# MODERN PHYSICS

*Last update 01/07/2020 20:28*

OVERVIEW

The course aims to provide a general picture of modern physics. Starting from statistical thermodynamics and from the theories of mechanical and electromagnetic waves we will proceed to the discussion of the transition from classical to quantum physics with some hints to the theory of relativity. Particular emphasis will be placed on the common features of classical wave theory and wave mechanics, on the discussion of experiments and on the solution of simple problems.

## AIMS AND CONTENT

LEARNING OUTCOMES

It is expected that the student will have a good knowledge of the main quantum phenomena in atoms, molecules, gas and solids.

AIMS AND LEARNING OUTCOMES

The aim of the course is to bring students to a good level of knowledge of the basic principles of modern physics. The student is expected to be able to apply mathematical techniques such as Fourier series and integrals to solving problems in modern physics .

Teaching methods

Traditional lectures for 70 hours.

SYLLABUS/CONTENT

Modern physics, developed since the second half of the nineteenth century, has marked a conceptual leap in various respects with respect to classical physics. Its roots are in statistical thermodynamics and classical eletromagnetism, and more generally in classical wave theory. These themes constitute the first two parts of the course. The third part will cover quantum mechanics. The course will highlight not only the conceptual leap, but also the profound continuity that links these three important chapters of physics.

**Statistical thermodynamics**

Classical mechanics in phase space. Energy and the Hamilton function. Microscopic meaning of energy and entropy. Heat propagation, local energy conservation and Fourier equation. Boltzmann's Entropy. Statistical significance of the second law of thermodynamics. Ehrenfest model. Thermodynamic fluctuations and Einstein's formula. Probability rudiments: random variables, continuos and discrete distributions, mean value and variance, statistical independence. Random walk and introduction to the theory of Brownian motion. Gibbs distribution and calculation of average values using the partition function; applications to the ideal gas and systems with discrte energy levels. Boltzmann equipartition theorem and calculation of specific heats for poly-atomic gases and comparison with experimental data.

**Classic waves**

Free harmonic oscillations, normal modes. Propagation by waves of physical perturbations. Elastic waves in fluids. D'Alembert equation. Stationary waves and vibrating string. Helmholtz equation. Local conservation laws and continuity equation. Electromagnetic waves. Plane and spherical waves. Wave packets and Fourier analysis. Bandwidth theorem ("Heisenberg classical principle"). Dispersive media. Dispersion relations and wave equations. Wave interference and coherence. The two-slit experiment. Diffraction. Fourier Optics. Principle of Fermat and of Huygens. Introduction to the special theory of relativity.

Quantum waves

Elements of crisis of the classical theories: black body, photoelectric effect. Electromagnetic field in a cavity as a set of independent oscillators. Einstein hypothesis E = h nu and determination of the Planck distribution. Stefan-Boltzmann law. De Broglie hypothesis mv = h / lambda and its experimental verification (from the Davidson and Germer experiment to recent experiments of interference of material waves). Minimum action principle and Fermat principle. Energy waves and matter waves. Basic concepts of quantum mechanics: wave equation in dispersive media and time-dependent Schroedinger equation, wave function and quantum state. Equation of continuity and probabilistic interpretation of the wave function. Uncertainty principle. Correspondence rules. Classical Hamilton function and quantum Hamilton operator. Stationary states and the time-independent Schroedinger equation. Free states and bound states. Gaussian packets. Schroedinger equation with potential, study of some one-dimensional cases: wells, potential barriers and harmonic oscillator. Calculation of the average energy of a quantum harmonic oscillator in contact with a thermostat. The Schroedinger equation in three dimensions. Well of infinite cubic potential and the notion of degeneration of levels. The hydrogen atom: levels and quantum numbers. The Stern-Gerlach experiment and the spin. Quantum mechanics of multi-particle systems. Bosons and Fermions. Entanglement.

RECOMMENDED READING/BIBLIOGRAPHY

-La fisica di Feynman*, Volume 3. “Meccanica quantistica” (Zanichelli* *2007); freely available on line in english**: http://www.feynmanlectures.info/*

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

**- **Chimica Fisica, Peter Atkins e Julio De Paula, (Zanichelli 2012);

**-** Introduction to Quantum Mechanics: S.M. Blinder (Elsevier, 2004).

- Introduction to Quantum Mechanics: David J. Griffiths (Benjamin Cumming, 2004);

- Modern Quantum Mechanics: J. J Sakurai (Addison Wesley, 1993);

- Lecture notes and problems collections available on-line: https://www.ge.infn.it/~imbimbo

## TEACHERS AND EXAM BOARD

Exam Board

PIERANTONIO ZANGHI' (President)

SILVANA TERRENI

PAOLO SOLINAS

CAMILLO IMBIMBO

NICODEMO MAGNOLI (President Substitute)

## LESSONS

Teaching methods

Traditional lectures for 70 hours.

ORARI

## EXAMS

Exam description

Written test and oral exam.