Physical and mathematical fundamentals for biophysics

06 Nov 2025 (2h 00m)

STEFANO Luin - Corso (attività didattica) - Mista

Functions. Cartesian plane, points, lines; line through two points. Derivatives, differentials; chain rule. Potential energy (U) and forces (F) in 1D: from the U to the F graphs. Kinetic and total energies for a mass point. Partial and total derivatives of functions of more than one variable. Dot-notation for total derivatives with respect to times. Complex numbers, conjugation, modulus; imaginary exponentials, trigonometric functions, unit circle, complex plane.

13 Nov 2025 (2h 00m)

STEFANO Luin - Corso (attività didattica) - Mista

Euler identity and identities in trigonometry. Many-valued root functions in the complex field; multiplication as "rotation" in the complex plane. Polynomials. Integral as the "inverse" of derivative; integrals of elementary functions. Linear differential equations: checking their solutions (and determining their parameters); solutions for the classical and damped harmonic oscillator.

20 Nov 2025 (2h 00m)

STEFANO Luin - Corso (attività didattica) - Mista

Fourier transforms, anti-Fourier transforms, translations in direct space, Dirac Delta for proving antiFourier transform. Approximations in physics and their validity. Quantum wells. Meaning of time and space phase changing in wavefunctions. Origin of boundary conditions in steps-potential 1D problems; validity of infinite well approximation. Dirac delta; boundary conditions with Dirac-delta-like potentials.

27 Nov 2025 (2h 00m)

STEFANO Luin - Corso (attività didattica) - Mista

Solutions for 1D Schrödinger equation with steps-potentials; exact solution for the ground state wavefunction of a negative-Dirac-delta potential. Theoretical basis on algebraic groups. Exercise on 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). Bra-Ket notation.

04 Dic 2025 (2h 00m)

STEFANO Luin - Corso (attività didattica) - Mista

Cosine of angle amongst two vectors; equivalence of definitions in R^2. Versors, orthonormal bases and orthonormalization of a basis. Determination of the components of a vector with orthonormal bases. Operators and functionals: fundamentals, nonlinear and linear operators; matrices as linear operators, product of matrices (hints); bra as a functional. Infinite dimensional vectorial spaces.

11 Dic 2025 (2h 00m)

STEFANO Luin - Corso (attività didattica) - Mista

Discussion of exam modalities. Hermitian operators and observables. 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.