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Advanced methods for physics and Quantum physics
01TUSOD
A.A. 2024/25
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 striking technological advancements of these years, as well as our response to the future challenges (protection of the environment, waste reduction, optimal exploitation of renewable energies and alternative routes of energy production, and so on) are essentially based on a worldwide research on physical phenomena that are very far from our everyday's experience, occur in "extreme" conditions (e.g. very small length scales, very high energies) and whose description is not possible without quantum mechanics and relativity. New solutions are always found by pushing forward the boundary of our present knowledge, and a Physical Engineer, who will work at the very forefront of technology, is expected to master today’s physical concepts prior to develop their practical applications. However, modern Physics is not disconnected from its ancestors, i.e. classical electromagnetism and Newtonian mechanics. Understanding these connections is fundamental to really grasp the novelty and the power of modern Physics theories and models, to understand the respective domains of validity, and to trace back the genesis and the evolution of today’s physical conceptions. The course of “Advanced Methods for Physics” has exactly this purpose. Its first part 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 ones due to Euler-Lagrange, Hamilton and Hamilton-Jacobi. Its second part will briefly describe how and why Special Relativity has been formulated, and explain its tremendous conceptual impact. The incredibly rich consequences of a very general, but radical, relativity principle will be shown. Then, kinematics, dynamics and electromagnetism will be re-analyzed within the formalism of Special Relativity. The two parts will be connected by some concepts of classical electromagnetism that are necessary for the subsequent treatment of special relativity, and concern, in particular, the potential equations, gauge transformations, retarded potentials. Examples and exercises will be suggested and analyzed during the lectures, in order to deepen the understanding of abstract concepts and stimulate the development of problem-solving skills that are fundamental for a Physical Engineer.
Advanced methods for physics and Quantum physics (Quantum physics)
The teaching objectives of the module concerning Quantum Physics aim 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 harmonic oscillator, 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 will: - know and understand the fundamentals of analytical mechanics, in the approaches by Lagrange, Hamilton and Hamilton-Jacobi, as well as their interconnections, and their analogies with the formalism used in Quantum Mechanics; -know and understand the (main) reasons why Special Relativity was developed, its basic concepts, their profound meaning, and their difference with respect to those used in Newtonian mechanics. - know the Lorentz transformations of coordinates and velocities, as well as the basics of the relativistic formulation of kinematics, optics, dynamics and electromagnetism. Moreover, the students are expected to become able to: - derive the main equations of analytical mechanics, discussing the assumptions made and the relevant limits of validity - derive the main equations of Special Relativity, properly define the relevant physical quantities, and show how they transform under changes of the reference frame - solve simple problems of mechanics by using the various approaches described during the course and discuss the results in comparison with those provided by Newton’s approach. - solve simple problems of Special relativity concerning: Lorentz transformations of coordinates and velocities; the Doppler effect; time dilation and length contraction; conservation of linear momentum and energy; transformation of the electromagnetic field, etc. This knowledge and these abilities will contribute to the acquisition of the technical (calculation) skills of a Physical Engineer but also to the critical thinking and the ability to establish connections between different aspects of Physics usually treated in different 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, 2) 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 (about 28 h) Survey of basic concepts of classical mechanics - Constraints and generalized coordinates - Principle of virtual work - D'Alembert principle - Generalized forces - Lagrangian function and Euler-Lagrange equations - Action and Hamilton's principle - Examples of applications of the Lagrangian approach - The method of Lagrange multipliers - Semi-holonomic constraints - Conservation theorems (linear momentum, angular momentum energy) and symmetry properties - The two-body problem and Kepler's problem - The theory of small oscillations and normal modes - Légendre transformations and the canonical equations of Hamilton - Poisson's brackets -Canonical transformations - The Hamilton-Jacobi equation for Hamilton's principal function and for Hamilton's characteristic function. Examples and exercises. 2) Complements of electromagnetism (about 7 h) Brief survey on Maxwell's equations - Electromagnetic potentials - Gauge transformations - Maxwell's equations in terms of potentials - Solution of Maxwell's equations with sources - Retarded potentials. Electromagnetism and analytical mechanics. 3) Special relativity (about 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. Exercises and examples.
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. Schroedinger equation and 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) (0.7 cr) 9) Symmetric and antisymmetric states of identical particles. Bosons, fermions and symmetrization principle. The exclusion principle. Quantization of Electromagnetic field. The Helium atom. (0.7 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)
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. A weekly 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 quantum 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 tutoring time is 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 format) will be uploaded in the web page of this course, accessible through the Teaching Portal of Politecnico. 2) After each lecture, the notes written on the whiteboard by the teacher will be made avaliable in PDF format 3) Examples of exercises and questions will be uploaded on the web page of the Course. 4) Suggested books (that can be found and bought online): H. Goldstein, J. L. Safko, C. P. Poole Classical mechanics III edition, Addison-Wesley 2002 (for Analytical Mechanics) M. Anselmino, S. Costa, E. Predazzi Origine classica della fisica moderna Levrotto&Bella, 1999 ISBN: 8882180352 (for Analytical Mechanics, Electromagnetism and Relativity) R. Gautreau, W. Savin Schaum's outline of Modern Physics -2nd edition Mc Graw-Hill, 1999 ISBN: 0-07-024830-3 (for exercises of Relativity)
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. Suggested handbooks: Franz Schwabl , Quantum Mechanics, Springer-Verlag 2007, K. Konishi, G. Paffuti, Quantum Mechanics: A New Introduction, Oxford University Press, 2009, D. J. Griffiths, Introduction to Quantum Mechanics, Addison-Wesley 2005;
Advanced methods for physics and Quantum physics (Advanced methods for physics)
Slides; Dispense; Esercizi; Esercizi risolti;
Advanced methods for physics and Quantum physics (Quantum physics)
Dispense; Libro di testo;
Advanced methods for physics and Quantum physics (Advanced methods for physics)
Lecture slides; Lecture notes; Exercises; Exercise with solutions ;
Advanced methods for physics and Quantum physics (Quantum physics)
Lecture notes; Text book;
Advanced methods for physics and Quantum physics (Advanced methods for physics)
Modalità di esame: Prova orale obbligatoria;
Advanced methods for physics and Quantum physics (Quantum physics)
Modalità di esame: Prova orale obbligatoria;
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.
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.
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 other 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".
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.