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Quantum phisycs 2

A.Y. 2020/2021

Learning objectives

This is an advanced quantum mechanics course that builds upon the

introductory course of the prevous semester, and specifically introduces

three-dimensional systems (in particular the hydrogen atom) and a

variety of theoretical developments, including the theory of angular

momentum, spin, path-integral methods, perturbation theory, scattering

theory, identical particles, and entanglement.

introductory course of the prevous semester, and specifically introduces

three-dimensional systems (in particular the hydrogen atom) and a

variety of theoretical developments, including the theory of angular

momentum, spin, path-integral methods, perturbation theory, scattering

theory, identical particles, and entanglement.

Expected learning outcomes

At the end of this course the student

1. will know how to deal with the Schroedinger equation for intera

cting two-particles systems (including the case of identical particles) 2. will be able to solve for the spectrum of the Hamiltonian for central problems using spherical coordinates

3. will be able to determine the spectrum of the hydrogen atom

4. will be able to determine the spectrum of the orbital angular momentum

operator and of intrinsic angular momentum (spin) operators, and will

be able to add angular momenta

5. will be capable of connecting classical and quantum equations of motion, using either the WKB approximation or a path-integral approach

6. will be able to compute time-independent perturbations to the spectrum of a known Hamiltonian

7. will be able to calculate a transition amplitude using time-dependent perturbation theory

8. will be able to compute a cross section in terms of an amplitude

9. will be able to write down the wave function for a system of identical particles

10. will be able to determine the density matrix for a statistical ensemble and use it to calculate expectation values.

1. will know how to deal with the Schroedinger equation for intera

cting two-particles systems (including the case of identical particles) 2. will be able to solve for the spectrum of the Hamiltonian for central problems using spherical coordinates

3. will be able to determine the spectrum of the hydrogen atom

4. will be able to determine the spectrum of the orbital angular momentum

operator and of intrinsic angular momentum (spin) operators, and will

be able to add angular momenta

5. will be capable of connecting classical and quantum equations of motion, using either the WKB approximation or a path-integral approach

6. will be able to compute time-independent perturbations to the spectrum of a known Hamiltonian

7. will be able to calculate a transition amplitude using time-dependent perturbation theory

8. will be able to compute a cross section in terms of an amplitude

9. will be able to write down the wave function for a system of identical particles

10. will be able to determine the density matrix for a statistical ensemble and use it to calculate expectation values.

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

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### CORSO A

Responsible

Lesson period

First semester

Lectures will be held remotely using zoom and a virtual blackboard (tablet), at the scheduled time. The zoom link will be available on the course website. Recordings and the printout of the virtual blackboards for all lectures will be made available from the course website. A google group will be available for discussions among students and with the course teaching assistants and instructors.

The learning objectives, expected outcomes, and syllabus of the course are unchanged

Written exams will be held in presence, but students who wish to take the exam remotely will be able to do so using exam.net, on the scheduled day and time. In case of large numbers of students taking the exam remotely the length of the test might be shortened to two hours.

The learning objectives, expected outcomes, and syllabus of the course are unchanged

Written exams will be held in presence, but students who wish to take the exam remotely will be able to do so using exam.net, on the scheduled day and time. In case of large numbers of students taking the exam remotely the length of the test might be shortened to two hours.

**Course syllabus**

A. Quantum systems in more than one dimension

1. Direct product spaces

2. Separable potentials

3. The two-body problem and central potentials

B. Angular momentum

1. Angular momentum and rotations

2. The angular momentum operator and its spectrum

3. Spin

4. Addition of angular momenta

C. Three-dimensional problems

1. The radial Schrödinger equation

2. The isotropic harmonic oscillator

3. The Coulomb potential and the hydrogen atom

D. The semiclassical limit of quantum mechanics

1. The action in quantum mechanics

2. The Lagrangian approach to quantum mechanics: the path integral

3. The semiclassical (or WKB) approximation

E. Perturbation theory

1. Time-independent perturbations

2. Time-dependent perturbations and the interaction representation

3. Introduction to scattering theory

F. Identical particles

1. Systems of many identical particles

2. Bose and Fermi statistics

3. The spin-statistics theorem

G. Entanglement

1. Density matrix, entanglement, partial measurements

2. The Einstein-Podolsky-Rosen paradox and local realism

3. Bell inequalities and the measurement problem

1. Direct product spaces

2. Separable potentials

3. The two-body problem and central potentials

B. Angular momentum

1. Angular momentum and rotations

2. The angular momentum operator and its spectrum

3. Spin

4. Addition of angular momenta

C. Three-dimensional problems

1. The radial Schrödinger equation

2. The isotropic harmonic oscillator

3. The Coulomb potential and the hydrogen atom

D. The semiclassical limit of quantum mechanics

1. The action in quantum mechanics

2. The Lagrangian approach to quantum mechanics: the path integral

3. The semiclassical (or WKB) approximation

E. Perturbation theory

1. Time-independent perturbations

2. Time-dependent perturbations and the interaction representation

3. Introduction to scattering theory

F. Identical particles

1. Systems of many identical particles

2. Bose and Fermi statistics

3. The spin-statistics theorem

G. Entanglement

1. Density matrix, entanglement, partial measurements

2. The Einstein-Podolsky-Rosen paradox and local realism

3. Bell inequalities and the measurement problem

**Prerequisites for admission**

Basics of quantum physics and quantum mechanics. Quantum mechanics in one space dimension. Basics of complex analysis.

**Teaching methods**

The course consists of lectures (40 hours) and recitations (30 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 20 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. A teaching assistant will be available and will take care of extra exercise sessions including a simulation of the written test.

**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 final exam is a 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. All written tests of the last several years are available (with solutions) from the instructors' website. The final grade is determined based on the outcome of the intermediate test at the end of part I and of the final exam at the end of part II.

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 8

Practicals: 36 hours

Lessons: 40 hours

Lessons: 40 hours

Professors:
Cruz Martinez Juan Manuel, Forte Stefano

### CORSO B

Responsible

Lesson period

First semester

The course will be delivered online

**Course syllabus**

--Quantum mechanics in more than one dimension

+Direct product spaces

+Separable potentials

+The two-bosy problem and central problems

--Angular momentum

+Elements of group theory

+Rotations and angular momentum

+The angular momentum operator and its spectrum

+Spin

+Addition of angular momenta

--Three-dimensional problems

+The radial Schroedinger equation

+The isotropic harmonic oscillator

+The Coulomb potential and the hydrogen atom

--The classical limit of quantum mechanics

+The action in quantum mechanics

+Lagrangean quantum mechaincs andf the path-integral approach

+The semiclassical (WKB) approximation

--Perturbation theory

+Time-independent perturbation theory

+Time-dependent perturbation theory and the interaction picture

+Introduction to scattering theory

--Identical particles

+Many-particle systems

+Bose and Fermi statistics

+The spin-statistics theorem

--Entanglement

+Quantum statistical mechanics and density matrix

+The Einstein-Podolsky-Rosen paradox and local realism

+Bell inequalities, the measurement problem and e paradoxes of quantum mechanics

+Direct product spaces

+Separable potentials

+The two-bosy problem and central problems

--Angular momentum

+Elements of group theory

+Rotations and angular momentum

+The angular momentum operator and its spectrum

+Spin

+Addition of angular momenta

--Three-dimensional problems

+The radial Schroedinger equation

+The isotropic harmonic oscillator

+The Coulomb potential and the hydrogen atom

--The classical limit of quantum mechanics

+The action in quantum mechanics

+Lagrangean quantum mechaincs andf the path-integral approach

+The semiclassical (WKB) approximation

--Perturbation theory

+Time-independent perturbation theory

+Time-dependent perturbation theory and the interaction picture

+Introduction to scattering theory

--Identical particles

+Many-particle systems

+Bose and Fermi statistics

+The spin-statistics theorem

--Entanglement

+Quantum statistical mechanics and density matrix

+The Einstein-Podolsky-Rosen paradox and local realism

+Bell inequalities, the measurement problem and e paradoxes of quantum mechanics

**Prerequisites for admission**

Non-relativistic quantum mechanics in one dimension. Basic knowledge of classical mechanics and analytical mechanics, mathematical analysis, geometry and linear algebra.

**Teaching methods**

The teaching method consists of theory lessons on the blackboard and in the solution of exercises and applications on the topics covered.

**Teaching Resources**

Libri di testo:

J.J. Sakurai, Meccanica Quantistica Moderna; Zanichelli;

S. Forte, L. Rottoli, Fisica quantistica, Zanichelli;

L.D. Landau, E.M. Lifshitz, Meccanica Quantistica. Teoria non-relativistica, Editori Riuniti;

R.P. Feynman et al., The Feynman Lectures on Physics III, Addison-Wesley Pub.;

P.A.M. Dirac, Principi della Meccanica Quantistica, Boringhieri;

S. Patri', M. Testa, Fondamenti di Meccanica Quantistica, Ed. Nuova Cultura;

L.E. Picasso, Lezioni di Meccanica Quantistica, Ed. ETS;

Raccolte di esercizi svolti:

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

L. Angelini, Meccanica quantistica: problemi scelti, Springer;

E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica, ETS;

A. Z. Capri, Promlems and Solutions in Nonrelativistic Quantum Mechanics, World Scientific;

K. Tamvakis, Problems and Solutions in Quantum Mechanics, Cambridge U.P.;

V. Galitski, B. Karnakov, V. Kogan e V. Galitski, Exploring Quantum Mechanics, Oxford U.P.;

J.J. Sakurai, Meccanica Quantistica Moderna; Zanichelli;

S. Forte, L. Rottoli, Fisica quantistica, Zanichelli;

L.D. Landau, E.M. Lifshitz, Meccanica Quantistica. Teoria non-relativistica, Editori Riuniti;

R.P. Feynman et al., The Feynman Lectures on Physics III, Addison-Wesley Pub.;

P.A.M. Dirac, Principi della Meccanica Quantistica, Boringhieri;

S. Patri', M. Testa, Fondamenti di Meccanica Quantistica, Ed. Nuova Cultura;

L.E. Picasso, Lezioni di Meccanica Quantistica, Ed. ETS;

Raccolte di esercizi svolti:

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

L. Angelini, Meccanica quantistica: problemi scelti, Springer;

E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica, ETS;

A. Z. Capri, Promlems and Solutions in Nonrelativistic Quantum Mechanics, World Scientific;

K. Tamvakis, Problems and Solutions in Quantum Mechanics, Cambridge U.P.;

V. Galitski, B. Karnakov, V. Kogan e V. Galitski, Exploring Quantum Mechanics, Oxford U.P.;

**Assessment methods and Criteria**

The exam at the end of the second part of the course is a written exam, three hours long.

Several problems of increasing complexity and covering the program of the course have to be solved. Also, a theory questions has to be answered.

Several problems of increasing complexity and covering the program of the course have to be solved. Also, a theory questions has to be answered.

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 8

Practicals: 36 hours

Lessons: 40 hours

Lessons: 40 hours

Professor:
Vicini Alessandro

Professor(s)