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 ↓


Uncertainty principle


The uncertainty principle is a fundamental concept in quantum mechanics that was formulated by Werner Heisenberg in 1927. It states that there is an inherent limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. This principle has profound implications for our understanding of the physical world, especially at the microscopic scale. It challenges classical intuition and highlights the inherent probabilistic nature of quantum systems.

Introduction to quantum mechanics

Before diving into the uncertainty principle, it is essential to understand the basics of quantum mechanics. Quantum mechanics is the branch of physics that deals with the behaviour of particles at the atomic and subatomic levels. Unlike classical physics, which describes the macroscopic world with great precision, quantum mechanics reveals a universe where particles exhibit both wave-like and particle-like properties and where certainty gives way to probability.

Wave–particle duality

One of the key concepts in quantum mechanics is wave-particle duality. Particles, such as electrons, can behave as both discrete particles and waves. This duality is clearly demonstrated through experiments such as the double-slit experiment, which shows an interference pattern when particles pass through two slits and behave like waves.

        // Double-slit experiment setup
        // ParticleSource emits individual particles
        // particles pass through two slits
        // Draw an interference pattern on the detection screen

What is the uncertainty principle?

The uncertainty principle is a statement about the limits of measurement in the quantum realm. It affects our ability to measure pairs of complementary properties with exact precision. The most famous example of such a pair is position (x) and momentum (p). Mathematically, the uncertainty principle is expressed by the inequality:

        Δx * Δp ≥ ħ/2

Here, Δx represents the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck constant, which is approximately 1.0545718 × 10^-34 Js. The principle implies that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.

Example of the principle

Representation of particles and waves

Think of the particle as a wave packet, a local wave with a wide range of positions and momentum. By shrinking the wave packet to get a precise position, the corresponding wavelength becomes less defined, which increases the uncertainty in momentum. This is like tuning a musical instrument: tightening a string reduces the range of possible vibrations (frequency), which is the same as tightening the certainty about position and loosening the certainty about momentum. Below is a visual representation:

Everyday analogies

Imagine you are trying to measure two properties simultaneously, such as the size and speed of a spinning wheel. The faster it spins, the harder it is to measure the size of any specific part because of blurriness. Heisenberg's insight was not just about our limitations in measurement, but about an intrinsic quality of quantum objects, such as particles behaving in an uncertain way.

Mathematical derivation of the theory

The uncertainty principle can be derived rigorously using the framework of quantum mechanics and can be understood intuitively through the Fourier transform of wave functions.

Wave function and Fourier transform

The wave function is a fundamental concept that describes the quantum state of a system. By taking the Fourier transform of the wave function, you convert from one description to another, often from position to momentum space, and vice versa. Precision in one domain leads to uncertainty in the other, due to the inherent properties of the Fourier transform.

        // Mathematical representation of wave function
        Ψ(x) = A * e^(i(kx - ωt))
        
        // Fourier transform relating Ψ in position and momentum spaces
        Φ(p) = 1/√(2πħ) ∫ Ψ(x) * e^(-ipx/ħ) dx

Implications of the etymology

The mathematical descriptions provide a vivid illustration of the interference of probabilities, expressing the fate of particles neither as classical objects nor as mere uncertainties, but as dynamical components evolving in time.

Implications of the uncertainty principle

Quantum world vs. classical world

Historically, classical mechanics assumed that every detail of a system could potentially be measured and determined. Newton's deterministic universe suffered a profound realization through the uncertainty principle: at the atomic and subatomic levels, precision and certainty are inherently limited.

Philosophical questions

The uncertainty principle challenges traditional notions of objectivity and certainty. In the quantum world, the statement "the cat is either dead or alive" in Schrödinger's famous thought experiment reflects not only our knowledge, but also the state of the system that remains unstable until it is observed.

Practical applications

The uncertainty principle is not just a theoretical construct; it also has practical applications. From refining the accuracy of electron microscopy to influencing the design of quantum computers, the principle is central to modern technology and science.

Quantum chemistry

Predictive understanding of atomic behavior and chemical bonding is affected by recognizing uncertainties in electron placement. In addition, tunneling, a quantum phenomenon made possible by the uncertainty principle, is important in processes such as nuclear fusion in stars.

Heisenberg microscope thought experiment

In Heisenberg's own thought experiment, attempting to measure the position of an electron using a photon actually changes the electron's momentum. Such interactions show how the process of observation not only reveals nature but sometimes substantially affects it.

Conclusion

As we delve deeper into the uncertainty principle, the complex dance of certainty and unpredictability in quantum mechanics challenges not just our scientific abilities but also our philosophical contemplation. It expands our horizons of understanding what is knowable, changing the landscape of physics forever.

Through understanding this concept, we begin to appreciate the dual character of quantum entities and the beautiful complexity of the universe, which does not seem intuitively understandable but turns out to be quite amazing when we explore its deeper levels.


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Uncertainty principle

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