Mathematical Physics 2018/19
Laurea magistrale in Fisica, I anno, II semestre, 6 CFU (48 ore di lezione)
NOTICE:
DATES OF THE EXAMS
Tuesday, May 7, at 14.30 (straordinario)
Tuesday, June 25, at 9.00
Tuesday, July 16, at 9.00
Tuesday, September 10, at 9.00
Thursday, September 19, at 9.00
Tuesday, November 19, at 14.30 (straordinario)
Syllabus
1. Ordinary differential equations
Existence, uniqueness and continuity w.r. to data.
Properties of the flow.
Liouville's equation.
Equilibria and stability.
2. Hamiltonian systems
Basic properties.
Hamilton-Jacobi equation.
Action-angle variables.
First order perturbation theory.
3. Topics in Quantum Mechanics
Elements of operator theory.
Formulation of the rules.
Free particle.
Scattering theory.
Detailed program (notice: w.p. means without proof)
1. Ordinary differential equations
Existence and uniqueness theorem (w.p.), Gronwall lemma, continuity w.r. to initial conditions.
Evolution of the derivative of the flow and of its determinant, Liouville equation.
Equilibrium position, stable and asymptotically stable equilibrium position, Lyapunov theorem, stability and instability recognized from the linearized system (w.p.), examples and applications in mechanical systems.
Prey/predator model.
[HS] ch. 8, 9 and 12 (sect. 2), [T1] sect 1.7 (see also [E] ch. 2 and sect. 10.5, or [B] ch. 2).
2. Hamiltonian systems
Derivation of Hamilton's equations, variational principle, constant of motions, Poisson brackets.
Definition of canonical and completely canonical transformations, examples.
Criterion via Poisson brackets, extended phase space, sufficient condition via differential form, generating functions, examples.
Hamilton-Jacobi equation for the time-dependent and time-independent case, separation of variables, action-angle variables.
Liouville and Arnold-Liouville theorems (w.p.).
First order perturbation theory.
Application: action-angle variables for the planar Kepler problem, restricted planar three body problem, precession of the perihelion of Mercury.
[T1] sect. 1.1,...,1.5, [FM] sect. 11.4, 11.5, notes on the Kepler problem and the restricted three body problem distributed to students (see also [E] ch. 12 and 13, sect. 14.1, 14.2; [Ce] ch. 7, sections MC1,...,MC5).
3. Topics in Quantum Mechanics
- Elements of operator theory
Def. of bounded and unbounded operator.
Def. of adjoint, symmetric and self-adjoint operators, self-adjointness criterion, examples: position and momentum operators Q, P.
Def. of perturbation of a self-adjoint operator, Kato-Rellich theorem (w.p.).
Def. of spectrum, the spectrum of a self-adjoint operator is real (w.p.), examples: spectrum of Q,P.
Def. of isometric and unitary operators, examples.
Spectral theorem (w.p.), function of a self-adjoint operator (w.p.), examples.
Def. of strongly continuous unitary group, solution of the Schroedinger equation.
Def. of point spectrum, absolutely continuous spectrum, singularly continuous spectrum.
[T1] sect. 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10.
- Formulation of the rules
[T1] sect. 5.1
- Free particle
Def. of the Hamiltonian.
Spectrum, resolvent (w.p.), unitary group, asymptotic behaviour of the unitary group for t large.
[T1] sect. 6.1, 6.2.
- Scattering theory
Formulation of the problem.
Wave operators.
Asymptotic completeness and scattering operator.
Scattering into cones.
Existence of wave operators.
Generalized eigenfunctions (w.p.) and eigenfunction expansion theorem (w.p.).
Proof of asymptotic completeness.
Representation of the S matrix.
[T2] sect. 1, 2, 3, 4, 5, 6, 7, 8, 9.
Exercises for the exam (see file pdf below)
References
[E] R. Esposito, Appunti dalle lezioni di meccanica razionale, ed. Aracne
[B] P. Butta', Note di fisica matematica http://www1.mat.uniroma1.it/~butta/didattica/note_FM.pdf
[HS] M. Hirsch and S Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press 1974
[FM] A. Fasano and S. Marmi, Analytical Mechanics, Oxford Univ. Press 2006
[T1] A. Teta, A Mathematical Primer on Quantum Mechanics, Springer 2018
[T2] A. Teta, Notes on Scattering Theory, see file in Appunti ed esercizi
[Ce] A. Celletti, Esercizi e complementi di meccanica razionale, ed. Aracne
[D] Dell'Antonio, Elementi di meccanica, ed. Liguori