Study the phenomena related to dark matter and dark energy in a specialized and up-to-date academic environment” 

The field of research in Quantum Physics offers a wide range of lines of development with enormous potential for engineering professionals who decide to enter this universe of scientific exploration. Areas such as energy production, ultracold atoms, trapped ions, or photonics represent only a portion of the possibilities that this discipline presents at both the theoretical and applied levels.  

Recent advances in physics have opened new avenues of study in fields as diverse as astrophysics, cosmology, chemistry, medicine, or artificial intelligence. For this reason, TECH has designed this Master's Degree in Quantum Physics, with the aim of enabling graduates to master key concepts of planetary and solar physics, the work of authors such as Paul Dirac or Richard Feynman, and the fundamentals of quantum field theory, among other highly relevant scientific contents.  

All knowledge is delivered through a 100% online program, allowing students to delve into aspects such as Einstein’s equations, the Schwarzschild solution, dark matter and dark energy, or the thermodynamics of the early universe. The practical cases included will serve to integrate what has been learned into everyday professional practice, becoming a valuable tool for intellectual and technical growth.  

Through this proposal, TECH provides a unique opportunity for engineers who wish to advance in their professional careers through high-quality university education, without barriers of time or place.  Only a device with an internet connection is required to access a flexible learning experience, adapted to each lifestyle. In addition, this academic pathway includes 10 exclusive Masterclasses delivered by a prestigious international expert, who serves as Guest Director. 

Thanks to exclusive Masterclasses delivered by TECH International Guest Director, you will be able to update all your research competencies in the field of Quantum Physics” 

This Master's Degree in Quantum Physics contains the most complete and up-to-date university program on the market. Its most notable features are:

• The development of practical cases presented by experts in quantum physics 
• The graphic, schematic, and practical contents with which they are created, provide scientific and practical information on the disciplines that are essential for professional practice 
• Practical exercises where the self-assessment process can be carried out to improve learning 
• Its special emphasis on innovative methodologies  
• Theoretical lessons, questions to the expert, debate forums on controversial topics, and individual reflection assignments 
• Content that is accessible from any fixed or portable device with an internet connection 

The library of multimedia resources of this course will allow you to learn the main contributions to Quantum Physics from Richard Feynman, Paul Dirac, Peter Higgs or Schrödinger” 

Its teaching staff includes professionals from the field of Quantum Physics, who contribute to this program the experience gained through their work, as well as renowned specialists from leading societies and prestigious universities. 

The multimedia content, developed with the latest educational technology, will provide the professional with situated and contextual learning, i.e., a simulated environment that will provide an immersive learning experience designed to prepare for real-life situations. 

This program is designed around Problem-Based Learning, whereby the student must try to solve the different professional practice situations that arise throughout the program. For this purpose, the professional will be assisted by an innovative interactive video system created by renowned and experienced experts. 

Explore the secrets of the expansion of the universe and its relationship with the theory of general relativity"

Study the impact of gravitational waves and their relevance in contemporary cosmology"

The teaching materials that make up this Master’s Degree have been developed by a group composed of experts in quantum physics, astrophysics, cosmology, quantum computing, and other related disciplines.  As a result, the syllabus rigorously and comprehensively addresses the theoretical and experimental foundations of quantum mechanics, field theories, general relativity, and emerging technologies.  This academic pathway will enable graduates to master the physical principles that govern the universe, as well as to apply advanced methodologies in research or technological development contexts.   

You will gain command of the postulates that govern the quantum world and the laws that rule the cosmos, thanks to a complete, in-depth syllabus oriented toward current scientific practice” 

Module 1. Introduction to Modern Physics 

1.1. Introduction to Medical Physics 

1.1.1. How to Apply Physics to Medicine 
1.1.2. Energy of Charged Particles in Tissues 
1.1.3. Photons through Tissues  
1.1.4. Applications 

1.2. Introduction to Particle Physics 

1.1.1. Introduction and Objectives 
1.1.2. Quantified Particles 
1.1.3. Fundamental Forces and Charges 
1.1.4. Particle Detection 
1.1.5. Classification of Fundamental Particles and Standard Model 
1.1.6. Beyond the Standard Model 
1.1.7. Current Generalization Theories 
1.1.8. High Energy Experiments 

1.3. Particle Accelerators 

1.3.1. Particle Acceleration Processes 
1.3.2. Linear Accelerators 
1.3.3. Cyclotrons 
1.3.4. Synchrotrons 

1.4. Introduction to Nuclear Physics 

1.4.1. Nuclear Stability 
1.4.2. New Methods in Nuclear Fission 
1.4.3. Nuclear Fusion 
1.4.4. Synthesis of Superheavy Elements 

1.5. Introduction to Astrophysics 

1.5.1. The Solar System 
1.5.2. Birth and Death of a Star 
1.5.3. Space Exploration 
1.5.4. Exoplanets 

1.6. Introduction to Cosmology 

1.6.1. Distance Calculation in Astronomy 
1.6.2. Velocity Calculations in Astronomy 
1.6.3. Dark Matter and Energy 
1.6.4. The Expansion of the Universe 
1.6.5. Gravitational Waves 

1.7. Geophysics and Atmospheric Physics 

1.7.1. Geophysics 
1.7.2. Atmospheric Physics 
1.7.3. Meteorology 
1.7.4. Climate Change 

1.8. Introduction to Condensed Matter Physics 

1.8.1. Aggregate States of Matter 
1.8.2. Matter Allotropes 
1.8.3. Crystalline Solids 
1.8.4. Soft Matter 

1.9. Introduction to Quantum Computing 

1.9.1. Introduction to the Quantum World 
1.9.2. Qubits 
1.9.3. Multiple Qubits 
1.9.4. Logic Gates 
1.9.5. Quantum Programs 
1.9.6. Quantum Computers 

1.10. Introduction to Quantum Cryptography 

1.10.1. Classic Information 
1.10.2. Quantum Information 
1.10.3. Quantum Encryption 
1.10.4. Protocols in Quantum Cryptography 

Module 2. Mathematical methods 

2.1. Prehibertian Spaces 

2.1.1. Vector Spaces 
2.1.2. Positive Hermitian Scalar Product 
2.1.3. Single Vector Module 
2.1.4. Schwartz Inequality 
2.1.5. Minkowsky Inequality 
2.1.6. Orthogonality 
2.1.7. Dirac Notation 

2.2. Topology of Metric Spaces 

2.2.1. Definition of Distance 
2.2.2. Definition of Metric Space 
2.2.3. Elements of Topology of Metric Spaces 
2.2.4. Convergent Successions 
2.2.5. Cauchy Successions 
2.2.6. Complete Metricl Space 

2.3. Hilbert Spaces 

2.3.1. Hilbert Spaces: Definition 
2.3.2. Herbatian Base 
2.3.3. Schrödinger versus Heisenberg. Lebesgue Integral 
2.3.4. Continuous Frames of a HIlbert Space 
2.3.5. Change of Basis Matrix 

2.4. Linear Operations 

2.4.1. Linear Operators: Basic Concepts 
2.4.2. Inverse Operator 
2.4.3. Adjoint Operator 
2.4.4. Self-Adjoint Operator 
2.4.5. Positive Definite Operator 
2.4.6. Unitary Operator  I Change of Basis 
2.4.7. Antiunitary Operator 
2.4.8. Projector 

2.5. Stumr-Liouville Theory 

2.5.1. Eigenvalue Theorem 
2.5.2. Eigenvector Theorem 
2.5.3. Sturm-Liouville Problem 
2.5.4. Important Theorems for Sturm-Liouville Theory 

2.6. Introduction to Group Theory 

2.6.1. Definition of Group and Characteristics 
2.6.2. Symmetries 
2.6.3. Study of SO(3), SU(2) and SU(N) Groups 
2.6.4. Lie Algebra 
2.6.5. Groups and Quantum Physics 

2.7. Introduction to Representations 

2.7.1. Definitions 
2.7.2. Fundamental Representation 
2.7.3. Adjoint Representation 
2.7.4. Unitary Representation 
2.7.5. Product of Representation 
2.7.6. Young Tables 
2.7.7. Okubo Theorems 
2.7.8. Applications to Particle Physics 

2.8.  Introduction to Tensors 

2.8.1. Definition of Covariant and Contravariant Tensors 
2.8.2. Kronecker Delta  
2.8.3. Levi-Civita Tensor 
2.8.4. Study of SO(N) i SO(3) 
2.8.5. Study of SO(N) 
2.8.6. Relation between tensors and representations 

2.9. Group Theory Applied to Physics 

2.9.1. Translation Group 
2.9.2. Lorentz Group 
2.9.3. Discrete Groups 
2.9.4. Continuous Groups 

2.10. Representations and Particle Physics 

2.10.1. Representations of SU(N) Groups 
2.10.2. Fundamental Representations 
2.10.3. Multiplication of Representations 
2.10.4. Okubo Theorem and Eightfold Ways 

Module 3. Quantum Physics 

3.1. Origins of Quantum Physics 

3.1.1. Blackbody Radiation 
3.1.2. Photoelectric Effect 
3.1.3. Compton Effect 
3.1.4. Atomic Spectra and Models 
3.1.5. Pauli Exclusion Principle 

3.5.1.1. Zeeman Effect 
3.5.1.2. Stern-Gerlach Experiment 

3.1.6. Broglie Wave Length and the Double Slit Experimient 

3.2. Mathematical Formulation 
3.2.1. Hilbert Spaces 
3.2.2. Dirac Nomenclature Bra - ket 
3.2.3. Internal and External Product 
3.2.4. Linear Operators 
3.2.5. Hermetic Operators and Diagonalization 
3.2.6. Sum and Tensor Product 
3.2.7. Density Matrix 

3.3. Quantum Mechanics Postulates 

3.3.1. Postulate 1: Definition of State 
3.3.2. Postulate 2: Definition of Observables 
3.3.3. Postulate 3: Definition of Measurements 
3.3.4. Postulate 4: Probability of Measurements 
3.3.5. Postulate 5: Dynamics 

3.4. Apply the postulates of quantum mechanics 

3.4.1. Probability of Results Statistics 
3.4.2. Indeterminism 
3.4.3. Temporary Evolution of the Expected Values 
3.4.4. Compatibility and Commuting of Observables 
3.4.5. Pauli Matrices 

3.5. Quantum Mechanics Dynamics  

3.5.1. Representation of Positions 
3.5.2. Momentum Representation 
3.5.3. Schrödinger Equation 
3.5.4. Ehrenfest Theorem 
3.5.5. Virial Theorem 

3.6. Potential Barriers 

3.6.1. Infinite Square Well 
3.6.2. Finite Square Well 
3.6.3. Potential Step 
3.6.4. Delta Potential 
3.6.5. Tunnel Effect 
3.6.6. Free Particle 

3.7. Simple Harmonic Oscillator 

3.7.1. Analogy with Classical Mechanics 
3.7.2. Hamiltonian and eigenvalues of energy 
3.7.3. Analytical Method 
3.7.4. Blurred Quantum 
3.7.5. Coherent States 

3.8. 3D Operators and Observables 

3.8.1. Review of Calculus Notions with Several Values 
3.8.2. Position Operator 
3.8.3. Linear Momentum Operator 
3.8.4. Orbital Angular Momentum 
3.8.5. Ladder Operators 
3.8.6. Hamiltonian 

3.9. Three-Dimensional Eigenvalues and Eigenfunctions 

3.9.1. Position Operator 
3.9.2. Linear Momentum Operator 
3.9.3. Orbital Angular Momentum and Spherical Harmonics Operator 
3.9.4. Angular Equation 

3.10. Three-Dimensional Potential Barriers 

3.10.1. Free Particle 
3.10.2. Particle in a Box 
3.10.3. Central Potentials and Radial Equations 
3.10.4. Infinite Spheric Well 
3.10.5. Hydrogen Atom 
3.10.6. 3D Harmonic Oscillator 

Module 4. Astrophysics 

4.1. Introduction 

4.1.1. Brief History of Astrophysics 
4.1.2. Instrumentation 
4.1.3. Observational Magnitude Scale 
4.1.4. Calculation of Astronomical Distances 
4.1.5. Color Index 

4.2. Spectral Lines  

4.2.1. Historical Introduction 
4.2.2. Kirchoff's Laws 
4.2.3. Relationship between Spectrum and Temperature 
4.2.4. Doppler Effect 
4.2.5. Spectrograph 

4.3. Radiation Field Study  

4.3.1. Prior Definitions 
4.3.2. Lens Opacity 
4.3.3. Optical Depth 
4.3.4. Microscopic Opacity Sources 
4.3.5. Total Opacity 
4.3.6. Extinction  
4.3.7. Structure of Spectral Lines 

4.4. Stars 

4.4.1. Classification of Stars 
4.4.2. Methods for Determining the Mass of a Star 
4.4.3. Binary Stars 
4.4.4. Classification of Binary Stars 
4.4.5. Determining the Masses of a Binary System 

4.5. Life of Stars 

4.5.1. Characteristics of a Star 
4.5.2. Birth of a Star 
4.5.3. Life of a Star. Hertzprung-Russell Diagrams 
4.5.4. Death of a Star 

4.6. Death of Stars  

4.6.1. White Dwarf 
4.6.2. Supernovas 
4.6.3. Neutron Stars 
4.6.4. Black Holes 

4.7. Study of the Milky Way 

4.7.1. Shape and Dimensions of the Milky Way 
4.7.2. Dark Matter 
4.7.3. Phenomenon of Gravitational Lensing 
4.7.4. Massive Particles of Weak Interaction 
4.7.5. Shape and Halo of the Milky Way 
4.7.6. Spiral Structure of the Milky Way 

4.8. Galaxy Clusters 

4.8.1. Introduction 
4.8.2. Classification of Galaxies 
4.8.3. Photometry of Galaxies 
4.8.4. Local Group: Introduction 

4.9. Distribution of Large-Scale Galaxies 

4.9.1. Shape and Age of the Universe 
4.9.2. Standard Cosmological Model 
4.9.3. Formation of Cosmological Structures 
4.9.4. Observational Methods in Cosmology 

4.10. Dark Matter and Energies 

4.10.1. Discovery and Characteristics 
4.10.2. Consequences on the Distribution of Ordinary Matter 
4.10.3. Dark Matter Problems 
4.10.4. Candidate Particles for Dark Matter 
4.10.5. Dark Energy, its Consequences 

Module 5. Quantum Physics II 

5.1. Quantum Mechanics Description: Images or Representations 

5.1.1. Schrödinger Picture 
5.1.2. Heisenberg Picture 
5.1.3. Dirac Picture or Interaction Picture 
5.1.4. Change of Pictures 

5.2. 3D Harmonic Oscillator 

5.2.1. Creation and annihilation operators 
5.2.2. Wave Functions of Fock States 
5.2.3. Coherent States 
5.2.4. States of Minimum Indeterminacy 
5.2.5. Squeezed States 

5.3. Angular Momentum 

5.3.1. Rotations 
5.3.2. Switches of Angular Momentum 
5.3.3. Angular Momentum Basis 
5.3.4. Scale Operators 
5.3.5. Matrix Representation 
5.3.6. Intrinsic Angular Momentum: the Spin 
5.3.7. Spin Cases 1/ 2, 1, 3/ 2 

5.4. Multi-Component Wave Functions: Spinorials 

5.4.1. Single-Component Wave Functions: Spin 0 
5.4.2. Double-Component Wave Functions: Spin 1/2 
5.4.3. Expected Value of Spin Observable 
5.4.4. Atomic States 
5.4.5. Addition of Angular Momentum 
5.4.6. Clebsch-Gordan Coefficient 

5.5. State of the Compound Systems 

5.5.1. Distinguishable Particles 
5.5.2. Indistinguishable Particles 
5.5.3. Case of Photons: Semitransparent Mirror Experiment 
5.5.4. Quantum Bonding 
5.5.5. Bell Inequalities 
5.5.6. EPR Paradox 
5.5.7. Bell Theorem 

5.6. Introduction to Approximate Methods: Variational Method 

5.6.1. Introduction to the Variational Method 
5.6.2. Linear Variations 
5.6.3. Rayleigh-Ritz Variational Method 
5.6.4. Harmonic Oscillator: a Study by Variational Methods 

5.7. Study of Atomic Models with the Variational Method 

5.7.1. Hydrogen Atom 
5.7.2. Helium Atom 
5.7.3. Ionized Hydrogen Molecule 
5.7.4. Discrete Symmetries 

5.7.4.1. Parity 
5.7.4.2. Temporary Inversion 

5.8. Introduction to Disturbance Theory 

5.8.1. Time-Independent Perturbations 
5.8.2. Non Degerate Case 
5.8.3. Degenerate Case 
5.8.4. Fine Structure of Hydrogen Atom 
5.8.5. Zeeman Effect 
5.8.6. Coupling Constant between Spins. Hyperfine Structure 
5.8.7. Time-Dependent Perturbation Theory 

5.8.7.1. Two-Level Atom 
5.8.7.2. Sinusoidal Perturbation 

5.9. Adiabatic Approximation 

5.9.1. Introduction to Adiabatic Approximation 
5.9.2. The Adiabatic Theorem 
5.9.3. Berry Phase 
5.9.4. Aharonov-Bohm Effect 

5.10. Wentzel-Kramers-Brillouin (WKB) Approximation 

5.10.1. Introduction to the WKB Method. 
5.10.2. Classical Region 
5.10.3. Tunnel Effect 
5.10.4. Connection Formulas 

Module 6. Nuclear and Particle Physics 

6.1. Introduction to Nuclear Physics 

6.1.1. Periodic Table of the Elements 
6.1.2. Important Discoveries 
6.1.3. Atomic Models 
6.1.4. Important Definitions. Scales and Units in Nuclear Physics 
6.1.5. Segré's Diagram 

6.2. Nuclear Properties 

6.2.1. Binding Energy 
6.2.2. Semiempirical Mass Formula 
6.2.3. Fermi Gas Model 
6.2.4. Nuclear Stability 

6.2.4.1. Alpha Decay 
6.2.4.2. Beta Decay 
6.2.4.3. Nuclear Fusion 

6.2.5. Nuclear Desexcitation 
6.2.6. Double Beta Decay 

6.3. Nuclear Scattering 

6.3.1. Internal Structure: Dispersion Study 
6.3.2. Effective Section 
6.3.3. Rutherford's Experiment: Rutherford's Effective Section 
6.3.4. Mott's Effective Section 
6.3.5. Momentum Transfer and Shape Factors 
6.3.6. Nuclear Charge Distribution 
6.3.7. Neutron Scattering 

6.4. Nuclear Structure and Strong Interaction 

6.4.1. Nucleon Scattering 
6.4.2. Bound States Deuterium 
6.4.3. Strong Nuclear Interaction 
6.4.4. Magic Numbers 
6.4.5. The Layered Model of the Nucleus 
6.4.6. Nuclear Spin and Parity 
6.4.7. Electromagnetic Moments of the Nucleus 
6.4.8. Collective Nuclear Excitations: Dipole Oscillations, Vibrational States and Rotational States 

6.5. Nuclear Structure and Strong Interaction II 

6.5.1. Classification of Nuclear Reactions 
6.5.2. Reaction Kinematics 
6.5.3. Conservation Laws 
6.5.4. Nuclear Spectroscopy 
6.5.5. The Compound Nucleus Model 
6.5.6. Direct Reactions 
6.5.7. Elastic Dispersion 

6.6. Introduction to Particle Physics 

6.6.1. Particles and Antiparticles 
6.6.2. Fermions and Baryons 
6.6.3. The Standard Model of Elementary Particles: Leptons and Quarks 
6.6.4. The Quark Model 
6.6.5. Intermediate Vector Bosons 

6.7. Dynamics of Elementary Particles 

6.7.1. The Four Fundamental Interactions 
6.7.2. Quantum Electrodynamics 
6.7.3. Quantum Chromodynamics 
6.7.4. Weak Interaction 
6.7.5. Disintegrations and Conservation Laws 

6.8. Relativistic Kinematics 

6.8.1. Lorentz Transformations 
6.8.2. Quatrivectors 
6.8.3. Energy and Linear Momentum 
6.8.4. Collisions 
6.8.5. Introduction to Feynman Diagrams 

6.9. Symmetries 

6.9.1. Groups, Symmetries and Conservation Laws 
6.9.2. Spin and Angular Momentum 
6.9.3. Addition of Angular Momentum 
6.9.4. Flavor Symmetries  
6.9.5. Parity 
6.9.6. Load Conjugation 
6.9.7. CP Violation 
6.9.8. Time Reversal 
6.9.9. CPT Conservation 

6.10. Linked States 

6.10.1. Schrödinger's Equation for Central Potentials 
6.10.2. Hydrogen Atom 
6.10.3. Fine Structure 
6.10.4. Hyperfine Structure 
6.10.5. Positronium 
6.10.6. Quarkonium 
6.10.7. Lightweight Mesons 
6.10.8. Baryons 

Module 7. Quantum Field Theory 

7.1. Classical Field Theory 

7.1.1. Notation and Conventions 
7.1.2. Lagrangian Formulation 
7.1.3. Euler Lagrange Equations  
7.1.4. Symmetries and Conservation Laws 

7.2. Klein-Gordon Field 

7.2.1. Klein-Gordon Equations 
7.2.2. Klein-Gordon Field Quantization 
7.2.3. Klein-Gordon Field  Lorentz Invariance 
7.2.4. Vacuum Vacuum  and Fock States 
7.2.5. Vacuum Energy  
7.2.6. Normal Ordering: Convention 
7.2.7. Energy and Momentum of States 
7.2.8. Study of Causality 
7.2.9. Klein-Gordon propagator 

7.3. Dirac Field 

7.3.1. Dirac Equation 
7.3.2. Dirac Matrices and their Properties 
7.3.3. Representation of Dirac Matrices 
7.3.4. Dirac Lagrangian 
7.3.5. Solution to Dirac Equation: Plane Waves 
7.3.6. Commuting and Anticommuting 
7.3.7. Quantification of Dirac Field 
7.3.8. Fock Space 
7.3.9. Dirac Propagator 

7.4. Electromagnetic Field 

7.4.1. Classical Field  Electromagnetic Theory 
7.4.2. Quantization of the Electromagnetic Field and its Problems 
7.4.3. Fock Space 
7.4.4. Gupta-Bleuler Formalism 
7.4.5. Photon Propagator  

7.5. S-Matrix Formalism 

7.5.1. Lagrangian and Hamitonian of Interaction 
7.5.2. S Matrix: Definition and Properties 
7.5.3. Dyson Expansion 
7.5.4. Wick Theorem 
7.5.5. Dirac Picture 

7.6. Feinman Diagrams in the Position Space 

7.6.1. How to Draw Feynman Diagrams? Standards. Utilities 
7.6.2. First Order 
7.6.3. Second Order 
7.6.4. Dispersion Processes with Two Particles 

7.7. Feynman Rules 

7.7.1. Normalization of States in Fock Space 
7.7.2. Feynman Amplitude 
7.7.3. Feynman Rules for QED 
7.7.4. Gauge Invariance in the Amplituides 
7.7.5. Examples 

7.8. Cross Section and Decay Rates 

7.8.1. Definition of Cross Sections 
7.8.2. Definition of Decay Rate 
7.8.3. Example with Two Bodies in Final State 
7.8.4. Unpolarized Cross Section 
7.8.5. Summation on Fermion Polarization 
7.8.6. Summation on Photon Polarization 
7.8.7. Examples 

7.9. Study of Muons and Other Charged Particles 

7.9.1. Muons 
7.9.2. Charged Particles 
7.9.3. Scalar Charged Particles 
7.9.4. Feynman Rules for Scalar Quantum Electrodynamics Theory 

7.10. Symmetries 

7.10.1. Parity 
7.10.2. Load Conjugation 
7.10.3. Time Reversal  
7.10.4. Violation of Some Symmetries 
7.10.5. CPT Symmetry 

Module 8. General Relativity and Cosmology 

8.1. Special Relativity 

8.1.1. Postulates 
8.1.2. Lorentz Transformations in Standard Configuration 
8.1.3. Impulses (Boosts) 
8.1.4. Tensors 
8.1.5. Relativistic Kinematics 
8.1.6. Relativistic Linear Momentum and Energy 
8.1.7. Lorentz Covariance 
8.1.8. Energy-Momentum Tensor 

8.2. Equivalence Principle 

8.2.1. Principle of Weak Equivalence 
8.2.2. Experiments on the Weak Equivalence Principle 
8.2.3. Locally Inertial Reference Systems 
8.2.4. Equivalence Principle 
8.2.5. Consequences on the Equivalence Principle 

8.3. Particle Motion in the  Gravitational Field 

8.3.1. Path of Particles under Gravity 
8.3.2. Newtonian Limit 
8.3.3. Gravitational Redshift and Tests 
8.3.4. Temporary Dilatation 
8.3.5. Geodesic Equation 

8.4. Geometry: Required Concepts 

8.4.1. Two-Dimensional Spaces 
8.4.2. Scalar, Vector, and Tensor Fields 
8.4.3. Metric Tensor: Concept and Theory 
8.4.4. Partial Derivative 
8.4.5. Covariant Derivative 
8.4.6. Christoffel Symbols 
8.4.7. Covariant Derivatives of Tensors 
8.4.8. Directional Covariant Derivatives 
8.4.9. Divergence and Lapacian 

8.5. Curved Space-Time 

8.5.1. Covariant Derivative and Parallel Transport: Definition 
8.5.2. Geodesics from Parallel Transport 
8.5.3. Riemann Curvature Tensor 
8.5.4. Riemann Tensor: Definition and Properties 
8.5.5. Ricci Tensor: Definition and Properties 

8.6. Einstein’s Equations: Derivation 

8.6.1.  Reformulation of the Equivalence Principle 
8.6.2. Applications of the Equivalence Principle 
8.6.3. Conservation and Symmetries 
8.6.4. Derivation of Einstein's Equations from the Equivalence Principle 

8.7. Schwarzschild Solution 

8.7.1. Schwartzschild Metrics 
8.7.2. Length and Time Elements 
8.7.3. Conserved Quantities 
8.7.4. Equation of Motion  
8.7.5. Light Deflection. Study of Schwartzschild Metrics 
8.7.6. Schwartzschild Radius 
8.7.7. Eddington– Finkelstein Coordinates 
8.7.8. Black Holes 

8.8. Linear Gravity Limits Consequences

8.8.1. Linear Gravity: Introduction 
8.8.2. Coordinate Transformation 
8.8.3. Linearized Einstein Equations 
8.8.4. General Solution of Linearized Einstein Equations 
8.8.5. Gravitational Waves 
8.8.6. Effects of Gravitational Waves on Matter 
8.8.7. Generation of Gravitational Waves 

8.9. Cosmology: Introduction 

8.9.1. Observation of the Universe: Introduction 
8.9.2. Cosmological Principle 
8.9.3. System of Coordinates 
8.9.4. Cosmological Distances 
8.9.5. The Hubble’s Law 
8.9.6. Inflation 

8.10. Cosmology: Mathematical Study 

8.10.1. First Equation of Friedmann 
8.10.2. second Equation of Friedmann 
8.10.3. Densities and Scale Factor 
8.10.4. Consequences of Friedmann Equations. Curvature of the Universe 
8.10.5. Thermodynamics of the Early Universe 

Module 9. High-Energy Physics 

9.1. Mathematical Methods: Groups and Representations 

9.1.1. Theory of Groups 
9.1.2. SO(3), SU(2), SU(3), and SU(N) Groups 
9.1.3. Lie Algebra 
9.1.4. Representations 
9.1.5. Multiplication of Representations 

9.2. Symmetries 

9.2.1. Symmetries and Conservation Laws 
9.2.2. C, P, T Symmetries 
9.2.3. CPT Symmetry Violation and Conservation 
9.2.4. Angular Momentum 
9.2.5. Addition of Angular Momentum 

9.3. Feynman Calculations: Introduction 

9.3.1. Mean Lifetime 
9.3.2. Cross Section 
9.3.3. Fermi’s Golden Rule for Decays 
9.3.4. Fermi’s Golden Rule for Scattering 
9.3.5. Two-Body Scattering in the Center-of-Mass Reference Frame 

9.4. Application of Feynman Calculations: Toy Model 

9.4.1. Toy Model: Introduction 
9.4.2. Feynman Rules 
9.4.3. Mean Lifetime 
9.4.4. Dispersion 
9.4.5. Higher Order Diagrams 

9.5. Quantum Electrodynamics 

9.5.1. Dirac Equation 
9.5.2. Solution for Dirac Equations 
9.5.3. Bilinear covariants 
9.5.4. The Photon 
9.5.5. Feynman Rules for Quantum Electrodynamics 
9.5.6. Casimir’s Trick 
9.5.7. Renormalization 

9.6. Electrodynamics and Chromodynamics of Quarks 

9.6.1. Feynman Rules 
9.6.2. Production of Hadrons in Electron-Positron Collisions 
9.6.3. Feynman Rules for Chromodynamics 
9.6.4. Color Factors 
9.6.5. Quark-Antiquark Interaction 
9.6.6. Quark-Quark Interaction 
9.6.7. Pair Annihilation in Quantum Chromodynamics 

9.7. Weak Interaction 

9.7.1. Weak Charged Interaction 
9.7.2. Feynman Rules 
9.7.3. Muon Decay 
9.7.4. Neutron Decay 
9.7.5. Pion Decay 
9.7.6. Weak Interaction between Quarks 
9.7.7. Weak neutral Interaction 
9.7.8. Electroweak Unification 

9.8. Gauge Theories 

9.8.1. Local Gauge Invariance 
9.8.2. Yang-Millis Theory 
9.8.3. Quantum Chromodynamics 
9.8.4. Feynman Rules 
9.8.5. Mass Term 

9.8.6. Spontaneous Symmetry Breaking 
9.8.7. Higgs Mechanism 

9.9. Neutrino Oscillation 

9.9.1. Solar Neutrino Problem 
9.9.2. Neutrino Oscillation 
9.9.3. Neutrino Masses 
9.9.4. Mixing Matrix 

9.10. Advanced Topics. Brief Introduction 

9.10.1. Higgs Boson 
9.10.2. Grand Unification 
9.10.3. Matter-Antimatter Asymmetry 
9.10.4. Supersymmetry, Strings, and Extra Dimensions 
9.10.5. Dark Matter and Energy 

Module 10. Quantum Information and Computing 

10.1. Introduction: Mathematics and Quantum Physics 

10.1.1. Complex Vector Spaces 
10.1.2. Linear Operators 
10.1.3. Scalar Products and Hilbert Spaces 
10.1.4. Diagonalization 
10.1.5. Tensor Product 
10.1.6. The Role of Operators 
10.1.7. Important Theorems on Operators 
10.1.8. Checked Quantum Mechanics Postulates 

10.2. Statistical States and Samples 

10.2.1. The Qubit 
10.2.2. Density Matrix 
10.2.3. Two-Part System 
10.2.4. Schmidt Decomposition 
10.2.5. Statistical Interpretation of the Mixing States 

10.3. Measurements and Temporary Evolution 

10.3.1. Von Neumann Measurements 
10.3.2. Generalized Measurements 
10.3.3. Neumark Theorem 
10.3.4. Quantum Channels 

10.4. interwoven and its Applications 

10.4.1. ERP States 
10.4.2. Dense Coding 
10.4.3. State Teleportation 
10.4.4. Density Matrix and its Representations  

10.5. Classic and Quantum Information 

10.5.1. Introduction to Probability 
10.5.2. Information 
10.5.3. Shannon Entropy and Mutual Information 
10.5.4. Communication 

10.5.4.1. The Bynary Symmettric Channel 
10.5.4.2. Channel Capacity 

10.5.5. Shannon Theorems 
10.5.6. Difference between Classic and Quantum Information 
10.5.7. Von Neumann Entropy 
10.5.8. Schumacher Theorem 
10.5.9. Holevo Information 
10.5.10. Accesible Information and Holevo Limit 

10.6. Quantum Computing 
10.6.1. Turing Machines 

10.6.2. Circuits and Classification of Complexity 
10.6.3. Quantum Computer 
10.6.4. Quantum Logic Gates 
10.6.5. Deutsch-Josza and Simon´s Algorithm 
10.6.6. Unstructured Search; Grover´s Algorithm 
10.6.7. RSA Encryption Method 
10.6.8. Factorizatión: Shor Algorithm 

10.7. Quantum Theory of the Light-Matter Interaction 

10.7.1. Two-Level Atom 
10.7.2. AC-Stark Splitting 
10.7.3. Rabi Oscillations 
10.7.4. Light dipole force 

10.8. Quantum Theory of Light-Matter Interaction 

10.8.1. Quantum States of the Electromagnetic Field 
10.8.2. Jaynes-Cummings Model 
10.8.3. The Problem of Decoherence 
10.8.4. Treatment of Weisskopf-Wigner Model of Spontaneous Emission 

10.9. Quantum Communication 

10.9.1. Quantum Cryptography: BB84 and Ekert91 Protocols 
10.9.2. Bell Inequalities 
10.9.3. Generation of Individual Photons 
10.9.4. Propagation of Individual Photons 
10.9.5. Detection of Individual Photons 

10.10. Quantum Computing and Simulation 

10.10.1. Neutral Atoms in  Dipolar Traps 
10.10.2. Cavity Quantum Electrodynamics 
10.10.3. Ions in Paul Traps 
10.10.4. Superconducting Cubits 

Master key concepts of atmospheric physics, meteorology, and their link to climate change”