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


Quantum mechanics is a fundamental theory in physics that describes the nature of matter at the smallest scales. Unlike classical mechanics, which is based on Newtonian laws, quantum mechanics is essential for understanding the behavior of atoms, electrons, photons, and other subatomic particles.

The birth of quantum mechanics

The development of quantum mechanics began in the early 20th century. Scientists observed phenomena that could not be explained by classical physics. A notable example of this was the photoelectric effect, where light falling on a metal surface knocks electrons out of it. Albert Einstein explained this by proposing that light exists in discrete packets called quanta or photons.

In 1900 Max Planck suggested that the energy of electromagnetic waves is quantized, and his work laid the groundwork for further quantum theories. Planck introduced the idea that energy can only be exchanged in discrete amounts called quanta. The formula that defines this relationship is:

E = hν

where E is the energy, h is the Planck constant (about 6.626 × 10-34 Js), and ν (nu) is the frequency of the radiation.

Example: If you have a photon with a frequency of 5 x 1014 Hz, the energy would be calculated as:

E = (6.626 × 10-34 Js) × (5 × 1014 Hz) = 3.313 × 10-19 J

Wave–particle duality

One of the cornerstones of quantum mechanics is wave-particle duality. This is the concept according to which every particle or quantum entity exhibits both particle and wave characteristics.

For example, an electron may exhibit behavior characteristic of a particle, such as when it scatters off another particle. However, it may also exhibit wave-like behavior, such as when it creates interference patterns.

Double slit experiment

To understand wave-particle duality, consider the double-slit experiment. When a beam of electrons passes through two closely spaced slits, it creates an interference pattern on a screen behind the slits. This is similar to what happens with water waves, illustrating the wave nature of electrons.

double slit experiment

Uncertainty principle

The uncertainty principle, introduced by Werner Heisenberg, is another fundamental concept of quantum mechanics. It states that it is impossible to accurately measure both the position and momentum of a particle at the same time.

The more precisely you know one of these values, the less precisely you can know the other. Mathematically, this is expressed as follows:

Δx * Δp ≥ ℏ / 2

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ℏ is the reduced Planck constant (h/2π).

Example: If the uncertainty in the position of an electron is 0.1 nm, then the minimum uncertainty in its momentum can be calculated as follows:

Δp ≥ ℏ / (2 * Δx)
Δp ≥ (1.0545718 × 10-34 Js) / (2 * 0.1 × 10-9 m) ≈ 5.2729 × 10-25 kg m/s

Quantum superposition

Quantum superposition is the principle according to which a quantum system can exist in multiple states simultaneously until it is measured. Only after a measurement does it collapse into one of the possible states. This leads to fascinating phenomena such as interference patterns and entanglement.

Quantum superposition |ψ₁⟩ |ψ₂⟩ |ψ> = c₁|ψ₁> + c₂|ψ₂>

Quantum entanglement

Quantum entanglement is a phenomenon in which particles become intertwined in such a way that the state of one particle cannot be described independently of the state of another, even when they are separated by large distances. This interesting aspect of quantum mechanics has been studied extensively for quantum computing and information theory.

Quantum mechanics and observers

In quantum mechanics the role of the observer is far more important than in classical physics. The act of measurement affects the system being observed. This idea suggests a kind of participatory universe where observations affect outcomes.

The famous thought experiment known as Schrödinger's cat illustrates this idea. In this experiment, a cat is placed in a sealed box containing a radioactive atom and a poison that will be released when the atom decays. According to quantum mechanics, the atom can exist in a superposition of decaying and not decaying, and so the cat is both alive and dead until someone opens the box and looks at it.

Quantum tunneling

Quantum tunneling is a phenomenon in which particles pass through a potential barrier that is not possible for them to cross. It is as if the particles "tunnel" through the barrier. This theory explains many processes in nature, such as nuclear fusion in stars.

Quantum tunneling

Quantization of energy levels

Quantum mechanics predicts that systems such as atoms and molecules have discrete energy levels. This is the basis for countless techniques and theories in chemistry.

For example, the energy levels of the hydrogen atom are given as follows:

E_n = -13.6 eV * (1/n^2)

where E_n is the energy of the n-th level. Electrons can move between these levels by absorbing or emitting photons of specific energies.

Applications of quantum mechanics

Quantum mechanics has led to the development of many technologies and branches of science that fundamentally shape modern life. These include:

  • Quantum computing: Taking advantage of quantum superposition and entanglement to perform computations that are beyond the capability of classical computers.
  • Medical imaging: Techniques such as MRI rely on the principles of quantum mechanics to function.
  • Semiconductors: Understanding electron behavior in materials has led to the development of all modern electronics, such as transistors and microchips.

As you can see, quantum mechanics explains and predicts a wide range of phenomena, providing a rich framework for understanding the universe at its most fundamental level. The continued exploration of quantum mechanics promises new technologies and deeper insights into the nature of reality.


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