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


Relativistic quantum mechanics is a fundamental theory that combines the principles of quantum mechanics with Einstein's theory of relativity. It attempts to describe the behavior of particles moving at speeds close to the speed of light and incorporates both the probabilistic nature of quantum mechanics and the deterministic nature of relativity.

To understand relativistic quantum mechanics, let's revisit quantum mechanics and special relativity independently. In quantum mechanics, one of the main equations is the Schrödinger equation, which describes how the quantum state of a physical system changes over time. For a single particle in a potential field, it is expressed as:

iħ ∂ψ/∂t = Ĥψ

Where:

  • i is the imaginary unit.
  • ħ is the decreasing Planck constant.
  • ∂ψ/∂t is the partial derivative of the wave function ψ with respect to time.
  • Ĥ is the Hamiltonian operator, corresponding to the total energy of the system.

However, the Schrödinger equation does not incorporate relativistic effects, which become important at speeds close to the speed of light. To address this, one has to consider Einstein's theories of special relativity, which suggest that the laws of physics are the same for all non-accelerating observers and that the speed of light in a vacuum is constant.

The cornerstone of the special theory of relativity is the famous energy–momentum relation, given as:

E² = p²c² + m₀²c⁴

Where:

  • E is the energy of the particle.
  • p is the momentum.
  • c is the speed of light.
  • m₀ is the rest mass of the particle.

To integrate quantum mechanics with special relativity, we turn to the concepts of relativistic quantum mechanics, primarily using the Dirac equation, which was developed by Paul Dirac in 1928. The Dirac equation accounts for the behavior of electrons and predicts phenomena such as antimatter and spin.

Dirac equation

The Dirac equation is a relativistic wave equation for fermions, which captures the essence of particles such as the electron. It provides a description that reflects the prerequisites of both relativity and quantum mechanics:

(iγⁿ∂ₙ - m)ψ = 0

In this equation:

  • γⁿ are gamma matrices, which encode the spinor structure of the particles.
  • ∂ₙ is the four-gradient, representing space-time derivatives.
  • m is the rest mass of the particle.
  • ψ is the wave function represented as a spinor.

The Dirac equation successfully emerges from Hermann Minkowski's four-dimensional space-time concept and provides additional solutions corresponding to antiparticles.

Antiparticle

One of the most paradigm-shifting predictions of the Dirac equation is the existence of antiparticles. These are particles that have the same mass as their corresponding ordinary particles, but have opposite electric charge and other quantum numbers.

For example, the antiparticle of the electron is the positron, which has a positive charge. The prediction and subsequent discovery of the positron were landmark achievements that verified the truth of the Dirac equation.

Spin and the Pauli exclusion principle

The Dirac equation also explains the concept of spin, a fundamental property of quantum particles. Spin is an intrinsic form of angular momentum, distinct from orbital angular momentum, and is described by the angular momentum quantum number.

The spin values for electrons are ±½, and this leads to the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state simultaneously. This principle is essential in explaining the structure of atoms and the properties of matter.

Below is given the symbolic representation of electron spin:

|↑> |↓>

Coulomb force correction - Lamb shift

In hydrogen atoms, the interaction between the electron and the nucleus is not fully accounted for by the energy levels. The Dirac equation provides a more accurate description with the Lamb shift, which corrects for anomalies arising from quantum electrodynamic interactions.

This effect is caused by the electron interacting with its own electromagnetic field and results in subtle differences in energy levels that would otherwise be predicted to be equal.

Feynman diagrams

Feynman diagrams are often used to represent processes in relativistic quantum mechanics. They depict the behavior of subatomic particles through lines and vertices in space-time, providing a pictorial shorthand for complex equations.

e- → e- + γ

This simple example must be understood within the rules that guide the interactions: electron (e-) and photon (γ).

Weaknesses and extensions

Despite covering many essential aspects, relativistic quantum mechanics cannot incorporate the principle of gauge invariance with the same strength as it is established in quantum field theory (QFT). QFT extends these ideas by considering particles as excited states of the underlying fields, thereby fully integrating the concepts of relativity and quantum mechanics.

Conclusion

In short, relativistic quantum mechanics provides essential insights into understanding the behavior of particles at high energies and connects them to observable phenomena, thereby adding depth to theoretical and applied physics. Although it represents a fundamental merger of quantum mechanics with relativity, continued efforts in quantum field theory and beyond indicate an ongoing journey towards a more unified understanding of the universe.

The vastness and subtlety of relativistic quantum mechanics are reflected through equations, examples and diagrams, thereby expanding our understanding and generating new directions of investigation in the field of science and reality.


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Relativistic quantum mechanics

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