Quantum Phisycs 2
A.Y. 2023/2024
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
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
CORSO A
Responsible
Lesson period
First semester
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 exercises (26 hours) or supplementary topics (10 hours). All lectures are done on the blackboard and involve the
presentation of theoretical and methodological arguments.
presentation of theoretical and methodological arguments.
Teaching Resources
Textbook
Stefano Forte e Luca Rottoli, Fisica Quantistica; Zanichelli.
Recommended books
J.J. Sakurai, Modern Quantum Mechanics , Pearson (general reference)
Stefano Forte e Luca Rottoli, Fisica Quantistica; Zanichelli.
Recommended books
J.J. Sakurai, Modern Quantum Mechanics , Pearson (general reference)
Assessment methods and Criteria
The final exam is a three-hour long written test that requires solving a number of quantum physics problems of increasing degree of complexity, that cover the main topics of the syllabus.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 8
Practicals: 36 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Archidiacono Maria, Zaro Marco
CORSO B
Responsible
Lesson period
First semester
Course syllabus
*) Quantum mechanics in more than one dimension
- Direct product spaces
- Separable potentials
- The two-body problem and central problems
*) Introduction to the theory of groups
- Rotation group and its representation
*) Angular momentum
- 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 and the path-integral approach
- The semiclassical (WKB) approximation
*) Perturbation theory
- Time-independent perturbation theory
- Time-dependent perturbation theory and the interaction picture
*) 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 and the measurement problem
- Direct product spaces
- Separable potentials
- The two-body problem and central problems
*) Introduction to the theory of groups
- Rotation group and its representation
*) Angular momentum
- 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 and the path-integral approach
- The semiclassical (WKB) approximation
*) Perturbation theory
- Time-independent perturbation theory
- Time-dependent perturbation theory and the interaction picture
*) 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 and the measurement problem
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 performance of exercises and applications of the topics covered.
Teaching Resources
Textbooks:
J.J. Sakurai, Modern Quantum Mechanics, Cambridge UP;
S. Forte, L. Rottoli, Fisica quantistica, Zanichelli;
L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Elsevier; R.P. Feynman et al., The Feynman Lectures on Physics III, Addison-Wesley Pub.; P.A.M. Dirac, The Principles Of Quantum Mechanics, Oxford Science Publications; S. Patri', M. Testa, Fondamenti di Meccanica Quantistica, Ed. Nuova Cultura;
L.E. Picasso, Lectures In Quantum Mechanics, Springer;
Collections of solved exercises:
G. Passatore, Problemi di meccanica quantistica elementare, Franco Angeli;
L. Angelini, Meccanica quantistica: problemi scelti, Springer;
E. d'Emilio, L. E. Picasso, Problems in Quantum Mechanics, Springer;
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, Modern Quantum Mechanics, Cambridge UP;
S. Forte, L. Rottoli, Fisica quantistica, Zanichelli;
L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Elsevier; R.P. Feynman et al., The Feynman Lectures on Physics III, Addison-Wesley Pub.; P.A.M. Dirac, The Principles Of Quantum Mechanics, Oxford Science Publications; S. Patri', M. Testa, Fondamenti di Meccanica Quantistica, Ed. Nuova Cultura;
L.E. Picasso, Lectures In Quantum Mechanics, Springer;
Collections of solved exercises:
G. Passatore, Problemi di meccanica quantistica elementare, Franco Angeli;
L. Angelini, Meccanica quantistica: problemi scelti, Springer;
E. d'Emilio, L. E. Picasso, Problems in Quantum Mechanics, Springer;
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 consists of a written test, which contributes together with the test of module 1 to the determination of the final vote. Alternatively an overall written test of both modules.
The exam will evaluate both the skills acquired and the ability to solve new problems.
The exam will evaluate both the skills acquired and the ability to solve new problems.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 8
Practicals: 36 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Ferrera Giancarlo, Röntsch Raoul Horst
Educational website(s)
Professor(s)