CODE 66559 2022/2023 16 cfu anno 3 FISICA 8758 (L-30) - GENOVA FIS/02 GENOVA Annual Per sostenere l'esame di questo insegnamento è necessario aver sostenuto i seguenti esami: PHYSICS 8758 (coorte 2020/2021) ANALYTICAL MECHANICS 25911 2020 PHYSICS II 57049 2020 GENERAL PHYSICS 3 57050 2020 AULAWEB

## OVERVIEW

The course discusses the main experimental evidences that motivate abandoning classical mechanics to describe atomic physics. The fundamental principles of non-relativistic quantum mechanics and its mathematical formalism are presented. Applications of quantum mechanics to the physics of particles, atoms, molecules, gases and condensed matter are described. In the second part of the course approximation methods and scattering theory are developed.

## AIMS AND CONTENT

### LEARNING OUTCOMES

At the end of this lecture course, the student will have developed a good understanding of quantum mechanics basic principles, as well as the theory's mathematical foundations. The student will be able to apply this knowledge to solve concrete problems involving physical phenomena at the atomic scale, both exactly and by using approximation methods

### AIMS AND LEARNING OUTCOMES

At the end of these courses the student will be able to

1. Identify the physical contexts which require a quantum description of the relevant phenomena;
2. Compute the probabilities of simple quantum mechanical processes involving systems with finite number of states;
3. Compute the tunnelling probabilities for potential barriers in one spatial dimension;
4. Compute the spectrum of angular momentum (spin) operators for both simple and composed systems;
5. Solve the Schrödinger equation for a system of two particles interacting with a central potential to compute the energy spectrum of hydrogen atom;
6. Compute wave functions and energy spectra of systems composed by identical particles;
7. Relate the laws of motion of classical mechanics to those of quantum mechanics, using both the WKB method, and the variational method;
8. Calculate the time-independent perturbation to the spectrum of a known Hamiltonian;
9. Determine a transition amplitude by the theory of time-dependent perturbations;
10. Express the cross section in terms of a transition amplitude;
11. Determine the density matrix for a given statistical mixture and use it to calculate an average value.

PREREQUISITES

### PREREQUISITES

Basic knowledge of classical mechanics and analytical mechanics, mathematical analysis, geometry and linear algebra.

### TEACHING METHODS

The course is delivered through frontal lectures that include

• Blackboard presentation
• Problems solved by the lecturers at the blackboard
• Problem solved by the students and discussed all together

### SYLLABUS/CONTENT

Part A:

A1 The crisis of classical physics

1.1 Atomic models

1.2 Photoelectric effect and photons

1.3 Compton effect

1.4 Atomic absorption and emission spectra

A2 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 Davisson e Germer experiment

2.7 Particle interference

A3  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 finite basis

A4  Particle Quantum mechanics

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

A5 Temporal evolution

5.1 Schrödinger and Heisenberg pictures

5.2 Time evolution of a Gaussian packet

5.3 Continuity equation

5.4 Collective interpretation of the wave function

A6 One-dimensional Schrödinger equation

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

A7 Symmetries

7.1 Translations and rotations

7.2 Discrete translations: Bloch theorem

7.3 Angular momentum and its representations

7.4 Spin

7.6 Scalar and vector operators

7.7 Harmonic polynomials and spherical harmonics

7.8 Schrödinger equation in central potential

7.9 The levels and the energy eigenfunctions of the hydrogenoides

A8 Atoms and Molecules

8.1 Fundamental level of Helium

8.2 Excited levels of Helium

8.3 Identical particles: bosons and fermions

8.4 Pauli principle

8.5 Zeeman effect

8.6 Hydrocarbons

Part B:

B.I: Advanced topics in QM (3 weeks)

1. Symmetries in Quantum Mechanics

2. Selection rules for scalar and vector operator. Wigner-Eckhart theorem.

3. Mixed-states, density matrix

B.II: Approximation methods (4 weeks)

1. Time-independent perturbation theory: degenerate and non-degenerate case.

2. Applications to hydrogenoids: fine and hyperfine structure, Zeeman effect.

3. Variational methods

4. WKB approximation

B.III: Time-dependent Hamiltonians and Scattering (4 weeks)

1. Formal solution of the Schrodinger equation with time-dependent hamiltonians. Dyson series. Exact solutions: spin resonance.

3. Elastic scattering: time-dependent and time-independent formalism. Lippmann Schwinger equation. Born approximation.

4. Partial waves expansion.

B.IV: Towards Relativistic Quantum Mechanics (1 week)

1. Klein-Gordon and Dirac equation.

•  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)
•  L.D. Landau, E.M. Lifsits, vol. 3: "Meccanica Quantistica", Editori Riuniti
• Griffiths, Schroeter, “An introduction to Quantum Mechanics", 3rd edition
• Sakurai and Napolitano, “Modern Quantum Mechanics", 3rd  edition
• S. Weinberg, "Lectures on Quantum mechanics", ed. Cambridge

## TEACHERS AND EXAM BOARD

### Exam Board

CAMILLO IMBIMBO (President)

STEFANO GIUSTO

NICOLA MAGGIORE

SIMONE MARZANI (President Substitute)

## LESSONS

### Class schedule

The timetable for this course is available here: Portale EasyAcademy

## EXAMS

### EXAM DESCRIPTION

The exam is made up a written test and an oral part. Rules and criteria and specified on the course aulaweb page.

### ASSESSMENT METHODS

The written test has a duration of 4 hours and contains two problems, one for each part of the course. The problems are divided into questions of variable difficulty, in order to more accurately evaluate the level of competence achieved by the student.

The oral exam, which lasts about 40 minutes,  is lead by two professors responsible for the two parts of the course. During the oral exam the student is asked to answer a few questions or to solve problems at the blackboard, in order to assess the student’s knowledge and understanding of the topic discussed. During the lectures, students will be given the opportunity to solve problems and exercises, as a means of self-evaluation.

### Exam schedule

Data appello Orario Luogo Degree type Note
11/01/2023 14:00 GENOVA Scritto
10/02/2023 14:00 GENOVA Scritto
05/06/2023 14:00 GENOVA Scritto
05/07/2023 14:00 GENOVA Scritto
11/09/2023 14:00 GENOVA Scritto