Lecture Notes 

Contents

   LECTURE-0   DESCRIPTION OF MECHANICAL MOTION WITHOUT USING TRAJECTORIES                                                                (09-26-2023)
                                Trajectories imply  atom's instability:   How to describe the motion of particles without using the concept of trajectories?                              

                                 The necessity of complex numbers to describe quantum mechanics    (Ref: MIT Lecture QM )                                      Reading assignment
                                            Nature works with complex numbers, no with real numbers               
                                Photons and the loss of determinism    If photons exist we lose predictability   (Ref: MIT Lecture QM )                    Reading assignment

                                How the complex numbers were invented                                                                                                                                                        (Interesting video)
                                            Its inventors assumed they were giving up the  connection between Mathematics and reality (by
                                            accepting areas of negative  values), however complex numbers are conceived now part of a deeper
                                            truth  (the Schrodinger Eq., which  describes quantum Mechanics. ) 

                                Additional Reading
                                          Bell's inequality    In 1964, Bell established a restriction to particles subjected to  locality and realism.
                                          Nobel prize in 1922 was awarded to Clauser, Aspect and Zeilinger who experimentally demonstrated that
                                          quantum particles  violate Bell's inequality. Quantum Mechanics is incompatible with locality and realism.
                                          Bell's Theorem and the Non-locality of the Universe  By Brian Greene, Columbia University
                                          General Relativity was designed for local theories and Quantum Mechanics is a non-local theory
                                         That is why it is hard to put together Quantum Mechanis with Realtivity.

LECTURE-1    OVERVIEW: CONTRASTING  CLASSICAL and QUANTUM MECHANICS                                                                                     (09-26-2023)
                                          The claasical Hamiltonian determines  how classical states evolve with time
                                          The Hamiltonian operator determines how quantum states evolve with time  

LECTURE-2    CLASSICAL PHYSICS: Electromagnetism and Relativity (Review)    .  
                                          Relativity corrects classical mechanics by introducing a velocity-dependent mass

LECTURE-3    THE ORIGIN OF QUANTUM PHYSICS            .                      
                                          Planck's introduces the hypothesis of quantized energy

LECTURE-4     HAMILTON'S VARIATIONAL PRINCIPLE and the HAMILTON JACOBI EQ    .
                                          Canonical transformations, Poisson brackets          
                                          Short version file                                                           

LECTURE-5    THE HEISENBERG UNCERTAINTY PRINCIPLE                                                                                           .          .                                        (10-03-2023)
                             Understanding the Uncertainty Principle  through (the Fourier analysis of) universal property of waves
                                          Fourier spectral-decomposition of the wave-function  
                                          The wavefunction of a free-particle  (the non-relativistic and the relativistic cases)
                                          Appendix: Plane waves 

LECTURE-6      THE PRINCIPLE of COMPLEMENTARY:  Quantum Behavior of Photons, Electrons and Atoms                                    (10-05-2023)     .   
                                  Complementary distinguishes the world of quantum phenomena from the realm of classical                                                     
                                          Two observables are ‘complimentary’ if precise knowledge of one of them implies that all
                                          possible outcomes of  measuring the  other one are equally probable.

LECTURE-7    THE AMPLITUDE PROBABILITY         .                                                                                                                                                                    (10-12-2023)   
                                          Amplitude probabilities in the two-slit experiment ( Attempts to watch electron’s "trajectories")
                                          When to add “probabilities” and not “amplitude-probabilities”
                                          Dirac's Bra and Ket notation

LECTURE-8    THE HAMILTONIAN EQUATIONS           .
                                The working Principle of the Laser: Quantum mechanics description of stimulated emission   
                                          How do quntum states evolve with time?  The Hamiltonian Equations                .                                                                     (10-16-2023)
                                          The Hamiltonian Matrix
                                         Two-State Systems:
                                                          The free ammonia molecule (stationary states)
                                                          Ammonia molecule in a static electric field                                                                                                                                       (10-26-2023)                                                                          
                                                          Ammonia molecule in a time dependent electric field ε(t)                                                                                                       (10-30-2023)                                                                     
                                                          Operation of the ammonia maser;  Quantum mechanics calculation of the Einstein’s coefficients
                                          Appendix-1:  Light-Matter Interaction: Einstein's Law of Radiation 
                                          Appendix-2:  Transformation of Coordinates Under the Rotation of Coordinates  

LECTURE-9    From the HAMILTONIAN EQUATIONS to the SCHRODINGER EQUATION               .                                                                    (11-02-2023)    
                                          Understanding the energy band of a crystal                              
                                                          Stationary States, energy bands, electron wave-packet and group velocity, effective mass (case of low energy electrons.)
                                          Hamiltonian equations in the limit when the lattice space tends to zero.
                                                          Hamiltonian Eqs. adopt the form of the Schrodinger Eq. 

LECTURE-10   OBSERVABLES and  OPERATORS                                                                                                                                                                                 (11-28-2023)
                                          Part-1     
                                                          Spatial and momentum coordinates basis
                                                          Construction of the QM operators: Operators associated to observables
                                          Part-2                
                                                          The adjoint operator, Hermitian operators, generalized uncertainty principle
                                                          How to build a quantum state from a given set of experimental results.
                                                          Complete set of commuting operators.

LECTURE-11   SOLVING the SCHRODINGER EQUATION
                                  One-dimension case 
                                          The Schrodinger Equation (Non Relativistic)

                                                          Born’s interpretation of the wavefunction,  normalization condition for the wavefunction

                                          Case: Time independent potential  V = V (x)

                                                          Graphical analysis: Energy quantization 

                                                          Analytical solution: The linear harmonic oscillator
                                Three-dimension case 
                                          Case: Potential depends only on the relative position of the particles

                                                           Decoupling the Center of Mass motion and the Relative Motion
                                          Central Potentials
                                                          The angular equation, the Legendre Eq.
                                                          The radial equation, using the Coulomb Potential
                                                          Radial wavefunctions of the bound states (E’ < 0 )

LECTURE  12    QUANTUM ENTANGLEMENT              .                                                                                                                                                            (11-09-2023)                
                                          The Realistic, Orthodox and Agnostic points of view of QM     
                                          Detailed illustration of entanglement process: Case of the Annihilation of the Positronium
                                                                                                                                                   Decay of the positronium into an odd-parity two-photon state |F ⟩
                                                          Disintegration of the spin-zero state of the positronium

                                                          Polarization of the photons (Right and left circular polarization)

                                                          Conservation of parity in the positronium decay

                                                          The inversion operator

                                                          Conservation of parity

                                                          Joint and Conditional Probabilities
                                          (This lecture follows closely the Feynman's lecture    https://www.feynmanlectures.caltech.edu/III_18.html   )

LECTURE-13  QUANTUM TELEPORTATION       .                                                                                                                                                                     (11-16-2023)      

                                          Working principle of the quantum teleportation method proposed by Bennett
                                          Light propagation in birefringent materials

                                                          Spontaneous parametric down-conversion process  

                                                          Matching conditions: Type-I and Type-II  nonlinear crystals

                                          Classical and Quantum Bits (QUBITS

                                                          Generation of a Pair of Qubits  (A two-photon system)

                                                          How to generate entangled  two-qbit  states  (EBITS)

                                                          Bell States

                                          Analytical description of the proposed quantum teleportation
                                          Implementation of  Bell-states measurement

LECTURE-14  QUANTIZATION of ANGULAR MOMENTUM        
                                          CASE: Spin one     (The Feynman Lectures on Physics Vol. III Ch. 5)                              .                      
                                          CASE: Spin one-half      (The Feynman Lectures on Physics Vol. III Ch. 6)       
     

LECTURE-17   DENSITY MATRIX
                                Spin 1/2 density matrix

LECTURE-16   IDENTICAL PARTICLES