Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

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Condensed matter physics


Condensed matter physics is a branch of physics that focuses on understanding the physical properties of the solid and liquid states of matter. It is a field that not only explores fundamental questions about the states of matter but also has practical implications in the development of new materials and technologies. It includes the study of crystals, semiconductors, metals, superconductors, and other complex materials.

Basic concepts

Condensed matter physics essentially attempts to understand how large numbers of atoms and molecules interact with each other and how they give rise to the properties of matter. The interactions between these atoms and molecules can result in various interesting phenomena.

Crystal structure

One of the main features considered in condensed matter physics is the crystal structure of solids. The crystal structure describes the ordered arrangement of atoms in a crystalline solid. For example, in a simple cubic structure, the atoms are arranged at the corners of the cube. Here is a visual representation of a simple cubic lattice:

There are also other types of lattices, including body-centered cubic and face-centered cubic lattices, each of which has specific properties that are useful in different applications.

Band theory

Understanding the electronic properties of solids is essential in condensed matter physics. Band theory explains the behavior of electrons in solids. According to this theory, electrons in solids occupy energy levels known as bands, and the difference in energy between the highest occupied band and the lowest unoccupied band is called the band gap.

Mathematically, band theory can be expressed using the Schrödinger equation for electrons in a periodic potential:

Hψ = Eψ

Here, H is the Hamiltonian operator, ψ is the wave function, and E is the energy. Band gaps are important in classifying materials as conductors, semiconductors, or insulators.

Fermi energy and Fermi surface

The Fermi energy is an important concept, especially in metals. It is the highest energy level that is filled with electrons at absolute zero temperature. The Fermi surface, an abstract concept, describes the collection of points in reciprocal space that separate occupied electron energy levels from empty electron energy levels at the Fermi energy.

Phase transition

Condensed matter physics also investigates phase transition phenomena. These are changes between different states of matter, such as solid to liquid or liquid to gas. Phase transitions can also occur within a solid, such as a transition from one crystal structure to another.

phase 1 phase 2

An example of this is the ferromagnetic to paramagnetic phase transition at the Curie temperature. Below the Curie temperature, a material exhibits ferromagnetism, where the magnetic moments are aligned in the same direction. Above this temperature, thermal agitation overcomes the magnetic interactions, and the material becomes paramagnetic.

Superconductivity

One of the most fascinating phenomena observed in condensed matter physics is superconductivity. Superconductors are materials that can conduct electricity without any resistance below a certain temperature. This is called the critical temperature. The electrical resistance becomes zero, allowing perfect conduction of electricity.

The Meissner effect is another important property of superconductors, where they expel magnetic fields from within them. It is represented as:

Superconductors

Superconductivity has important technological implications, including magnetic resonance imaging (MRI) and maglev trains.

Quantum Hall effect

The quantum Hall effect is a quantum mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. It is characterized by the quantization of the Hall resistance.

R_H = frac{h}{e^2n}

Here, R_H denotes the Hall resistance, h is the Planck constant, e is the elementary charge, and n is an integer. Quantizing the Hall resistance in this way makes the effect highly precise, useful for defining standards of electrical resistance.

Applications and technical effects

Condensed matter physics has far-reaching implications for technology and industry. Its principles are crucial to the development of semiconductors, which are the foundation of modern electronics, including computers, smartphones, and GPS devices.

Quantum materials, which exhibit quantum mechanical effects on the macroscopic scale, are another hot topic. They are essential for the development of quantum computing and quantum sensing devices, which promise to revolutionize technology.

In conclusion, condensed matter physics is a versatile field that combines theoretical concepts with practical applications. It is a constantly growing area of research whose implications span many areas of science and technology. Understanding this field involves exploring both fundamental aspects, such as electron interactions and crystal structure, and applied dimensions, including novel material design and technological innovations.


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Condensed matter physics

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