# 00539 - Fundamentals of Theoretical Physics

## Learning outcomes

At the end of the course the student gets a basic knowledge of the principles of quantum mechanics, useful for the comprehension of the microscopic structure of the physical world.

## Course contents

Part 1: Mathematical Methods

• Analytical functions, some special functions in use in QM
• Distributions and Dirac delta function
• Hilbert spaces, linear operators, eigenstates and eigenvalues
• Fourier series and transforms
• Orthogonal Polynomials
• Ordinary Linear Homogeneous Differential Equations of 2nd order and their hypergeometric solutions

Part 2: Quantum Mechanics

Microscopical structure of matter

• Black body radiation, photoelectric effect, Compton effect
• Bohr atom
• Interference experiments; particle - wave dualism
• De Broglie hypotesis, wave function
• Schoedinger equation and temporal evolution

General principles of Quantum Mechanics

• Quantum Mechanics postulates.
• Mean values. Ehrenfest theorem.
• Position-momentum commuting relations
• Heisenberg indeterminacy relations
• Eigenvalue problem for the Hamiltonian
• Fourier transforms and momentum representation.

One-dimensional problems

• Potential wells
• Potential barriers, tunnel effect
• Delta potential
• Harmonic oscillator
• WKB method

Angular momenta

• 3D Spatial rotations and angular momentum in QM, its
• eigenvalues and eigenvectors.
• Half-integer eigenvalues and electron spin.
• Angular momentum sums

Central symmetry problems

• Spherical potential well
• Spherical harmonic oscillator
• Two-body problem
• Hydrogen atom

Symmetries in QM

• Symmetries, infinitesimal transformations and their generators.
• Translations and momentum.
• Rotations and angular momentum.
• Parity.

Identical particles and statistics

• Bosons and fermions
• Pauli exclusion principle

Approximate methods

• Perturbation theory: first and second order
• Degenerate perturbation theory
• Semiclassical approximation and WKB method
• Elements of the variational principle approach: Helium atom
• Electron interaction with the electromagnetic field
• Zeeman effect: normal and anomalous
• Fine structure of hydrogen atom

The topics of the course are treated in notes written by the teacher and deposited on the course IOL website.

A further encouraged reading is the book:
Griffiths, D.J. - Introduction to quantum mechanics - Ed. Prentice Hall

Exercises to train for the preparation of the written exam

• Problems solved during the tutoring can be found on IOL website
• Exercices and examples are proposed on the Griffiths book cited above
• Costantinescu F., Magyari E. - Problems in Quantum Mechanics - Ed. Pergamon Press
• Also take a look at the exam problems of the past years, available on IOL website

Other suggested books, suitable for deepening of knowledge on single arguments:

• Cohen-Tannoudji C., Diu B., Laloe F. - Quantum mechanics, vol. 1 - Wiley Ed.
• Sakurai J.J. - Modern Quantum Mechanics - Addison, Wesley Ed.
• Schiff L.I. - Quantum Mechanics - Mc Graw, Hill Ed.
• Phillips A.C. - Introduction to quantum mechanics - Wiley Ed.
• L.D. Landau, E.M. Lifshitz - Theoretical Physics, vol.3: Quantum mechanics: non relativistic theory - MIR Ed.
• Dirac P.A.M. - The principles of Quantum Mechanics - Clarendon Press

## Teaching methods

• Blackboard lectures, sometimes integrated by table of figure presentations with the projector.
• Exercises presented and commented at the blackboard
• Further exercises proposed as homework. Althogh not compulsory, they are often crucial for the training for the final written part of the exam.

## Assessment methods

The exam consists in written and oral parts.

• One cannot access the oral exam if a previous written exam has not been passed and got a sufficient vote (18/30).
• 6 exam sessions (written + oral) are orgainzed within the solar year: 3 in January /February, 2 in June/July, 1 in September. No other exam session will be organized in different dates.
• One can try the written and the oral exams also at different sessions. The vote of the written exam will be retained for 14 months. Later, a new written exam has to be passed before going to an oral session.

Written exam:

• the time is 3 hours and the exam consists in:
- a problem in Mathematical Methods
- a problem (quite elaborated) in QM
both to be fully solved.
• texts and notes can be consulted at will
• the written exam is valid only if both problems have been solved in a sufficient (18/30) way.
• The final vote is a weighted mean:
Vote_written_exam = (Vote_Math + 2 Vote_QM)/3
• the results are published on the Alma Esami site and the solutions are available after the exam on the AMS Campus site.
• the written exam has a validity of 14 months and can be repeated if the evaluation is unsatisfactory. In such case, the best vote is kept.

Oral exam:

• can be given only after a written exam has been passed
• 3 questions, chosen by the examiners
• alternatively to the first question, the candidate can choose to organise a presentation of an argument more in-depth than what explained during the lectures. The teacher is keen to offer directives for texts or other material for the deepening.
• the final vote is the mean of the written and oral exam evaluation. In case of in-depth argument, a better final judgement (up to 2 more points, according to the excellence of presentation) can be accessed.

## Teaching tools

To communicate, the section "Avvisi" of the teacher web-site will be used.
Learning material will be made available through the course website.

## Office hours

See the website of Francesco Ravanini