00539 - Fundamentals of Theoretical Physics

Academic Year 2023/2024

Learning outcomes

At the end of the course the student gets a basic knowledge of the principles of quantum mechanics and of the connected mathematical techniques, 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-Virtuale 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-Virtuale 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-Virtuale 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 a vote of 18/30 minimum is reached. 
  • The final vote is a weighted mean:
    Vote_written_exam = (1.3 * Vote_Math + 1.7 * 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, as well as the News section of the Virtuale web page of the course, will be used.
Learning material will be made available through the course website.

Office hours

See the website of Francesco Ravanini

See the website of Francisco Manuel Soares Verissimo Gil Pedro