# Physical and mathematical fundamentals for biophysics

## Registro delle lezioni

Docente Stefano Luin

## Lecture

• 10 Nov 2023 (2h 00m)

Stefano Luin - Corso (attività didattica) - In presenza

Cartesian plane, points, lines; line passing through two points. Derivatives, differentials, derivatives of elementary functions and of composition of functions (chain rule). Integral as the "inverse" of derivative. Potential energy (U) and forces (F) in 1D: from the U to the F graphs. Partial and total derivatives of functions of more than one variable. Dot-notation for total derivatives with respect to times. Exercises. Complex numbers, conjugation, modulus; exercises on imaginary exponentials, trigonometric functions and identities, unit circle, complex plane.

• 14 Nov 2023 (2h 00m)

Stefano Luin - Corso (attività didattica) - In presenza

Many-valued root functions in the complex field; multiplication as "rotation" in the complex plane. Solutions for the classical harmonic oscillator; derivatives of sums and products of functions; complex "waves" for representing classical ones; checking the solution (and determining their parameters) of differential equation. Total energy of a mass point. Scalar and vectorial fields; gradient and divergence. Isovalue surfaces and field lines.

• 28 Nov 2023 (2h 00m)

Stefano Luin - Corso (attività didattica) - Mista

Exercises on gradient and divergence, derivative operation on compositions and products of functions. Coulomb force and potential. Meaning of time and space phase changing in wavefunctions. Approximations in physics and their validity; origin of boundary conditions in steps-potential 1D problems; validity of infinite well approximation. Dirac delta and Heaviside theta. Boundary conditions with Dirac-delta-like potentials; exact solution for the ground state wavefunction of a negative-Dirac-delta potential.

• 05 Dic 2023 (2h 00m)

Stefano Luin - Corso (attività didattica) - Mista

Exercise: unbound states for negative Dirac-delta potential; trasmittivity and reflectivity. Checking results of differential equations. Theoretical basis on algebraic groups. Exercises (made interactively) on: demonstration of unicity; invertible functions on a domain as a group with the operator of function composition, inverse. Vectorial spaces: formal properties; scalar product (also in complex field); norm. Space spanned by a set of vectors. Vectorial space dimension and independent vectors (exercise). Definition of cosine of angle amongst two vectors.

• 12 Dic 2023 (2h 00m)

Stefano Luin - Corso (attività didattica) - Mista

Equivalence of scalar product definitions (by components and by cosine between two vectors) in R^2 (exercise). Orthonormal bases and orthonormalization of a basis (exercise). Kronecker delta and Dirac delta. Determination of the components of a vector with orthonormal bases. Infinite dimensional vectorial spaces, with discrete and continuous bases (hints). Fourier series as a basis for L^2 periodic functions. Bra-ket notation; operators and functionals: fundamentals, nonlinear and linear operators; matrices (hints). Sum with Einstein notation.

• 19 Dic 2023 (2h 00m)

Stefano Luin - Corso (attività didattica) - Mista

Hermitian operators and observables: an "historical" point of view. Recall of measurements in quantum mechanics, expected values. Exercise: sequential measurements of non-commuting observables, probabilities of the results of measurements. Negative-delta potential: estimate of ground state energy and typical wavefunction dimension (exercise); average value of an operator on a wavefunction.

• 16 Gen 2024 (2h 00m)

Stefano Luin - Corso (attività didattica) - In presenza

Calculating the potential knowing an eigenfunction. Vectorial product, Levi-Civita tensor, determinant of a matrix (hints). Curl. Antisymmetrization and symmetrization as projection operators. Double quantum wells (delta-like potential) solved as in perturbative theory. Sx in Sz eigenvector basis.

• 30 Gen 2024 (2h 00m)

Stefano Luin - Corso (attività didattica) - In presenza

Properties of commutators. Commutators of the components of angular momentum, and of the components of position with the ones of angular momentum; definition of vectorial operator (hints). Average value and probability distribution of p on the wavefunction theta(x)exp(-kx). Lorentzian distribution, recall on Dirac delta, Heavyside theta, and Fourier transform.

• 21 Mar 2024 (2h 00m)

Stefano Luin - Corso (attività didattica) - In presenza

Exam "simulation": Stern-Gerlach experiment; transmission and reflection from 1D barrier; expectation value and eigenvalues in measurement process.

• 10 Apr 2024 (2h 00m)

Stefano Luin - Corso (attività didattica) - In presenza

Exam simulation and recalls: Hydrogen molecule; Pauli exclusion principle. Raman and infrared absorption selection rules. Photovoltaic experiment. Born-Oppenheimer approximation. Estimation of energy and length scale for the hydrogen atom.