01TUSOD

A.A. 2020/21

2020/21

Advanced methods for physics and Quantum physics (Advanced methods for physics)

The course is aimed at providing the students with some important notions of "classical" (in the sense of non-quantum mechanical) physics that are necessary for the continuation of their studies. In particular, the course will allow the student to fill the conceptual gap between the formalism of classical mechanics taught in the basic courses of Physics 1 and the formalism of quantum mechanics, by describing the approaches to classical mechanics alternative to Newton's - in other words, the Lagrangian and Hamiltonian approaches. Similarly, the course will cover special relativity and some complements of electromagnetism that are not included in the program of the basics Physics courses but are needed, for instance, in the description of atomic and nuclear physics and solid state physics.

Advanced methods for physics and Quantum physics (Quantum physics)

The course aims to provide the basic mathematical tools and formalism necessary to obtain a quantum description of physical systems, and to discuss their application in predicting quantum phenomena, effects and properties characterizing systems such as the hydrogen and helium atoms, elementary molecule and matter-radiation models, the electromagnetic field, and systems of identical particles.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

The course is aimed at providing the students with some important notions of "classical" (in the sense of non-quantum mechanical) physics that are necessary for the continuation of their studies. In particular, the course will allow the student to fill the conceptual gap between the formalism of classical mechanics taught in the basic courses of Physics 1 and the formalism of quantum mechanics, by describing the approaches to classical mechanics alternative to Newton's - in other words, the Lagrangian and Hamiltonian approaches. Similarly, the course will cover special relativity and some complements of electromagnetism that are not included in the program of the basics Physics courses but are needed, for instance, in the description of atomic and nuclear physics and solid state physics.

Advanced methods for physics and Quantum physics (Quantum physics)

The module of this course concerning Quantum Physics aims to provide the basic mathematical tools and formalism necessary to develop a quantum description of physical systems, and to discuss their application in predicting quantum phenomena, effects and properties characterizing systems such as the hydrogen and helium atoms, elementary molecule and matter-radiation models, the electromagnetic field, and systems of identical particles.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

At the end of the course, the students should: - become familiar with the fundamentals of analytical mechanics (Lagrangian and Hamiltonian approach), understand the classical roots of the formalism they are going to use in Quantum Mechanics, and be able to apply the relevant methods to selected problems. - become acquainted with special relativity, understand the meaning of its postulates as well as their deep consequences, and be able to solve simple problems. - acquire the knowledge about some topics of electromagnetism (e.g. the potential equations, gauge transformations, retarded potentials), that are part of a solid background in classical physics and are not included in the introductory courses.

Advanced methods for physics and Quantum physics (Quantum physics)

Students are expected 1) to achieve an effective understanding of the fundamental concepts and formal tools of Quantum Mechanics, 3) to develop ability in using them in the study of quantum systems, and 3) to develop a correct approach to the investigation of quantum systems. These achievements are certainly important for subsequent courses such as solid-state and condensed-matter physics, field theory, quantum statistical mechanics, nuclear physics.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

At the end of the course, the students should: - become familiar with the fundamentals of analytical mechanics (Lagrangian and Hamiltonian approach), understand the classical roots of the formalism they are going to use in Quantum Mechanics, and be able to apply the relevant methods to selected problems. - become acquainted with special relativity, understand the meaning of its postulates as well as their deep consequences, and be able to solve simple problems. - acquire the knowledge about some topics of electromagnetism (e.g. the potential equations, gauge transformations, retarded potentials), that are part of a solid background in classical physics and are not included in the introductory courses.

Advanced methods for physics and Quantum physics (Quantum physics)

Students are expected 1) to achieve an effective understanding of the fundamental concepts and formal tools of Quantum Mechanics, 3) to develop ability in using them in the study of quantum systems, and 3) to develop a correct approach to the investigation of quantum systems. These achievements are certainly important for subsequent courses such as solid-state and condensed-matter physics, field theory, quantum statistical mechanics, nuclear physics.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

- good knowledge of classical mechanics (from the Physics I course) and, in particular: kinematics and relative motion; dynamics of systems of particles and of rigid bodies. - knowledge of basic electromagnetism (from the Physics II course) and in particular Maxwell's equations and wave propagation - good knowledge of mathematical analysis (from Analysis I and Analysis II courses)

Advanced methods for physics and Quantum physics (Quantum physics)

The knowledge of differential calculus, linear algebra concepts, vector-space theory and matrix diagonalization is required as well as the knowledge of the principles, laws and theorems of classical Mecahnics and electromagnetism.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

- good knowledge of classical mechanics (from the Physics I course) and, in particular: kinematics and relative motion; dynamics of systems of particles and of rigid bodies. - knowledge of basic electromagnetism (from the Physics II course) and in particular Maxwell's equations and wave propagation - good knowledge of mathematical analysis (from Analysis I and Analysis II courses) and linear algebra.

Advanced methods for physics and Quantum physics (Quantum physics)

The knowledge of differential calculus, linear algebra concepts, vector-space theory and matrix diagonalization is required as well as the knowledge of the principles, laws and theorems of classical Mecahnics and electromagnetism.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

1) Analytical mechanics (35 h) Survey of basic concepts of classical mechanics - Constraints and generalized coordinates - Principle of virtual work - D'Alembert principle - Generalized forces - Lagrangian function and Lagrange's equations - Generalized potentials - Examples of applications of the Lagrangian approach - Action and Hamilton's principle - Euler-Lagrange equations - Conservation theorems and symmetry properties - Cyclic coordinates and canonical momenta - Energy conservation - Legendre transformations and the canonical equations of Hamilton - Canonical transformations - Poisson's brackets - The Hamilton-Jacobi equations - Action variables. 2) Complements of electromagnetism (10 h) Brief survey on Maxwell's equations - Maxwell stress tensor - Angular momentum of the EM field - Electromagnetic potentials - Gauge transformations - Maxwell's equations in terms of potentials - Solution of Maxwell's equations with sources - Kirchhoff integral theorem - Retarded potentials. 3) Special relativity (15 h) Introduction: Galilean relativity and electromagnetism - The ether hypothesis and its crisis - Lorentz transformations - Length contraction and time dilation - Simultaneity and causality - Relativistic transformations of velocity - Proper time - Mass, impulse and energy - Doppler effect - Compton effect - Relativistic formulation of electromagnetism.

Advanced methods for physics and Quantum physics (Quantum physics)

1) The Lagrangian and Hamiltonian approaches to classical mechanics. (0.5 cr) 2) The postulates of Quantum Theory. Time evolution of quantum systems. The Eherenfest theorem. The canonical-quantization scheme. (0.5 cr.) 3) The Hilbert space of quantum states. Scalar product, completeness relation. Properties of Hermitian operators and eigenvalue equations. (0.7 cr.) 4) Raising and lowering operators, solution of the harmonic-oscillator problem. The Heisenberg uncertainty relation. Coherent states and semiclassical picture of quantum systems. (0.6 cr). 5) Spectrum and eigenstates of the angular momentum. The two-body problem. The hydrogen atom. (1.0 cr.) 6) Dirac's formulation of quantum states and operators. The Schroedinger and Heisenberg representations of quantum Mechanics. (0.6 cr) 7) The spin operator and spin states. Addition of angular momenta. Time-independent perturbation theory. (0.7 cr) 8) Charged particles in the electromagnetic field. The Zeeman effect for Hydrogen atoms (weak and strong-field limit) (0.8 cr) 9) Symmetric and antisymmetric states of identical particles. Bosons, fermions and symmetrization principle. The exclusion principle. The Helium atom. (0.6 cr)

Advanced methods for physics and Quantum physics (Advanced methods for physics)

1) Analytical mechanics (25 h) Survey of basic concepts of classical mechanics - Constraints and generalized coordinates - Principle of virtual work - D'Alembert principle - Generalized forces - Lagrangian function and Lagrange's equations - Generalized potentials - Examples of applications of the Lagrangian approach - Action and Hamilton's principle - Euler-Lagrange equations - Conservation theorems and symmetry properties - Cyclic coordinates and canonical momenta - Energy conservation - Legendre transformations and the canonical equations of Hamilton - Canonical transformations - Poisson's brackets - The Hamilton-Jacobi equation. 2) Complements of electromagnetism (10 h) Brief survey on Maxwell's equations - Maxwell stress tensor - Angular momentum of the EM field - Electromagnetic potentials - Gauge transformations - Maxwell's equations in terms of potentials - Solution of Maxwell's equations with sources - Kirchhoff integral theorem - Retarded potentials. 3) Special relativity (25 h) Introduction: Galilean relativity and electromagnetism - The ether hypothesis and its crisis - Lorentz transformations - Length contraction and time dilation - Simultaneity and causality - Relativistic transformations of velocity - Doppler effect - Introduction to tensor calculus - Four-vectors - Spacetime graphs - The light cone - Relativistic dynamics - Mass, linear momentum and energy - Relativistic formulation of electromagnetism.

Advanced methods for physics and Quantum physics (Quantum physics)

1) The Lagrangian and Hamiltonian approaches to classical mechanics. (0.2 cr) 2) The postulates of Quantum Theory. Time evolution of quantum systems. The Eherenfest theorem. The canonical-quantization scheme. (0.6 cr.) 3) The Hilbert space of quantum states. Scalar product, completeness relation. Properties of Hermitian operators and eigenvalue equations. (0.7 cr.) 4) Raising and lowering operators, solution of the harmonic-oscillator problem. The Heisenberg uncertainty relation. Coherent states and semiclassical picture of quantum systems. (0.6 cr). 5) Spectrum and eigenstates of the angular momentum. The two-body problem. The hydrogen atom. (1.0 cr.) 6) Dirac's formulation of quantum states and operators. The Schroedinger and Heisenberg representations of quantum Mechanics. (0.7 cr) 7) The spin operator and spin states. Addition of angular momenta. Time-independent perturbation theory. (0.8 cr) 8) Charged particles in the electromagnetic field. The Zeeman effect for Hydrogen atoms (weak and strong-field limit) (0.8 cr) 9) Symmetric and antisymmetric states of identical particles. Bosons, fermions and symmetrization principle. The exclusion principle. The Helium atom. (0.6 cr)

Advanced methods for physics and Quantum physics (Advanced methods for physics)

Advanced methods for physics and Quantum physics (Quantum physics)

Advanced methods for physics and Quantum physics (Advanced methods for physics)

Advanced methods for physics and Quantum physics (Quantum physics)

Advanced methods for physics and Quantum physics (Advanced methods for physics)

The course will consist of theoretical lectures, including the derivation of most important results and the description of some typical examples. The application of the theoretical concepts to the solution of relevant problems will be illustrated as well. Exercises for self-training of the students will be proposed.

Advanced methods for physics and Quantum physics (Quantum physics)

The teaching approach of this course is the traditional one based on frontal lectures. Each part of the program and the relevant calculations are discussed, step by step, and showed on the blackboard. As shown in the syllabus, the course is essentially divided in two sections: 1) Introduction to quantum physics, definition and/or derivation of the new formalism, methods and new theoretical tools necessary to develop the quantum-mechanical approach to physical systems, 2) study of various fundamental physical systems with the goal to achieve an effective modeling of such systems and to highlight their properties and significant physical effects. Exercise are proposed in the form of simple applications of the theory that each student must solve on his own. Solutions are provided in the subsequent lectures. Weekly meetings will be scheduled to give the students participating in this course the possibility to ask for questions, express doubts and clarify unclear points.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

Irrespective of the online /onsite teaching mode, the course will mainly consist of theoretical lectures, including the step-by-step derivation of the important results (about 48 h) The application of the theoretical concepts to the solution of relevant problems will be illustrated as well, and exercises for self-training of the students will be proposed. The time devoted to examples and exercises will be approximately equal to 12 h. The structure of the course will not be affected by the teaching mode (online or onsite). In the case of "blended" teaching mode, a weakly session of examples and exercises in classroom can be foreseen, if allowed by the rules of Politecnico and the general regulations for the limitation of the spread of Covid-19. In case of online teaching, a weekly VC session of questions and answers will be organized to improve the interaction between students and teacher.

Advanced methods for physics and Quantum physics (Quantum physics)

The teaching approach of this course is the traditional one based on frontal lectures. Each part of the program and the relevant calculations are discussed, step by step, and showed on the blackboard. As shown in the syllabus, the course is essentially divided in two sections: 1) Introduction to quantum physics, definition and/or derivation of the new formalism, methods and new theoretical tools necessary to develop the quantum-mechanical approach to physical systems, 2) study of various fundamental physical systems with the goal to achieve an effective modeling of such systems and to highlight their properties and significant physical effects. Exercise are proposed in the form of simple applications of the theory that each student must solve on his own. Solutions are provided in the subsequent lectures. Weekly meetings will be scheduled to give the students participating in this course the possibility to ask for questions, express doubts and clarify unclear points.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

H. Goldstein, J. L. Safko, C. P. Poole Classical mechanics III edition, Addison-Wesley 2002 M. Anselmino, S. Costa, E. Predazzi Origine classica della fisica moderna Levrotto&Bella, 1999 ISBN: 8882180352

Advanced methods for physics and Quantum physics (Quantum physics)

The PDF file containing the notes relevant to the course lectures are available in the web page of this course accessible through the Teaching Portal of Politecnico. Franz Schwabl , Quantum Mechanics, Springer-Verlag 2007 David J. Griffiths, Introduction to Quantum Mechanics, Addison-Wesley 2005;

Advanced methods for physics and Quantum physics (Advanced methods for physics)

1) The teacher's notes (in pdf) will be uploaded in the web page of this course, accessible through the Teaching Portal of Politecnico. 2) the recordings of the lectures of the academic year 2019/2020 will be made available on the course webpage 3) Suggested books (that can be found and bought online): H. Goldstein, J. L. Safko, C. P. Poole Classical mechanics III edition, Addison-Wesley 2002 M. Anselmino, S. Costa, E. Predazzi Origine classica della fisica moderna Levrotto&Bella, 1999 ISBN: 8882180352

Advanced methods for physics and Quantum physics (Quantum physics)

The PDF file containing the notes relevant to the course lectures are available in the web page of this course accessible through the Teaching Portal of Politecnico. Franz Schwabl , Quantum Mechanics, Springer-Verlag 2007, David J. Griffiths, Introduction to Quantum Mechanics, Addison-Wesley 2005;

Advanced methods for physics and Quantum physics (Advanced methods for physics)

**Modalitą di esame:** Prova orale obbligatoria;

Advanced methods for physics and Quantum physics (Quantum physics)

**Modalitą di esame:** Prova orale obbligatoria;

Advanced methods for physics and Quantum physics (Advanced methods for physics)

The exam is oral and is aimed at testing: - the knowledge of the topics listed in the program - the ability of the student to connect different concepts and to apply the theoretical notions to the solution of selected problems. The exam will thus consist in: 1 problem of analytical mechanics or relativity, to be solved completely (if particularly simple) or partially, by describing the main steps and the strategy of solution (max 10 points) 2 theoretical/conceptual questions about the topics of the course as listed in the detailed program that will be made available to students by the end of the lectures (max 10 points per each question) In general, the students will not be allowed to use books, notes or any online material during the exams. The teacher can however allow the access to the students' notes if particularly complicated formulas (difficult to remember) are necessary for the solution of the problems. The minimum mark to pass the exam is 18/30. The mark will be averaged with that obtained in the module "Quantum Physics". Online oral exams will take place according to the guidelines of the Decreto Rettorale n. 271, February 28, 2020. In particular, video-communications tools such as Virtual Classroom (preferred) or others will be used.

Advanced methods for physics and Quantum physics (Quantum physics)

The exam is oral and consists of three questions concerning different topics of the course. Each question corresponds to a score of 10 out of 30 marks. The goal, in general, is to test the knowledge of the course program. These questions aim to evaluate the understanding of students concerning 1) the methodology used to quantize classical systems, 2) the significant properties and effects that characterize physical systems as a consequence of the quantization process and 3) the mathematical tools used to develop quantum mechanics. To this end one of the three question is focused on proving/discussing some theorem or general property characterizing quantum systems and the mathematical formalism of quantum mechanics. The other questions are devoted to discuss the quantization of physical systems for some specific case and/or to determine quantum effects and physical properties of interest emerging from this process. Oral exams based on the video-communication tools will take place according to the guide lines of the Decreto Rettorale n. 271, February 28, 2020.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

**Exam:** Compulsory oral exam;

Advanced methods for physics and Quantum physics (Quantum physics)

**Exam:** Compulsory oral exam;

Advanced methods for physics and Quantum physics (Advanced methods for physics)

The exam is oral and is aimed at testing: - the knowledge of the topics listed in the program - the ability of the student to connect different concepts and to apply the theoretical notions to the solution of selected problems. The exam will thus consist in: - one problem of analytical mechanics or relativity, to be solved completely (if particularly simple) or partially, by describing the main steps and the strategy of solution (max 10 points) - two theoretical/conceptual questions about the topics of the course as listed in the detailed program that will be made available to students by the end of the lectures (max 10 points per each question). The duration of the oral exam is approximately 45 minutes. In general, the students will not be allowed to use books, notes or any online material during the exams. The teacher can however allow the access to the students' notes if particularly complicated formulas (difficult to remember) are necessary for the solution of the problems. The minimum mark to pass the exam is 18/30. The mark will be averaged with that obtained in the module "Quantum Physics". Online oral exams will take place according to the guidelines of the Decreto Rettorale n. 271, February 28, 2020. In particular, video-communications tools such as Virtual Classroom (preferred) or others will be used.

Advanced methods for physics and Quantum physics (Quantum physics)

The exam is oral and consists of three questions concerning different topics of the course. Each question corresponds to a score of 10 out of 30 marks. The goal, in general, is to test the knowledge of the course program. These questions aim to evaluate the understanding of students concerning 1) the methodology used to quantize classical systems, 2) the significant properties and effects that characterize physical systems as a consequence of the quantization process and 3) the mathematical tools used to develop quantum mechanics. To this end one of the three question is focused on proving/discussing some theorem or general property characterizing quantum systems and the mathematical formalism of quantum mechanics. The other questions are devoted to discuss the quantization of physical systems for some specific case and/or to determine quantum effects and physical properties of interest emerging from this process. Oral exams based on the video-communication tools will take place according to the guide lines of the Decreto Rettorale n. 271, February 28, 2020.

Advanced methods for physics and Quantum physics (Advanced methods for physics)

**Modalitą di esame:** Prova orale obbligatoria;

Advanced methods for physics and Quantum physics (Quantum physics)

**Modalitą di esame:** Prova orale obbligatoria;

Advanced methods for physics and Quantum physics (Advanced methods for physics)

The rules for the oral exam, either online or onsite, are identical. No

Advanced methods for physics and Quantum physics (Quantum physics)

No blended exam are planned for the current course (Quantum Physics)

Advanced methods for physics and Quantum physics (Advanced methods for physics)

**Exam:** Compulsory oral exam;

Advanced methods for physics and Quantum physics (Quantum physics)

**Exam:** Compulsory oral exam;

Advanced methods for physics and Quantum physics (Advanced methods for physics)

The rules for the oral exam, either online or onsite, are identical. Therefore, should onsite teaching be allowed, or online teaching be still necessary, the rules of the exam will be those described in the previous field.

Advanced methods for physics and Quantum physics (Quantum physics)

No blended exam are planned for the module on Quantum Physics

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