors, angular momentum, gradient, examples

7. 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, Dirac delta function, position states, representation of a state in the position space and in the momentum space, geometrical interpretation of the uncertainty principle, position and momentum operators in the two spaces, time evolution of a wavefunction, superposition of eigenstates and their evolution, wavepacket and its time evolution in a constant potential, time dependence of the average value of an operator, Ehrenfest theorem, exercises

8. 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, eigenvalues for potentials with symmetries, parity and translation operators and their commutation with the Hamiltonian, cyclic boundary conditions, Bloch theorem, Kronig-Penney model for a particle in a periodic potential, its matrix formulation and general solution, allowed energy bands and their physical interpretation, energy gap, limiting cases, hyperbolic functions

9. The harmonic oscillator

Classical oscillator as a local approximation of an inharmonic potential, Taylor series, quantum formulation of the harmonic oscillator, operator method solution, creation and annihilation operators, number operator, derivation of the ground state eigenfunction and its energy, considerations on the uncertainty principle, derivation of higher energy states using the creation operator, exercises with operators, calculation of average values and derivation of selection rules using creation and annihilation operators, adjoint operators and their properties, comparison between classical and quantum probability, correspondence principle, tunnel effect in the harmonic oscillator, virial theorem and its classical and quantum derivations, time evolution of a wavepacket in an harmonic oscillator.

10. Coherent states

Coherent or “Glauber” states as eigenfunctions of the annihilation operator, demonstration of their “minimal” dispersion, their series expansion of eigenstates of the Hamiltonian, internal product of two coherent states, average energy of a coherent state and its dispersion, Poisson distribution, physical meaning of the number operator in a coherent state, time evolution of a coherent state, comparison between the complex eigenvalue of a coherent state and its classical analogue, derivation of the analytical expression for the wavefunction of a coherent state, physical considerations, coherent state as a quantum description of a classical state.

RECOMMENDED READING/BIBLIOGRAPHY

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

## TEACHERS AND EXAM BOARD

Exam Board

LIBERATO MANNA (President)

MASSIMO OTTONELLI

FABIO MICHELE CANEPA

## LESSONS

Teaching methods

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.

## EXAMS

Exam description

The exam is based on an oral examination

Assessment methods

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