MATHEMATICAL METHODS FOR MATERIALS SCIENCE
Learning outcomes of the course unit
To provide the student with the mathematical basis for the study of vector spaces and linear operators in classical physics and quantum mechanics
Course contents summary
Finite dimensional vector spaces.
Linearity and non linearity in physics. Vector spaces and subspaces. Linear dependence, basis, components and dimension. Change of basis. Sum, direct sum and intersection of subspaces. Direct products.
Matrices and determinants
Definitions. Algebra of matrices. Commutators and anticommutators. Determinants, properties and calculation techniques. Permutations and cycles decomposition. Characteristic polynomial.
Euclidean and unitary spaces.
Distances, norms, scalar and unitary products. Metric spaces, normed vector spaces, unitary and Euclidean spaces. Euclidean and chemical distance. Basic inequalities, angles, ortogonality and ortonormal systems. Grahm-Schmidt procedure. Metrical matrix, change of basis between ortonormal systems. Projections. Ortogonal complement.
Linear functionals and operators
Omomorphisms between vector spaces. Operators algebra. Matrix representation of a linear operator. Similarity and change of basis. Invariants in the characteristic polynomial. Invariant subspaces. Eigenvectors and eigenvalues. Spectral representation. Projectors.
Linear functionals and operators on unitary spaces
Dual space, Dirac notation. Bras and kets. Reisz-Fisher theorem. Complete systems. Hermitian conjugate. Hermitian, unitary and normal operators and their spectral representation. Commuting operators. Operators that can be diagonalized on the same basis.
Infinite dimensional vector spaces
Definitions. Separable Hilbert spaces. Complete ortonormal systems. Fourier series. Bounded and non bounded functionals and operators. Hermitian, self-adjoint, isometric and unitary operators. Eigenvectors and eigenvalues. Resolvent. Discrete, continuous and residual spectrum.
Number of fermions and bosons, eigenvalues and eigenvectors, creations and annihilation operators. Harmonic oscillations of a network. Spectrum of the Laplacian. Regular lattices and translation invariance. Huckel theory. Oscillation of a chain with simple and composite basis. Normal modes in biatomic molecules. Dispersion relations. Optical and acustic modes. Ortonormal polynomials. Recurrence relation e Rodrigues relation. Legendre, Hermite and Laguerre polynomials.
E. Onofri. 'Lezioni sulla teoria degli operatori Lineari', Edizioni Zara (1984)
H. Margenau, G.M. Murphy 'The mathematics of Physics and Chemistry', Van Nostrand, Princeton (1961)