Programa de Pós-Graduação em Física (Eng)

COURSES

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The student must integrate 24 credits. The master dissertation is equivalent to 4 credits. Among the 20 credits not related to the dissertation, 14 are relative to the required courses, and the remaining 6  credits to elective and/or optative courses. A course with x credits has x classes of 50 min duration per week.

Article acceptance in a scientific journal with an impact factor greater than or equal to 1,000 can count 2 credits.

The student must enroll in “Dissertation” in the semesters in which he has no courses to attend.

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• Courses

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Courses details

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Required courses

 

COURSE: Electromagnetism I

CREDITS: 4

Fundamentals of Electromagnetism. Electric and magnetic multipoles. Boundary problems in material media. Electromagnetic Field's equations. Radiation by systems of charges and currents. Electromagnetic waves.

BIBLIOGRAPHY

1. J. Jackson, Classical Eletrodynamics, 3a Ed. Wiley 1998.
2. J. Frenkel, Princípios de Eletrodinâmica Clássica, EDUSP, 1996.

 

COURSE: Statistical Mechanics I

CREDITS: 4

Review of Thermodynamics. Ensembles of Statistical Physics: microcanonical, canonical and grand-canonical. Classic fluids. Quantum gases: fermions, bosons and photons. Phase transitions and critical phenomena. 

BIBLIOGRAPHY

1. Silvio R.A. Salinas, Introdução à Física Estatística, EDUSP, 1997.
2. R.K. Pathria, Statistical Physics, Pergamon, 1978, second edition, 1996.
3. K. Huang, Statistical Mechanics, Wiley, 1963, second edition, 1987.
4. L. Landau, E. Lifshitz, Mecânica Estatística, ed Mir , 1980.

 

COURSE: Quantum Mechanics I

CREDITS: 4

Foundations of quantum theory. Quantum Dynamics. Schrödinger and Heisenberg pictures. Simple quantum systems. Approximation methods. Rotations. Angular momentum. The hydrogen atom. Spin. Fine and hyperfine structure. Time-independent perturbation.

BIBLIOGRAPHY

1. J.J. Sakurai - Modern Quantum Mechanics, Addison-Wesley, 1994.
2. C. Cohen-Tanoudji, B. Diu e F. Laloê, Quantum Mechanics, 2 vols, Wiley, 1977.
3. A. Messiah: Quantum Mechanics, Dover. A.F.R. de Toledo Piza: Mecânica Quântica, EDUSP 2003.
4. P. A M. Dirac – The International Series of Monographs on Physics – 27: The Principles of Quantum Mechanics, 4a Ed Oxford Science pub, 1989.

 

COURSE: Teaching Practicum

CREDITS: 2

Teaching classes in undergraduate courses under supervision. 

BIBLIOGRAPHY

Specific.

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Elective courses

 

COURSE: Classical Mechanics

CREDITS: 4

Variational principle. Lagrangean and Hamiltonian formalism. Conservation laws, angle and action variables. Canonical transformations. Poisson brackets. Hamilton-Jacobi theory. Canonical perturbation theory. Integrability.

BIBLIOGRAPHY

1. H. Goldstein, C. P. Poole Jr., J. L. Safko. Classical Mechanics, 3a ed. Pearson 2001.
2. W. F. Wreszinski,  Mecânica Clássica Moderna - EDUSP 1996.
3. L. Landau, E. Lifchitz, Mecânica, MIR, 1980.
4. N. Mukunda, E.G. Sudarshan, "Classical Dynamics: A Modern Perspective", John Wiley (1974).
5. C. Lanczos, “The Variational Principles of  Mechanics” 4th Ed.,  Dover, NY, 1986.

 

COURSE: Mathematical Methods I

CREDITS: 4

Functions of one complex variable. Analytical functions. Taylor and Laurent series. Séries de Taylor e Laurent. Multiform functions. Residuals calculation. Analytic continuation. Gamma function. Elliptic functions. Asymptotic series: saddle point / stationary phase method. Conformal mapping. Fourier integrals. Laplace transform. 

BIBLIOGRAPHY

1. Philip M. Morse e  Herman Feshbach, Methods of Theoretical Physics, McGraw-Hill Book Company Inc., New York, 1953.
2. Frederick W. Byron, Jr. and Robert W. Fuller, Mathematics of Classical and Quantum Physics, Dover Publications inc., New York, 1969.
3. Jerrold E. Marsden e Michael J. Hoffman, Basic Complex Analysis, W. H. Freeman Co., 3rd ed., 1999.
4. H. Moysés Nussenzveig, Métodos Matemáticos III, Notas de Aula.

 

COURSE: Electromagnetism II

CREDITS: 4

Relativistic Eletrodynamics. Dynamics of relativistic particles in electromagnetic fields. Collisions of charged particles with the matter. Liénard-Wiechert potentials and the fields of moving charges. Radiation of accelerated charges. Scattering. Radiation dispersion and reaction Lagrangean formulation of Maxwell's equations. Quantization of electric charge. 

BIBLIOGRAPHY

1. J. Jackson, Classical Eletrodynamics, 3a Ed. Wiley 1998.
2. J. Frenkel,  Princípios de Eletrodinâmica Clássica, EDUSP, 1996.
3. J. Schwinger, Classical Eletrodynamics.
4. L. Landau, E. Lifchitz, Teoria do Campo, MIR, 1972.

 

COURSE: Quantum Mechanics II

CREDITS: 4

Time-dependent perturbation theory. Scattering. Interaction picture. Density matrix. Interaction of radiation with matter. Identical particles. Symmetry and conservation laws. The method of second quantization.

BIBLIOGRAPHY

1. J.J. Sakurai - Modern Quantum Mechanics, Addison-Wesley, 1994.
2. C. Cohen-Tanoudji, B. Diu e F. Laloê, Quantum Mechanics, 2 vols, Wiley, 1977.
3. A. Messiah : Quantum Mechanics, Dover.
4. P. A M. Dirac – The International Series of Monographs on Physics – 27: The Principles of Quantum Mechanics, 4a Ed Oxford Science pub, 1989.
5. A.F.R. de Toledo Piza: Mecânica Quântica (notas de aula, disponíveis na rede).

 

COURSE: Spinorial Calculus in General Relativity

CREDITS: 4

The real Minkowski space: topological characterization. The Minkowskian metric tensors. Contravariant and covariant world vectors. Mixed world vectors. Causal structure. The Lorentz group and its components. The homomorphism between SL(2,C) and the restricted ortochronous Lorentz subgoup. The fundamental space of two-component spinors. The metric spinors. Dual spaces and complex conjugate.  Conections between world tensors and spinors. Spinorial representations of the Minkowskian metric tensor and null vectors. Bivectors: definition and spinoral representation. Reduction formulas and computational manipulations. 

BIBLIOGRAPHY

1. W. L. Bade, H. Jehle, Rev. Mod. Phys., Vol. 25, 714 (1953).
2. H. Weyl, Z. Physik 56, 330 (1929). E. M. Corson, Introduction to Tensors, Spinors and Relativistic Wave Equations, Glasgow, Blackie 1953.
3. L. D. Landau. L. Lifchitz,, Théorie Quantique Relativiste, Première Partie, Edn. MIR, Moscou 1972.
4. R. Penrose, W. Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, Cambridge 1984.
5. M. Carmeli, S. Malin, Theory of spinors, An Introduction, World Scientific, Singapore, New Jersey, London, Hong Kong 2000.
6. R. Penrose, Ann. Phys. (N.Y.) 10, 171 (1960).
7. L. Witten, Phys. Rev., Vol. 113, 357 (1959).
8. J. G. Cardoso, Dissertação de Mestrado, UnB. (1979).

 

COURSE: Spinors in General Relativity

CREDITS: 4

Spinorial structure in general relativity. The metric spinors and conector objects. Tensors and spin densities. Spinorial affinities e covariant derivatives. Eigenvalue equations and metric expressions. Weyl gauge tranformations. Spinorial curvature. Covariant comutators and curvature spinors. Delta derivatives of tensors and spin densities.  

BIBLIOGRAPHY

1. W. L. Bade, H. Jehle, Rev. Mod. Phys., Vol. 25, 714 (1953).
2. R. Penrose, W. Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, Cambridge 1984.
3. R. Penrose, Ann. Phys. (N.Y.) 10, 171 (1960).
4. L. Witten, Phys. Rev., Vol. 113, 357 (1959).
5. R. Penrose, W. Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, Cambridge 1984.
6. J. G. Cardoso, Czech Journal of Physics, Vol. 55, 4, 401-462 (2005).
7. M. Carmeli, S. Malin, Theory of spinors, An Introduction, World Scientific, Singapore, New Jersey, London, Hong Kong 2000.

 

COURSE: Physics of Electrical Discharges I

CREDITS: 4

Electrical discharge system and generation of ionized gas. Collisional processes in ionized gases.  Townsend theory for gas breakdown. Paschen curve. Low pressure electrical discharges. Electrostatic sheaths. Langmuir probe. The positive column and its plasma properties. Radio-frequency electrical discharges. Discharges with external magnetic field. Introduction to plasma chemistry. 

BIBLIOGRAPHY

1. M. A. Lieberman, A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2a ed., Wiley Interscience, New York (2005).
2. B. Chapman, Glow Discharge Processes, John Wiley & Sons, New York (1980).
3. A. Fridman e L. A. Kennedy, Plasma Physics and Engineering, Taylor & Francis, New York (2004).

 

COURSE: Solid State Physics

CREDITS: 4

Crystal lattices. Phonons. Electronic structure in crystals. Semiconductors. Electron-electron interaction. Transport properties. Optical properties. Magnetism. Superconductivity. 

BIBLIOGRAPHY

1. J.M. Ziman, Principles of the Theory of Solids, 2a Ed., Cambridge Univ. Press, 1972.
2. N.W.Ashcroft, N.D. Mermin, Solid State Physics, Saunders College, 1976.
3. O. Madelung, Introduction to Solid-State Theory, Springer-Verlag, 2000.
4. C. Kittel, Introduction to Solid State Physics, 7a Ed., Wiley Text Books, 1995.
5. W. A. Harrison, Solid State Theory, Dover, 1980.

 

COURSE: Introduction to Classical Molecular Dynamics

CREDITS: 4

Atomic and molecular systems, crystal lattices. Interatomic potentials. Boundary conditions, minimum image criteria and truncation of potential. Verlet algorithm. Nosé and Nosé-Hoover dynamics. Determination of statistical quantities in microcanonical, canonical and isobaric-isothermal ensembles. Methods of energy minimization. Applications in different systems. 

BIBLIOGRAPHY

1. M. P. Allen e D. J. Tildesley, Computer Simulations of Liquids, Clarendon Press, Oxford (1987).
2. F. Ercolessi, Molecular Dynamics Primer, Spring College in Computational Physics, ICTP (1997). 
3. J. D. M. Vianna, A. Fazzio e S. Canuto, Teoria Quântica de Moléculas e Sólidos, Ed. Livraria da Física-SP (2004).
4. Manual do LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator): http://lammps.sandia.gov/.

 

COURSE: Introduction to Functional Density Theory

CREDITS: 4

Fundamental wave mechanics. Hartree-Fock approximation. Thomas-Fermi model. Density functional theory. Time-dependent density functional theory.

BIBLIOGRAPHY

1. R. G. Parr e W. Yang, Density-Functional Theory of Atoms and Molecules. Oxford University Press, 1989.
2. E. Engel e R. M. Dreizler, Density Functional Theory: An advanced course. Springer, 2011.
3. M. A. L. Marques et al., Time-Dependent Density Functional Theory. Springer, 2006.
4. C. Fiolhais, F. Nogueira e M. A. L. Marques, A Primer in Density-Functional Theory. Springer, 2003.

 

COURSE: Magnetism

CREDITS: 4

Paramagnetism. Interacting spins. Mean field approximation. Spin waves. Green functions methods. Dipolar interactions. Itinerant magnetism. RKKY interaction. Kondo effect. Spin glasses.

BIBLIOGRAPHY

1. R. M. White, Quantum Theory of Magnetism, Springer-Verlag 1983.
2. N. Majlis, Quantum Theory of Magnetism, World Scientific 2000.
3. K. Yosida, Theory of Magnetism, Springer-Verlag 1988.
4. A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge Univ. Press 1997.
5. K. H. Fischer e J. A. Hertz, Spin Glasses, Cambridge Univ. Press 1993.

 

COURSE: Relativistic Quantum Mechanics

CREDITS: 4

Special relativity review. Klein-Gordon equation. Non-covariant and covariant forms of Dirac equation. Electron spin. Plane wave solutions of Dirac equation. Representations of the homogeneous Lorentz group. Solutions of positive and negative energies. Particles and antiparticles. Hole theory. Vacuum polarization. Conjugation of electric charge. Parity and temporal reversion. Dirac current density. Helicities of electrons and positrons. Propagation of Dirac particles in electromagnetic fields. 

BIBLIOGRAPHY

1. M. E. Rose, Relativistic Electron Theory, Wiley, New York 1961.
2. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, Mc Graw-Hill, New York 1964.
3. L. D. Landau et L. Lifchitz, Théorie Quantique Relativiste, Première Partie, Edn. MIR, Moscou 1972.

 

COURSE: General Relativity

CREDITS: 4

Curved space-time. World tensors. Symmetry operations. Parallel displacement of world tensors. Affinities and covariant derivatives of world tensors. Geodesics. World tensorial densities. Covariant derivatives of world densities. Ricci comutators. Condition to torsion absence. Tensors of curvature. Mach principle. Postulates of the General Relativity. Einstein equations. The variational principle of Einstein-Hilbert. Theory predictions. Schwarzschild solution and analytical extension. Kerr solution. Correspondence principle. Linear approximation. Gravitational waves. Cosmological model of Einstein. Cosmological model of Friedman-Robertson-Walker.

BIBLIOGRAPHY

1. Robert Wald, General Relativity, University of Chicago Press, 1984.
2. L. Landau e E. Lifschtz, Teoria do Campo, Física Teórica, vol. 2, Mir,1972.
3. S. Weinberg, Gravitation and Cosmology : Principles and Applications of the General Theory of Relativity, John Wiley & Sons, 1972.
4. Wolfgand Rindler, Relativity: Special General & Cosmological, Oxford University Press, 2001.

 

COURSE: Dynamical Systems

CREDITS: 4

Unidimensional maps: logistic map, bifurcations. Periodic, quasi-periodic and chaotic attractors. Fractal borders. Continuous time systems: conservatives and dissipatives, fixed points, notions of stability, Lyapunov exponents. Stable and unstable manifolds.

BIBLIOGRAPHY

1. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, 1993.
2. Alligood, T.D. Sauer, J.A. Yorke, Chaos, An lntroduction to Dynamical Systems, Springer, 1997.
3. Luiz Henrique Alves Monteiro, Sistemas Dinâmicos, Livraria da Física, 2002.
4. N. Fiedler-Ferrara e C.P. Cintra do Prado, Caos: Uma Introdução, Edgard Blücher, 1994.
5. W.F. Wreszinski, Mecânica Clássica Moderna , EDUSP, São Paulo, 1997.

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Optative courses

 

COURSE: Special Topics in Physics I

CREDITS: 2

Experimental and/or theoretical techniques used in the study of frontier areas in contemporary physics or in topics not covered in the list of elective courses. Each time this discipline is taught, the professor must present the course topics with the bibliography to be adopted.

 

 

COURSE: Special Topics in Physics II

CREDITS: 4

​Experimental and/or theoretical techniques used in the study of frontier areas in contemporary physics or in topics not covered in the list of elective courses. Each time this discipline is taught, the professor must present the course topics with the bibliography to be adopted.