Unit QUANTUM MECHANICS
- Course
- Physics
- Study-unit Code
- GP005464
- Curriculum
- In all curricula
- Teacher
- Gianluca Grignani
- CFU
- 12
- Course Regulation
- Coorte 2019
- Offered
- 2021/22
- Type of study-unit
- 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 study-unit | 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 wave-particle duality. Schroedinger wave mechanics. Eigenvalue equation for Hermitian operators. Pure states. Properties of the discrete spectrum of non-degenerate 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. One-dimensional problems. Potential well. One-dimensional 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 one-dimensional harmonic oscillator. The main skills acquired will be: Knowing how to solve the eigenvalue equation for the Hamiltonian. Knowing how to treat one-dimensional 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. Stefan-Boltzmann's theorem. Wien’s result. Rayleigh-Jeans 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 Rydberg-Ritz. 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. Wilson-Sommerfeld rules for the hydrogen atom. Bohr’s correspondence principle. Heisenberg's uncertainty principle. Wave packet and wave-particle duality. Bohr and Einstein experiments. Bohr’s complementarity principle. Schroedinger’s wave mechanics and Heisenberg’s matrix mechanics. Superposition principle. Time-dependent 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 non-degenerate 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 time-independent Hamiltonian. Stationary states and their properties. Eigenvalue equation for free particles in space. Hamiltonian with separable variables. Study of one-dimensional 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 one-dimensional 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. One-dimensional 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 study-unit | Obbligatorio (Required) |
| Language of instruction | Italian |
| Contents | Eigenvalue equation for angular momenta. Three-dimensionlal 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-. Three-dimensional 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. Clebsh-Gordan 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. |