Lecture Notes in Quantum Mechanics - BGU

1 Lecture Notes in Quantum MechanicsDoron CohenDepartment of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel(arXiv:quant-ph/0605180)These are the Lecture notesof Quantum Mechanics coursesthat are given by DC at Ben-Gurion University. They cover textbook topics that are listed below, and also additional advancedtopics (marked by *) at the same level of I The classical description of a particle Hilbert space formalism A particle in anNsite system The continuum limit (N= ) Translations and rotationsFundamentals II Quantum states / EPR / Bell The 4 postulates of the theory The evolution operator The rate of change formula Finding the Hamiltonian for a physical system The non-relativistic Hamiltonian The classical equation of motion Symmetries and constants of motionFundamentals III Group theory , Lie algebra Representations of the rotation group Spin 1/2, spin 1 andY`,m Multiplying representations Addition of angular momentum (*) The Galilei group (*) Transformations and invariance (*)

2 Dynamics and driven systems Systems with driving The interaction picture The transition probability formula Fermi golden rule Markovian master equations Cross section / Born The adiabatic equation The Berry phase theory of adiabatic transport (*) Linear response theory and Kubo (*) The Born-Oppenheimer picture (*)The Green function approach (*) The evolution operator Feynman path integral The resolvent and the Green function Perturbation theory for the resolvent Perturbation theory for the propagator Complex poles from perturbation theoryScattering theory (*) Scattering:Tmatrix formalism Scattering:Smatrix formalism Scattering:Rmatrix formalism Cavity with leads mesoscopic geometry Spherical geometry, phase shifts Cross section, optical theorem, resonancesQuantum Mechanics in practice The dynamics of a two level system Fermions and Bosons in a few site system (*) Quasi 1D network systems (*) Approximation methods forHdiagonalization Perturbation theory forH=H0+V Wigner decay, LDOS, scattering resonances The Aharonov-Bohm effect Magnetic field (Landau levels, Hall effect) Motion in a central potential, Zeeman The Hamiltonian of spin 1/2 particle, implicationsSpecial Topics (*) Quantization of the EM field Fock space formalism The Wigner Weyl formalism theory of Quantum measurements theory of Quantum computation The foundations of Statistical Mechanics2 Opening remarksThese Lecture Notes are based on 3 courses in non-relativistic Quantum Mechanics that are given at BGU.

3 Quantum 2 (undergraduates), Quantum 3 (graduates), and Selected topics in Quantum and Statistical Mechanics (graduates).The Lecture Notes are self contained, and give theroad mapto Quantum Mechanics . However, they do not intend tocome instead of the standard textbooks. In particular I recommend:[1] , Quantum Mechanics (library code: QC ).[2] Sakurai, Modern Quantum Mechanics (library code: QC ).[3] Feynman Lectures Volume III.[4] A. Messiah, Quantum Mechanics . [for the graduates]The major attempt in this set of lectures was to give a self contained presentation of Quantum Mechanics ,which is notbased on the historical quantization approach. The main inspiration comes from Ref.[3] and Ref.[1]. The challengewas to find a compromise between the over-heuristic approach of Ref.[3] and the too formal approach of Ref.[1].Another challenge was to give a presentation of scattering theory that goes well beyond the common undergraduatelevel, but still not as intimidating as in Ref.

4 [4]. A major issue was toavoid the over emphasis on spherical language that I use is much more suitable for research with mesoscopic those who look for original or advanced pedagogical pieces: The EPR paradox, Bell s inequality,and the notion of Quantum state; The 4 postulates of Quantum Mechanics ; Berry phase and adiabatic processes; Linearresponse theory and the Kubo formula; Wigner-Weyl formalism; Quantum measurements; Quantum computation;The foundations of Statistical Mechanics . Note also the following example problems: Analysis of systems with 2or 3 or more sites; Analysis of the Landau-Zener transition; The Bose-Hubbard Hamiltonian; Quasi 1D networks;Aharonov-Bohm rings; Various problems in scattering topics are covered by:[5] D. Cohen, Lecture Notes in Statistical Mechanics and Mesoscopic, first drafts of these Lecture Notes were prepared and submitted by students on a weekly basis during students were requested to use HTML with ITEX formulas.

5 Typically the text was written in were requested to use Latex. The drafts were corrected, integrated, and in many cases completely re-writtenby the lecturer. The English translation of the undergraduate sections has been prepared by my former studentGiladRosenberg. He has also prepared most of the illustrations. The current version includes further contributions by myPhD studentsMaya ChuchemandItamar Sela. I also thank my colleague Bandfor some commentson the text. The arXiv versions are quite remote from the original (submitted) drafts, but still I find it appropriateto list the names of the students who have participated: Natalia Antin, Roy Azulai, Dotan Babai, Shlomi Batsri,Ynon Ben-Haim, Avi Ben Simon, Asaf Bibi, Lior Blockstein, Lior Boker, Shay Cohen, Liora Damari, Anat Daniel,Ziv Danon, Barukh Dolgin, Anat Dolman, Lior Eligal, Yoav Etzioni, Zeev Freidin, Eyal Gal, Ilya Gurwich, DavidHirshfeld, Daniel Hurowitz, Eyal Hush, Liran Israel, Avi Lamzy, Roi Levi, Danny Levy, Asaf Kidron, Ilana Kogen,Roy Liraz, Arik Maman, Rottem Manor, Nitzan Mayorkas, Vadim Milavsky, Igor Mishkin, Dudi Morbachik, ArielNaos, Yonatan Natan, Idan Oren, David Papish, Smadar Reick Goldschmidt, Alex Rozenberg, Chen Sarig, Adi Shay,Dan Shenkar, Idan Shilon, Asaf Shimoni, Raya Shindmas, Ramy Shneiderman, Elad Shtilerman, Eli S.

6 Shutorov,Ziv Sobol, Jenny Sokolevsky, Alon Soloshenski, Tomer Tal, Oren Tal, Amir Tzvieli, Dima Vingurt, Tal Yard, UziZecharia, Dany Zemsky, Stanislav (part I)1 Introduction52 Digression: The classical description of nature93 Hilbert space134 A particle in anNsite system205 The continuum limit226 Rotations28 Fundamentals (part II)7 Quantum states / EPR / Bell / postulates338 The evolution of Quantum mechanical states439 The non-relativistic Hamiltonian4710 Getting the equations of motion52 Fundamentals (part III)11 Group representation theory6012 The group of rotations6613 Building the representations of rotations7014 Rotations of spins and of wavefunctions7315 Multiplying representations8116 Galilei group and the non-relativistic Hamiltonian9017 Transformations and invariance92 Dynamics and Driven Systems18 Transition probabilities9819 Transition rates10220 The cross section in the Born approximation10421 Dynamics in the adiabatic picture10722 The Berry phase and adiabatic transport11123 Linear response theory and the Kubo formula11724 The Born-Oppenheimer picture120 The Green function approach25 The propagator and Feynman path integral12126 The resolvent and the Green function12527 Perturbation theory13528 Complex poles from perturbation theory1404 Scattering Theory29 The plane wave basis14330 Scattering in theT-matrix formalism14631 Scattering in theS-matrix formalism15332 Scattering in quasi 1D geometry16333 Scattering in a

7 Spherical geometry171QM in Practice (part I)34 Overview of prototype model systems18035 Discrete site systems18136 Two level dynamics18237 A few site system with Bosons18638 A few site system with Fermions18939 Boxes and Networks191QM in Practice (part II)40 Approximation methods for finding eigenstates19641 Perturbation theory for the eigenstates20042 Beyond perturbation theory20643 Decay into a continuum21044 Scattering resonances219QM in Practice (part III)45 The Aharonov-Bohm effect22346 Motion in uniform magnetic field (Landau, Hall)23147 Motion in a central potential24048 The Hamiltonian of a spin 1/2 particle24449 Implications of having spin 247 special Topics50 Quantization of the EM Field25151 Quantization of a many body system25652 Wigner function and Wigner-Weyl formalism26753 Quantum states, operations and measurements27554 theory of Quantum computation28755 The foundation of statistical mechanics2975 Fundamentals (part I)[1] Introduction======[ ] The building blocks of the universeThe universe consists of a variety of particles which are described by the standard model.

8 The known particles aredivided into two groups: Quarks: constituents of the proton and the neutron, which form the 100 nuclei known to us. Leptons: include the electrons, muons, taus, and the interaction between the particles is via fields (direct interaction between particles is contrary to the principlesof the special theory of relativity ). These interactions are responsible for the way material is organized . Weshall consider in this course the electromagnetic interaction. The electromagnetic field is described by the Maxwellequations. Within the framework of the standard model there are additional gauge fields that can be treated onequal footing. In contrast the gravity field has yet to be incorporated into Quantum [ ] A particle in an electromagnetic fieldThis section is inteted for 3rd year BSc Physics students: its purpose is to place this course in the context of classicalanalytical Mechanics . Those who do not have this eduction can skip sections [ ]-[ ]-[ ].

9 A terse summary ofclassical Mechanics is provided in Lecture [2], and can be skipped as the framework of classical electromagnetism, the electromagnetic field is described by the scalar potentialV(x) and the vector potential~A(x). In addition one defines:B= ~A( )E= 1c ~A t VWe will not be working with natural units in this course, but from now on we are going to absorb the constantscandein the definition of the scalar and vector potentials:ecA A,eV V( )ecB B,eE EIn classical Mechanics , the effect of the electromagnetic field is described by Newton s second law with the Lorentzforce. Using the above units convention we write: x=1m(E B v)( )The Lorentz force dependents on the velocity of the particle. This seems arbitrary and counter intuitive, but we shallsee in the future how it can be derived from general and fairly simple analytical Mechanics it is customary to derive the above equation from a Lagrangian. Alternatively, one can use aLegendre transform and derive the equations of motion from a Hamiltonian: x= H p( ) p= H x6where the Hamiltonian is:H(x,p) =12m(p A(x))2+V(x)( )The Hamiltonian that describes a system of several charged particles in 3 dimensional space, including the electro-magnetic field in the Coulomb gauge, can be written as follows:H(r,p,A,E) = i12mi(pi eiA(ri))2+ ij eiej|ri rj|+18 (E2 +c2( A)2)d3x( )The canonical coordinates of the particles are (ri,pi), and the canonical coordinates of the radiations field are (A,E ).

10 The magnetic field is defined asB= A. One can define an electrostatic electric fieldE , and express the secondterm as an integral overE2 /(8 ).The units ofEas well as the prefactor 1/(8 ) are determined via Coulomb law as in the Gaussian CGI units ofAare determined as in the SI convention, namely, we do not make here the replacementA7 (1/c)A,and therefore the equations of motion for the radiation field imply thatE = A. AccordinglyBandEdo not havethe same units, and the Lorentz force formula does not included (1/c) the absence of particles the radiation term of the Hamiltonian describes waves that have a dispersion relation =c|k|. The strength of the interaction is determined by the coupling constantsei. Assuming that all the particleshave elementary chargeei= e, it follows that after canonical quantization (see below) the above Hamiltonian ischaracterized by a single dimensionless coupling constante2/(~c), which is knows as the fine-structure constant.