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 entanglement and measurement ↓


Quantum entanglement and measurement in quantum mechanics


Quantum mechanics is a fundamental theory of physics that describes the physical properties of nature on the scale of atoms and subatomic particles. Its many interesting aspects involve quantum entanglement and the process of measurement, which reveals the unique properties of the quantum world.

Introduction to quantum entanglement

Quantum entanglement is the phenomenon in which two or more particles become intertwined, and the state of one particle instantaneously affects the state of the other, no matter how far apart they are. This unique relationship occurs because the particles involved share quantum properties that result in a collective state.

To illustrate entanglement, consider two particles, A and B. If A and B are entangled, then measuring a property such as spin on particle A allows us to predict with certainty the outcome of the same measurement on particle B. This relationship holds even when the particles are extremely far from one another.

Basic concepts

Entanglement challenges the classical notion of locality, which holds that objects are only directly affected by their immediate surroundings. In classical physics, an interaction must travel through space to affect an object at a distance. Entangled particles challenge this by exhibiting instantaneously correlated behaviour.

Mathematically, consider two particles in a simple entangled state:

|ψ> = α|00> + β|11>

Here, |ψ> denotes the entangled quantum state, α and β are complex numbers, and |00> and |11> are the possible states of the particle pair. The coefficients α and β are as follows:

|α|^2 + |β|^2 = 1

This indicates that measuring one particle gives precise knowledge of the state of the other particle.

Visual example: Particle A Particle B

This illustration shows two entangled particles, with the blue line indicating their quantum link.

Quantum Measurement

Quantum measurement disrupts the quantum state, forcing it into a specific configuration. This effect is deeply connected to entanglement. While classical measurement simply observes an already existing state, quantum measurement fundamentally alters the state being measured.

Measurement procedure

When measuring a quantum system, its wave function, which describes all possible states of the system, 'collapses' into one of the possible eigenstates. The probability of the system collapsing into a specific state is given by the square of the amplitude associated with that state.

For a given wave function:

Ψ = c1|ψ1> + c2|ψ2> + ... + cn|ψn>

The probability that the system is found in the state |ψi> after measurement is given by:

P(ψi) = |ci|^2

Here, ci is the coefficient for the state |ψi> before the measurement.

Measurement implications

The measurement process forces the quantum system to 'choose' a particular state, thereby removing any prior superposition or entanglement. For example, if two particles are entangled, measuring one instantly reveals the state of the other, even if they are far apart.

Text example:

Imagine a coin toss but with a quantum twist: unless you look, the coin will be both heads and tails simultaneously. This illustrates the principle of superposition. Now, if entangled, two coins tossed in different rooms, checking one, instantly informs you of the state of the other.

Conclusion

Quantum entanglement and measurement reflect the non-intuitive principles of quantum mechanics that clearly distinguish it from classical physics. These concepts are not only fundamental to our understanding of the quantum realm, but are also essential for technologies such as quantum computing and quantum cryptography.

In quantum computing, entanglement enables qubits to perform complex calculations more efficiently than classical bits. Moreover, this phenomenon forms the backbone of secure quantum communication protocols, where entanglement can be used to detect eavesdropping, due to the unique properties of measurements.

Final thoughts

The study of entanglement and measurement in quantum mechanics is an emerging field that promises insights and technologies that could transform our understanding and interaction with the world. While challenging, these theories provide a glimpse into the profound and often bizarre workings of the universe at its most fundamental levels.

Continued research in these areas is essential, as they hold the key to unraveling quantum mysteries and ushering in a new era of technological advancement.

In short, quantum mechanics invites us into a universe where reality is not always what it seems. Understanding the role of entanglement and measurement requires an open mind and a willingness to engage with ideas far removed from our everyday experiences.


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