Università Roma Tre

Part I

Invitation

Fundamental physics as a theory of motion. The revolution of the XX century: quantum mechanics, special and general relativity, quantum field theory. The challenge of quantum gravity.
Basic constants: h, c and G. The cube of theories.

Introduction: the status of physics at the end of the XIX century

Precession of Mercury's perihelion. The discovery of radioactivity. Michelson and Morley and the speed of light. Black body radiation. Photoelectric effect. Compton diffusion. Compton wavelength. De Broglie and matter waves. Davisson and Germer and electron diffraction. Neglected hints from the past: equivalence of inertial and gravitational masses. Action at a distance. Similarities between gravitostatic and electrostatic forces.

Mathematical interlude: Waves.

Part II: Special Relativity

Reference systems and observers. Galilean Relativity. Principles of Newtonian mechanics. Galilean transformations. Maxwell's equations and electromagnetic waves.
The aether hypothesis and Michelson-Morley's experiment. Constancy of c and principles of Special Relativity. Space-time diagrams. Simultaneity. Boosts. c as the maximal signal speed. Length contraction and time dilation. Atmospheric muon lifetime. Invariance of the space-time interval. Minkowski space. Time-like, null and space-like intervals. Proper time. Causal structure of Minkowski space. Composition of velocities. The Doppler effects. The twin paradox. The garage paradox. Time dilation and gravitational time delay. Isometries in Minkowski space: pseudo-orthogonality and Lorentz transformations. Four-vectors: relativistic velocity and acceleration. Relativistic dynamics: action principle. Energy, momentum and conservation of the four-momentum. Dispersion relation. Particles of zero mass. Relativistic force. Covariant formulation of electromagnetism: Maxwell's tensor and inhomogeneous equations. The Lorentz force. Homogeneous Maxwell's equations: the vector potential; gauge invariance and its meaning.

Part III: Quantum Mechanics

Spin and qubits. Stern-Gerlach experiments. Physical states. Spin states: basis and normalization. Global phases. Bras and kets. Principles of QM: states, observables, measures, probabilistic interpretation. Pauli matrices and spin observables. Spectrum of the spin operator along an arbitrary direction. Time evolution. The operator U(t, t_0). Unitarity and its meaning. Schrödinger's equation. Hamiltonian. Time evolution of averages. Classical mechanics and Poisson brackets. Conservation laws. Spin in magnetic field. Compatible and incompatible observables. The uncertainty principle. Two-spin systems. Product states and entangled states. Singlet and triplet states. Observables on composite systems. Pure states and mixed states: the density matrix. Entanglement and density matrix. Tests of entanglement. Waves and particles. Position and momentum operators and their eigenfunctions. Basics of canonical quantization. Free particle Hamiltonian and its spectrum. Quantum Newton's second law. The harmonic oscillator. Creation and annihilation operators. Energy levels. Classically forbidden regions: the tunnel effect. Pathway towards relativistic quantum mechanics.

Mathematical interludes: Complex vector spaces. Scalar product. Hermitian operators. Exterior products and projectors. Completeness relation. Operators for continuous spectra.

-Susskind L and Friedman A, Quantum Mechanics. The Theoretical Minimum (2014)
-Susskind L and Friedman A, Special Relativity and Classical Field Theory. The Theoretical Minimum} (2017)