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Quantum phisycs 1

A.Y. 2019/2020

Learning objectives

This course proivdes an introduction to the ideas and methods of quantum physics. In ths first part, the motivations for quantum physics are presented, its basic principles are introduced and the formalisim of non-relativistic quantum mechanics is developed, specifically in one dimension.

Expected learning outcomes

At the end of part 1 the student:

1. will be able to justify the need for a quantum description of Nature;

2. Will be able to construct the quantum operators associated to physical observables, to understand the probabilistic nature of the masurement of observables, and to classify (also using the density matrix formalism) the information content of a quantum state;

3. Will be able to quantize a one-dimensional mechanical system

4. Will be able to determine the time evolution of quantum systems, using the Heisenberg or the Schroedinger picture;

5. Will understand the properties of plane waves and wave packets;

6. Will be able to solve the Schroedinger equation with a variety of one-dimensional potentials (steps, wells, etc) which give rise to discrete or continuous spectra;

7. Will be able to determine the spectrum of the harmonic oscillator and will be able to manpulate creation and annihilation operators, also for the construction of cohernet states.

1. will be able to justify the need for a quantum description of Nature;

2. Will be able to construct the quantum operators associated to physical observables, to understand the probabilistic nature of the masurement of observables, and to classify (also using the density matrix formalism) the information content of a quantum state;

3. Will be able to quantize a one-dimensional mechanical system

4. Will be able to determine the time evolution of quantum systems, using the Heisenberg or the Schroedinger picture;

5. Will understand the properties of plane waves and wave packets;

6. Will be able to solve the Schroedinger equation with a variety of one-dimensional potentials (steps, wells, etc) which give rise to discrete or continuous spectra;

7. Will be able to determine the spectrum of the harmonic oscillator and will be able to manpulate creation and annihilation operators, also for the construction of cohernet states.

**Lesson period:** Second semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:**

**Assessment result:**

Course syllabus and organization

### CORSO A

Responsible

Lesson period

Second semester

**Course syllabus**

A. The experimental foundations of quantum physics

1. Waves and particles

2.Superposition, interference, measurement

B.Foundations

1.State vectors

2.Operators and observables

3.Uncertainty

4.Information

C. Canonical quantization

1.The coordinate basis

2. Momentum translations

3.The canonical commutator

D. Time evolution

1.The generator of time translations

2.The Schrödinger equation

3. The Heisenberg formulation of quantum mechanics

E. The free particle

1.Plane waves

2.Wave packets and minimum uncertainty

3.The motion of a wave packet

F. One-dimensional problems

1.The potential well and bound states

2.Scattering: the potential step

3.The potential barrier and tunneling

5. The harmonic oscillator

1.Spectrum and creation and annihilation operators

2. Eigenfunctions and the Schrödinger approach

3. Time evolution and coherent states

1. Waves and particles

2.Superposition, interference, measurement

B.Foundations

1.State vectors

2.Operators and observables

3.Uncertainty

4.Information

C. Canonical quantization

1.The coordinate basis

2. Momentum translations

3.The canonical commutator

D. Time evolution

1.The generator of time translations

2.The Schrödinger equation

3. The Heisenberg formulation of quantum mechanics

E. The free particle

1.Plane waves

2.Wave packets and minimum uncertainty

3.The motion of a wave packet

F. One-dimensional problems

1.The potential well and bound states

2.Scattering: the potential step

3.The potential barrier and tunneling

5. The harmonic oscillator

1.Spectrum and creation and annihilation operators

2. Eigenfunctions and the Schrödinger approach

3. Time evolution and coherent states

**Prerequisites for admission**

Calculus of one variable; linear algebra; waves and oscillations.

**Teaching methods**

The course consists of lectures (40 hours) and recitations (20 hours). All lectures are done on the blackboard and involve the presentation of theoretical and methodological arguments. Of the recitations, 10 hours are devoted to explaining standard applications, while 10 hours are devoted to the discussion of problem sets which have been assigned at the end of each lecture, directly involving students in the solution. The lecture record and all problems (with solution hints) are published on the instructors' website along the way.

**Teaching Resources**

Textbook

Stefano Forte e Luca Rottoli, Fisica Quantistica; Zanichelli.

Recommended books

J.J. Sakurai, Modern Quantum Mechanics , Pearson (general reference)

F. Schwabl, Quantum Mechanics; Springer (useful for explicit computations)

S. Weinberg, Lectures on Quantum Mechanics; Cambridge U.P. (for deepening and special topics)

K. Gottfried e T.M. Yan, Quantum Mechanics: Fundamentals; Springer (for deepening and special topics)

J. Binney e D. Skinner, The Physics of Quantum Mechanics; Oxford U.P. (for deepening and special topics)

Collections of problems and exercises

G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli (elementary)

L. Angelini, Meccanica quantistica: problemi scelti; Springer (elementary)

E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica; ETS (elementary and intermediate)

A. Z. Capri, Problems and Solutions in Nonrelativistic Quantum Mechanics; World Scientific (elementary, intermediate and avanced)

K. Tamvakis, Problems and Solutions in Quantum Mechanics; Cambridge U.P. (intermediate and avanced)

V. Galitski, B. Karnakov, V. Kogan e V. Galitski, Exploring Quantum Mechanics; Oxford U.P. (700 problems, mostly intermediate and advanced)

Stefano Forte e Luca Rottoli, Fisica Quantistica; Zanichelli.

Recommended books

J.J. Sakurai, Modern Quantum Mechanics , Pearson (general reference)

F. Schwabl, Quantum Mechanics; Springer (useful for explicit computations)

S. Weinberg, Lectures on Quantum Mechanics; Cambridge U.P. (for deepening and special topics)

K. Gottfried e T.M. Yan, Quantum Mechanics: Fundamentals; Springer (for deepening and special topics)

J. Binney e D. Skinner, The Physics of Quantum Mechanics; Oxford U.P. (for deepening and special topics)

Collections of problems and exercises

G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli (elementary)

L. Angelini, Meccanica quantistica: problemi scelti; Springer (elementary)

E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica; ETS (elementary and intermediate)

A. Z. Capri, Problems and Solutions in Nonrelativistic Quantum Mechanics; World Scientific (elementary, intermediate and avanced)

K. Tamvakis, Problems and Solutions in Quantum Mechanics; Cambridge U.P. (intermediate and avanced)

V. Galitski, B. Karnakov, V. Kogan e V. Galitski, Exploring Quantum Mechanics; Oxford U.P. (700 problems, mostly intermediate and advanced)

**Assessment methods and Criteria**

The exam at the end of part I is an intermediate three-hour long written test which requires solving a number of quantum physics problem of increasing degree of complexity, that cover the main topics of the syllabus, and answering a simple theory question. All written tests of the last several years available from the instructors' website.

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 7

Practicals: 20 hours

Lessons: 40 hours

Lessons: 40 hours

Professors:
Cruz Martinez Juan Manuel, Forte Stefano

### CORSO B

Responsible

Lesson period

Second semester

**Course syllabus**

A. Experimental facts at the roots of Quantum Mechanics

1. Waves and particles

2. Superposition, interference, measurement

B. Foundations

1. State vectors

2. Operators and observables

3. Indetermination

4. Information

C. Canonical quantisation

1. Coordinates representation

2. Momentum and translations

3. Canonical commutators

D. Time evolution

1. The time-evolution generator

2. The Schroedinger's equation

3. Heisenberg formulation

E. The free particle

1. Plane waves

2. Wave packets and states with minimal indetermination

3. Wave-packet motion

F. One-dimensional problems

1. Potential well and bound states

2. Step potential and scattering problems

3. Potential barrier and tunnelling effect.

G. The harmonic oscillator

1. Creation and destruction operators

2. Eigenfunctions and Schroedinger formulation

3. Time evolution and coherent states

1. Waves and particles

2. Superposition, interference, measurement

B. Foundations

1. State vectors

2. Operators and observables

3. Indetermination

4. Information

C. Canonical quantisation

1. Coordinates representation

2. Momentum and translations

3. Canonical commutators

D. Time evolution

1. The time-evolution generator

2. The Schroedinger's equation

3. Heisenberg formulation

E. The free particle

1. Plane waves

2. Wave packets and states with minimal indetermination

3. Wave-packet motion

F. One-dimensional problems

1. Potential well and bound states

2. Step potential and scattering problems

3. Potential barrier and tunnelling effect.

G. The harmonic oscillator

1. Creation and destruction operators

2. Eigenfunctions and Schroedinger formulation

3. Time evolution and coherent states

**Prerequisites for admission**

Basic knowledge of classical mechanics, calculus and linear algebra.

**Teaching methods**

Theoretical lectures and explicit solution of problems, at the blackboard.

**Teaching Resources**

Recommended textbooks:

J.J. Sakurai, Modern Quantum Mechanics, Cambridge University Press (reference textbook);

F. Schwabl, Quantum Mechanics; Springer

S. Weinberg, Lectures on Quantum Mechanics; Cambridge U.P.

K. Gottfried e T.M. Yan, Quantum Mechanics: Fundamentals; Springer

J. Binney e D. Skinner, The Physics of Quantum Mechanics; Oxford U.P.

Exercises with solutions can be found in:

G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli (elementary level)

L. Angelini, Meccanica quantistica: problemi scelti; Springer (elementari level)

E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica; ETS (elementary ed intermediate level)

A. Z. Capri, Problems and Solutions in Nonrelativistic Quantum Mechanics; World Scientific (elementary, intermediate and advanced level)

K. Tamvakis, Problems and Solutions in Quantum Mechanics; Cambridge U.P. (intermediate and advanced level)

V. Galitski, B. Karnakov, V. Kogan e V. Galitski, Exploring Quantum Mechanics; Oxford U.P. (700 problems, mostly intermediate and advanced level)

J.J. Sakurai, Modern Quantum Mechanics, Cambridge University Press (reference textbook);

F. Schwabl, Quantum Mechanics; Springer

S. Weinberg, Lectures on Quantum Mechanics; Cambridge U.P.

K. Gottfried e T.M. Yan, Quantum Mechanics: Fundamentals; Springer

J. Binney e D. Skinner, The Physics of Quantum Mechanics; Oxford U.P.

Exercises with solutions can be found in:

G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli (elementary level)

L. Angelini, Meccanica quantistica: problemi scelti; Springer (elementari level)

E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica; ETS (elementary ed intermediate level)

A. Z. Capri, Problems and Solutions in Nonrelativistic Quantum Mechanics; World Scientific (elementary, intermediate and advanced level)

K. Tamvakis, Problems and Solutions in Quantum Mechanics; Cambridge U.P. (intermediate and advanced level)

V. Galitski, B. Karnakov, V. Kogan e V. Galitski, Exploring Quantum Mechanics; Oxford U.P. (700 problems, mostly intermediate and advanced level)

**Assessment methods and Criteria**

The exam is only written, with two tests, one at the end of the first part (Modulo 1) and one a the end of the second part (Modulo 2).

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 7

Practicals: 20 hours

Lessons: 40 hours

Lessons: 40 hours

Professor:
Vicini Alessandro

Educational website(s)

Professor(s)