Unit QUANTUM MECHANICS
 Course
 Physics
 Studyunit Code
 GP005464
 Curriculum
 In all curricula
 Teacher
 Gianluca Grignani
 CFU
 12
 Course Regulation
 Coorte 2019
 Offered
 2021/22
 Type of studyunit
 Obbligatorio (Required)
 Type of learning activities
 Attività formativa integrata
QUANTUM MECHANICS  MOD. I
Code  GP005471 

CFU  6 
Teacher  Gianluca Grignani 
Teachers 

Hours 

Learning activities  Caratterizzante 
Area  Teorico e dei fondamenti della fisica 
Academic discipline  FIS/02 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  The crisis of classical physics. Black body theory. Aspects of corpuscular radiation. Atomic models. Heisenberg's uncertainty principle. Wave packet and waveparticle duality. Schroedinger wave mechanics. Eigenvalue equation for Hermitian operators. Pure states. Properties of the discrete spectrum of nondegenerate Hermitian operators. Degenerate discrete spectrum. Continuous spectrum. Mixed spectrum. Unitarity transformations. Time evolution of the mean values ¿¿of physical observables. Stationary states and their properties. Onedimensional problems. Potential well. Onedimensional quantum harmonic oscillator. 
Reference texts  author: Cesare Rossetti, title: Rudimenti di Meccanica Quantistica, editor: Levrotto & Bella. alternatively one can also use the following book: authors: L.D. Landau and E.M. Lifsits, title: Meccanica quantistica, teoria non relativistica, Vol. III, editor: Editori Riuniti. (There is also the english version). 
Educational objectives  The course is the first module of Quantum Mechanics. The main objective is to provide students with the basic knowledge necessary to address the study of quantum mechanics. The main knowledge gained will be: Basic facts about the crisis of classical physics and the theory of black body radiation. Atomic models. Concept of wave function. Schroedinger wave mechanics. Eigenvalue equations. Properties of spectra for Hermitian operators. Time evolution of mean values ¿¿of physical observables. Unidimensional problems. Quantum treatment of the onedimensional harmonic oscillator. The main skills acquired will be: Knowing how to solve the eigenvalue equation for the Hamiltonian. Knowing how to treat onedimensional problems. Knowing how to solve the eigenvalue equation for the Hamiltonian of the harmonic oscillator. 
Prerequisites  Mathematical methods for physicists 
Teaching methods  face to face lecture and exersises 
Other information  none 
Learning verification modality  Written and oral exam 
Extended program  The crisis of classical physics. Study of electromagnetic radiation in a isotherm cavity. Kirchhoff theorem. StefanBoltzmann's theorem. Wien’s result. RayleighJeans law. Planck's quantization theory. Black body theory. Aspects of corpuscular radiation. Photoelectric effect and Einstein’s interpretation. Compton effect. Emission spectrum of hydrogenic atoms and the combination’s principle of RydbergRitz. Atomic models. Bohr’s atomic model. Frank and Hertz experiment. Stern and Gerlach experiment. Wave aspects of particles. Matter waves. Davisson and Germer experiment. Specific heat at constant volume of crystalline solids and Einstein model. WilsonSommerfeld rules for the hydrogen atom. Bohr’s correspondence principle. Heisenberg's uncertainty principle. Wave packet and waveparticle duality. Bohr and Einstein experiments. Bohr’s complementarity principle. Schroedinger’s wave mechanics and Heisenberg’s matrix mechanics. Superposition principle. Timedependent Schroedinger equation. Analogy between wave mechanics and the Schroedinger’s wave optics. Analogy between geometrical optics and classical mechanics. Solution of the Schroedinger equation for free particles. Interpretation of the Fourier transform of the wave function. Wave function representation in k space. Mean values ¿¿of physical observables. Properties of quantum commutators. Fundamental commutations relations. Eigenvalue equation for Hermitian operators. Pure states. Properties of the discrete spectrum of nondegenerate Hermitian operators. Degenerate discrete spectrum. Continuous spectrum. Mixed spectrum. Simultaneous measurement of two physical observables and common system of eigenfunctions. Unitarity transformations. Time evolution of mean values ¿¿of physical observables. Time evolution of the wave packet. Time evolution operator and solutions of the Schroedinger equation in the case of timeindependent Hamiltonian. Stationary states and their properties. Eigenvalue equation for free particles in space. Hamiltonian with separable variables. Study of onedimensional problems: the case of the free particle. Scattering of the wave packet for free particle. Scattering of the wave packet of minimum uncertainty. Generic potential. General properties of onedimensional systems. Transmission coefficient and reflection coefficient. Step potential of infinite height. Barrier potential. Tunnel effect. Alpha decay of heavy nuclei. Gamow factor. Potential well of infinite depth. Calculation of the mean values ¿¿of x, x ^ 2, p and p ^ 2. Onedimensional quantum harmonic oscillator. Algebraic method for the harmonic oscillator. Lowering and raising operators. Matrix representation of physical observables on the basis of eigenstates of the harmonic oscillator. 
QUANTUM MECHANICS  MOD. II
Code  GP005470 

CFU  6 
Teacher  Gianluca Grignani 
Teachers 

Hours 

Learning activities  Caratterizzante 
Area  Teorico e dei fondamenti della fisica 
Academic discipline  FIS/02 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  Eigenvalue equation for angular momenta. Threedimensionlal problems, central potentials, isotropic harmonic oscillator and hydrogenoid atoms. Time independent and time dependent perturbation theory. Fine structure of hydrogenoid atoms. Zeeman effect 
Reference texts  L. D. Landau and E. M. Lifshitz, Meccanica Quantistica, Editori Riuniti (2010) C. Rossetti, Rudimenti di Meccanica Quantistica, Levrotto e Bella, Torino, (2011) 
Educational objectives  This course represents the second part of Quantum Mechanics. The main aim of this teaching is to provide students with the bases needed to address and solve the most important problems in quantum mechanics. Main knowledge acquired will be: Knowledge of the solutions of the eigenvalues equations for angular momentum operators. Knowledge of series solutions of second order differential equations. Knowledge of the exact solutions of the Schrödinger equation for central potentials as the isotropic harmonic oscillator and the hydrogenoid atom. Perturbative and variational methods. Fine structure of hydrogenoid atoms. The main competence (i. e. the ability to apply the acquired knowledge) will be: Solutions of the eigenvalue equation for three dimensional Hamiltonians. Treatment of the eigenvalue problem with central potentials. Evaluation of the solution of the Schrödinger equation for the hydrogenoid atom. 
Prerequisites  Quantum Mechanics Module I and Mathematical Methods for Physicists. 
Teaching methods  Lectures and exercises 
Other information  none 
Learning verification modality  written and oral exam. 
Extended program  Angular momentum operators Li and their commutators. Eigenvalues and eigenvectors of L^2 and Lz. Derivation of the eigenvalues of J^2 and Jz (for a general angular momentum J¿) with the matrix method: operators J+ and J. Threedimensional problems. Separation of variables in Cartesian and spherical coordiantes. Radial equation and its treatment for a generic potential. Isotropic harmonic oscillator. Two body problem. Separation of the center of mass motion. Hydrogenoid atoms: energy eigenvalues and eigenfunctions. Intrinsic angular momentum: spin. Pauli's theory of spin. Angular momentum composition. ClebshGordan coefficients. Identical particles and their indistinguibility in a quantum theory. Bosons and Fermions. Pauli exclusion principle. Exclusion principle and periodic table of the elements. Time independent perturbation theory. Eigenvalues and eigenfunctions at the lowest perturbative order. An introduction to variational methods. Time dependent perturbation theory. Transition probability and Fermi's golden rule. Fine structure of hydrogenoid atoms. Selection rules. Semiclassic approximation and W.K.B method. Zeeman effect. 