LEARNING OUTCOMES

The purpose of the course is providing the basics and main analytic tools of non-relativistic Quantum Mechanics.

AIMS AND LEARNING OUTCOMES

The first part of the course discusses the difficulties of the classical descriptions of atomic phenomena and describes the Bohr-Sommerfeld quantization rules of the so-called old quantum theory as a partial solution to such difficulties. After an analysis of the fundamental principles of the new theory  -  uncertainty principle and probabilistic interpretation - the course will introduce the formalism of quantum mechanics in terms of Hilbert space of states and operators associated to observables. The concept of representation is emphasized and the coordinate and momentum representations described. Spin is introduced in the context of the discussion of rotational symmetry. Several applications of the Schrodinger equation to atomic and molecular physics will be discussed, such as radioactive decay, tunnel effect, Coulomb problem, and Zeeman effect. It is expected that the student will acquire a familiarity with the basic principles which characterize the quantum description of physical phenomena and an understanding of the physical contexts in which such description is required. It is also expected that he will learn to solve simple problems involving quantum particles or simple quantum mechanical systems.

TEACHING METHODS

Traditional lectures at the blackboard.

SYLLABUS/CONTENT

1 The crisis of classical physics

1.1 Atomic models
1.2 Photoelectric effect and photons
1.3 Compton effect
1.4 Microscopic theory of specific heats
1.5 Atomic absorption and emission spectra

2 The old quantum theory

2.1 Bohr atomic model
2.2 De Broglie wavelength and wave-particle duality
2.3 Bohr-Sommerfeld quantization rule
2.4 Quantum theory of specific heats
2.5 Black body and Planck spectrum
2.6 Davisson e Germer experiment
2.7 Particle interference

3 The formalism of quantum mechanics

3.1 Superposition principle: states and vectors
3.2 Scalar products and transition probabilities
3.3 Observables, operators and eigenvector basis
3.4 Compatible and incompatible observables
3.5 Equivalent representations and unitary transformations
3.6 Quantum systems with a finite basis

4 Quantum mechanics of a particle

4.1 Uncertainty relations
4.2 Canonical relations
4.3 Continuous spectrum: generalized eigenstates and observables
4.4 Coordinates and momenta representations

4.5 Wave packets
4.6 Schrödinger equation

5 Temporal evolution

5.1 Schrödinger and Heisenberg pictures
5.2 Time evolution of a Gaussian packet
5.3 Continuity equation

6 Schrödinger equation in one dimension

6.1 Free particle
6.2 Particle in a box
6.3 General properties of energy eigenfunctions in 1 dimension
6.4 Step potential
6.5 Square potential well
6.6 Potential barrier: transmission and reflection coefficients
6.7 Tunnel effect: semi-classical limit. Alpha decay
6.8 Harmonic oscillator: creation and destruction operators
6.9 Periodic potentials: bands

7 Symmetries

7.1 Translations and rotations
7.2 Discrete translations: Bloch theorem
7.3 Angular momentum and its representations
7.4 Spin
7.5 Addition of angular momenta
7.6 Spherical harmonics
7.7 Schröedinger equation in a central potential
7.8 The levels and the energy eigenfunctions of the hydrogenoids

RECOMMENDED READING/BIBLIOGRAPHY

• L. E. Picasso, Lezioni di Meccanica Quantistica (Edizioni ETS, Pisa,
2000)

• Richard Phillips Feynman, Robert B. Lieghton and Matthew Sands,
The Feynman Lectures on Physics Vol 3 (Quantum Mechanics),(1966)
(edizione on-line http://www.feynmanlectures.caltech.edu)

• D. J. Griffith, Introduction to Quantum Mechanics, ed. Pearson

• S. Weinberg, Lectures on Quantum mechanics, ed. Cambridge

•  J.J. Sakurai, J. Napolitano, Meccanica quantistica moderna, Zanichelli

•   L.D. Landau, E. Lifsits, Meccanica Quantistica, Editori Riuniti