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 → Quantum mechanics → Relativistic quantum mechanics ↓


Dirac equation


The Dirac equation is an essential concept in relativistic quantum mechanics, bridging the gap between the quantum world and the theory of relativity by providing a framework for understanding particles moving close to the speed of light. This equation, formulated by British physicist Paul Dirac in 1928, revealed the existence of antimatter and deeply influenced the development of modern physics.

Introduction to relativistic quantum mechanics

To understand the Dirac equation, it is important to understand the context in which it was developed. Classical physics as formulated by Newton worked well for centuries, but it had limitations, particularly in explaining very high velocities close to the speed of light or phenomena on the atomic scale. The theory of relativity and quantum mechanics addressed these challenges differently.

The need for a new equation

When physicists combined these two revolutionary theories in the early 20th century, they faced a major problem: the Schrödinger equation, which was fundamental to nonrelativistic quantum mechanics, did not align with Einstein's special relativity. One of the primary objectives in developing the Dirac equation was to find a wave equation compatible with both quantum mechanics and the theories of relativity.

Understanding the Dirac equation

The Dirac equation is a relativistic wave equation that describes how quantum states of matter behave when moving at a speed comparable to that of light. It is represented as:

iγμ∂μψ - mψ = 0

The symbols here have the following meanings:

  • i : imaginary unit, √(-1).
  • γμ : gamma matrices obeying some algebraic rules.
  • ∂μ : a four-gradient operator, including time and spatial derivatives.
  • ψ : a Dirac spinor, representing the position of a particle in this equation.
  • m : rest mass of the particle.

Exploration of gamma matrices

Gamma matrices are important to the Dirac equation. They are constructed such that their products satisfy anti-commutation relations. Specifically, the matrices are defined as:

{γμ, γν} = 2gμνI

Where:

  • {γμ, γν} denotes the anticommutator of two matrices.
  • gμν is the Minkowski metric of spacetime.
  • I is the identity matrix.

Gamma matrices are typically expressed as 4x4 matrices, which serve to ensure the compatibility of the Dirac equation with the theory of relativity by incorporating spin degrees of freedom.

Spinners and their importance

The Dirac equation uses spinors, which differ from vectors because they describe additional degrees of freedom related to the particle spin. Particles such as electrons have spins that can take half-integer values, which distinguishes them from classical point particles.

A spinor can be represented in two-component form as follows, although in four-component form in the Dirac equation:

Ψ = | ψ₁ | | ψ₂ |

Discovery of antimatter

One of the most remarkable results of the Dirac equation was the theoretical prediction of antimatter. Dirac found that his equation contained solutions with negative energy, which initially puzzled physicists. However, Paul Dirac proposed that these might correspond to particles with opposite charge, or antimatter. In 1932, the positron – the antiparticle of the electron – was discovered, which confirmed his prediction.

The Klein–Gordon equation: a predecessor to Dirac's work

Prior to Dirac's work, the Klein–Gordon equation was an early attempt to describe relativistic particles:

(□ + m²)ψ = 0

This second-order wave equation is suitable for scalar particles, but faced problems with negative probability densities. Dirac resolved these issues by designing a first-order equation in both time and space, introducing a more rigorous treatment of particles with intrinsic spin, and providing probabilistic interpretations compatible with quantum mechanics.

Visualization of relativistic spin

The concept of spin can be represented in a simple, visual example. Consider arrows indicating spin direction. Particles can act like tops, spinning along specific orientations. In Dirac's framework, these orientations can be positive or negative, corresponding to the spin pointing "up" or "down."

In this visualization, the red arrow may indicate the positive spin orientation while the blue arrow indicates the negative spin orientation. This simple representation helps to understand the complex concept of quantum mechanical spin.

Mathematical properties and implications

Dirac matrices have unique algebraic properties that enable calculations within the relativistic regime. They represent transformations concisely, integrate special relativity into quantum mechanics, and seamlessly incorporate Dirac's notion of electron spin.

Dirac Sea

Dirac proposed a theoretical model known as the Dirac sea to accommodate particles with "negative energy". This model posited that the vacuum is an infinite sea of negative-energy states. When energy is introduced, such as from a photon, it can lift a particle out of this sea, creating a visible "hole" that is interpreted as a positron. Although not fully corrected by later developed quantum field theories, this idea spurred advances in understanding particles and antiparticles within quantum physics.

Dirac equation in quantum field theory

Modern physics often incorporates the Dirac equation within the broader spectrum of quantum field theory. In this framework, particles are treated as excitations in a field, rather than as individual entities. The Dirac field describes fermions, particles with half-integer spin values, which influence countless areas, from fundamental interactions to technological advancements such as semiconductor devices.

Conclusion

The Dirac equation is a powerful synthesis of quantum mechanics and special relativity, fundamentally changing our understanding of particle physics and paving the way for the discovery of antimatter. The introduction of spinors and the prediction of antimatter are just a few of its many contributions. With applications extending beyond theoretical frameworks to practical technologies, the Dirac equation remains a cornerstone of modern physics.


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Dirac equation

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