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 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
  
• Hilbert spaces, linear operators, eigenstates and eigenvalues
• Fourier series and transforms
  
• 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

Readings/Bibliography

The topics of the course are treated in notes written by the lecturer 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
  
• 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.
  
• Exercises presented and commented at the blackboard
  
• Further exercises proposed as homework (not compulsory).

Assessment methods

The exam consists in a compulsory written and an optional oral parts.

• 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 organised in different dates.
  
• The written and the oral exams must be undertaken in the same exam session.

Written exam:

• The exam lasts  3 hours and the  consists in:
  - a problem in Mathematical Methods
  - a problem (quite elaborated) in QM
  both to be fully solved.
  
• Students may consult a 
  
• 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
  

Oral exam:

• The optional oral exam can only be undertaken by those students with a mark greater or equal to 26 in the written exam.
  
• The oral exam can induce a change of the written exam's mark by at most +/- 4 points.

Teaching tools

Learning material will be made available through the course website.

To communicate, the section "Avvisi" of the lecturer's website, as well as the News section of the Virtuale web page of the course, will be used.

Office hours

See the website of Francisco Manuel Soares Verissimo Gil Pedro

See the website of Silvia Pascoli