# QUANTUM PHYSICS (A)

OVERVIEW

The course "Quantum Physics" aims at illustrating the experimental evidences which prove the need for a modification of classical physics in atomic and subatomic contexts, and at introducing, in a systematic and self-consistent way, the relevant formalism.

## AIMS AND CONTENT

LEARNING OUTCOMES

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

AIMS AND LEARNING OUTCOMES

At the beginning, wave mechanics is iontroduced as a tool to overcome the difficulties of classical mechanics when applied to atomic models. At this stage, the concept of interference is introduced in the context of the position framework. After illustrating the main features of the new mechanics (quantum levels, uncertainty, probabilistic interpretation) the formalism of quantum mechanics is introduced in terms of Hilbert spaces of states and operators associated to observables. Problem solving and approximation methods are emphasized.

Teaching methods

Traditional lectures at the blackboard.

SYLLABUS/CONTENT

1 Crisis of classical physics

1.1 Particle behaviour of electromagnetic radiation

1.1.1 Black-body radiation and Planck's constant

1.1.2 Photoelectric effect

1.1.3 Compton effect

1.2 Wave behaviour of matter

1.2.1 Classical atomic models and their limitations

1.2.2 Spectral lines

1.2.3 Bohr's quantum theory

1.2.4 The interpretation of De Broglie

1.2.5 The expoeriment of Davisson and Gehrmer

2 Wave mechanics

2.1 Physical meaning of the wave function

2.2 IQualitative illustration of uncertainty

2.3 Schroedinger's equation

2.4 The vector space of square-integrable functions

2.5 Scalar product, metric spaces and Hilbert spaces

2.6 Continuity equation

2.7 Mean values of position and momentum

3 Quantum mechanics in one dimension

3.1 Separation of variables and stationary states

3.2 General properties of stationary states in one dimension

3.3 Infinetely deep potential well

3.3.1 Continuity of the wave function

3.3.2 Energy spectrum

3.3.3 Eigenfunctions of the hamiltonianl orthonormality

3.4 The free particle

3.4.1 Gussian wave packet

3.4.2 Phase velocity and group velocity

3.4.3 Uncertainty relation for the gaussian wave packet

3.5 The harmonic oscillator

3.5.1 Algebraic method

3.5.2 Analytic method

3.6 Finite potential well

3.7 Bound states and scattering states; continuum and discrete spectrum

Non-normalizable wave functions. The Dirac delta function.

3.8 Different representations of quantum states. Position and momentum representations.

3.9 The collective interpretation.

3.10 Potential barriers; transmition and reflection coefficients. Semiclassical limit. Thin wall limit.

4 General formulation oif quantum mechanics.

4.1 Hilbert space of state vectors

4.2 Observables and hermitian operators.

4.3 Properties of eigenvectors and eigenvalues of hermitian operators.

4.4 Matrices.

4.5 Uncertainty relations and minimum uncertainty.

5 Quantum mechanics in three dimensions.

5.1 Schroedinger's equation for a central potential.

5.2 Angular equation and spoherical harmonics.

5.3 Radial equation; centrifugal barrier, asymptotic behaviours of the solution.

5.4 The hydrogen atom.

5.5 Angular momentum. Algebra of commutators. Composition of angular momenta. Rotation invariance. Spin.

5.6 Sopherical well:bound states and scattering states.

5.7 Cubic box.

5.8 Three-dimensional harmonic oscillator. Degeneration of energy eigenvalues.

6 Symmetries in quantum mechanics.

6.1 Invariance properties and conservation laws in classical and quantum physics.

6.2 Unitary transformations.

6.3 Angular momentum as the generator of infinitesimal rotations.

6.4 An introduction to group theory: Lie groups, Lie algebras, representations.

6.5 Representations of the rotation group by the tensor method, the relation with spherical harminics.

RECOMMENDED READING/BIBLIOGRAPHY

D.J. Griffiths, Quantum Mechanics, Pearson

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

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

P.A.M. Dirac, The principles of quantum mechanics, Boringhieri

## TEACHERS AND EXAM BOARD

**Ricevimento:** By appointment.

Exam Board

CAMILLO IMBIMBO (President)

NICOLA MAGGIORE (President)

GIOVANNI RIDOLFI

NICODEMO MAGNOLI

CARLA BIGGIO

## LESSONS

Teaching methods

Traditional lectures at the blackboard.

LESSONS START

September 26th, 2016

## EXAMS

Exam description

Written test (4 problems, 2 for each section, in 4 hours). Oral test (approximately 30 minutes).