PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD ↓


Condensed matter physics


Condensed matter physics is a branch of physics that deals with the physical properties of condensed phases of matter. The most familiar condensed phases are solids and liquids, which arise from electromagnetic forces between atoms. The discipline is concerned with understanding the behavior of these phases using the principles of quantum mechanics, electromagnetism, and statistical mechanics.

Introduction

Condensed matter physics is a field that touches many aspects of modern technology and research. It is concerned with understanding the intrinsic properties and behaviours of condensed phases, often with the aim of explaining how they arise from the fundamental laws of elementary particles. It is a field where large-scale phenomena are explained in terms of their microscopic components and interactions.

Key concepts

Some of the key concepts in condensed matter physics include crystal structure, atomic bonding, and quantum mechanics. The behavior of solids and liquids can be very different from that of the low-density gas phases because atoms in these phases are in close contact with one another.

Atomic lattice

An important concept is the atomic lattice, which is a periodic arrangement of atoms in a crystalline solid. Imagine atoms as points in a repeating, three-dimensional pattern. This pattern determines many properties of the solid, such as electrical conductivity and hardness.

Let's imagine a simple lattice structure: 

+---+---+---+---+
| o | o | o | o |
+---+---+---+---+
| o | o | o | o |
+---+---+---+---+
| o | o | o | o |
+---+---+---+---+
| o | o | o | o |
+---+---+---+---+

Types of condensed matter

Condensed matter physics investigates many states of matter, the most familiar of which are:

  • Solid: Characterized by structural rigidity and resistance to change in shape.
  • Liquids: They have a definite volume, but they take the shape of their container.

Other strange states of matter can occur under extreme conditions, such as extremely low temperatures or extreme pressures, including:

  • Superconductors: Materials that can conduct electricity without resistance below a certain temperature.
  • Bose-Einstein condensate: A state of matter that forms at temperatures close to absolute zero.

Quantum mechanics and condensed matter

Quantum mechanics plays an important role in explaining the properties of condensed matter. The electrons in an atom reside in quantized energy levels. When these atoms come together, as in a metal, these energy levels form bands.

A simplified view of the electron bands in a metal can be represented as follows:

Filled Band conduction band Blank Band

Conductivity and band theory

Conductivity in solids can be explained through band theory, which holds that electrons occupy energy bands. When electrons have available energy states (in an adjacent band), they can move around freely, making the solid a conductor.

Metals have partially filled bands that allow electrons to flow easily, while insulators have filled bands that are separated by energy gaps, which prevent electron flow. 

Gap ----------- -----------
Filled Band    Empty Band

Electrons have to gain energy to cross the gap, which doesn't happen easily in insulators, making them poor conductors.

Emerging phenomena

A fascinating aspect of condensed matter physics is emergent phenomena, where collective behaviours arise from simple interactions between many particles.

Superconductivity

Superconductivity is an emerging property where, below a critical temperature, electrons pair up into Cooper pairs and flow without resistance. This state enables electric current to flow endlessly in a loop without dissipating power as heat.

Here is a representation of electron pairing in a superconductor:

The Cooper couple

Bose–Einstein condensate

At temperatures close to absolute zero, a collection of atoms can collapse into the lowest quantum state, forming a Bose-Einstein condensate (BEC). This state exhibits wave-like properties on a macroscopic scale.

Research and technology

Research in condensed matter physics has led to technological advances. For example, the understanding of semiconductor physics paved the way for the development of the transistor and eventually modern computing.

Semiconductors

Semiconductors have a band gap smaller than insulators, but large enough to prevent easy electron flow at low energies. By doping them with impurities, their conducting properties can be closely controlled, which is important in electronics. 

Intrinsic Semiconductor:
----------- Gap ------------
Valence Band    Conduction Band

N-type Doping (Adding electrons):
---→   ---→   ---→
----------- -----------
            Valence Band (Electrons)

P-type Doping (Creating holes):
←---   ←---   ←---
----------- -----------
         (Holes)     Conduction Band

Magnetic materials

Magnetic properties arise due to the arrangement of magnetic moments in materials. Ferromagnetism, where the spins are aligned parallel, leads to a strong magnetic field. Materials such as iron, cobalt and nickel are known to have these properties.

Topological materials

Recent research has focused on topological insulators, materials with an insulating interior but a conducting surface. Their electronic states are robust and resistant to disturbances, promising for applications in quantum computing.

Conclusion

Condensed matter physics is a broad and dynamic field that explores diverse phases and their properties. It provides fundamental understanding that drives technological advancement and helps describe a wide range of phenomena, from the mundane to the exotic. Its broad scope and depth ensure that it remains a field ripe for discovery and innovation.


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