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 ↓


Foundations of quantum mechanics


Introduction

Quantum mechanics is a fundamental theory in physics that describes the nature of matter at small scales, such as atoms and subatomic particles. It is a mathematical framework that provides a consistent and comprehensive description of the arrangement of energy and matter in our universe. Unlike classical mechanics, which is deterministic and intuitive, quantum mechanics introduces probability and uncertainty, leading to results that sometimes challenge common sense.

Wave–particle duality

One of the most important concepts in quantum mechanics is wave-particle duality. Particles such as electrons exhibit both wave-like and particle-like properties. This duality is evident in the famous double-slit experiment.

Double-slit experiment

Consider a barrier with two narrow slits and a screen behind it. When light (or electrons) strikes the barrier, it passes through the slits and creates an interference pattern on the screen, which reflects wave behavior. If one slit is blocked, the interference pattern disappears, which reflects particle behavior. This experiment shows that the same entity can behave as both a wave and a particle.

Uncertainty principle

Heisenberg's uncertainty principle is a key concept in quantum mechanics, emphasizing a fundamental limit to how precisely we can know certain pairs of properties of a particle, such as position and momentum.

Δx * Δp ≥ ħ/2

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck constant. The more precisely one property is measured, the less precisely the other can be known.

Superposition

The principle of superposition is another interesting aspect of quantum mechanics. It states that a quantum system can exist in multiple states simultaneously until it is measured. A famous thought experiment illustrating this principle is Schrödinger's cat, which is both alive and dead until it is observed.

alive dead

Quantum entanglement

Quantum entanglement is a phenomenon in which particles become intertwined in such a way that the state of one particle instantly affects the state of another particle, no matter how much distance there is between them. This instantaneous correlation between particles was described by Einstein as "spooky action at a distance."

Bell's theorem

Bell's theorem provides a way to test the predictions of quantum mechanics against classical physics through inequalities. Experiments have consistently validated the predictions of quantum mechanics, showing that entangled particles exhibit correlations that cannot be explained by classical physics.

Mathematical framework

The mathematical framework of quantum mechanics is primarily built around linear algebra. Below are some basic mathematical concepts:

State vector and Hilbert space

The state of a quantum system is represented by a state vector, usually denoted by |ψ⟩. These vectors exist in a complex vector space called Hilbert space, which provides a home for all possible states of the system.

Operators

Physical quantities in quantum mechanics are represented by operators acting on the state vector. Observables such as position, momentum, and energy correspond to specific operators. For example:

Ĥ|ψ⟩ = E|ψ⟩

Here, h is the Hamiltonian operator, corresponding to the total energy of the system, and E is the energy eigenvalue.

Schrödinger equation

The Schrödinger equation is the fundamental equation of motion in quantum mechanics. It describes how the quantum state of a physical system changes over time:

iħ ∂|ψ⟩/∂t = Ĥ|ψ⟩

This equation is important for predicting the behavior of a system in quantum mechanics and allows the determination of wavefunctions, which are probability amplitudes that describe the distribution of a particle in space.

Quantum mechanics interpretations

Due to the paradoxical nature of quantum mechanics, many interpretations have been proposed to understand its philosophical implications. Here are a few:

The Copenhagen interpretation

The Copenhagen interpretation states that a quantum system remains in a superposition until it is observed, at which point it collapses into a single state. This is perhaps the most traditional and widely taught interpretation.

Many-worlds interpretation

Proposed by Hugh Everett III, this interpretation holds that all possible outcomes of a quantum measurement become real, each in its own universe. This eliminates the concept of wave function collapse but leads to infinite branches of universes.

Conclusion

Quantum mechanics is a field of physics that challenges intuition, yet governs the fundamental forces and particles that make up the universe. Its principles of wave-particle duality, superposition, and entanglement challenge classical physics, paving the way for breakthroughs in technology and our understanding of the universe. Although its interpretations and implications can be complex and varied, they continue to fascinate and fascinate scientists and philosophers alike, pointing to a universe that is far richer and more complex than classical physics could ever explain.


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Foundations of quantum mechanics

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