Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate ↓


Quantum mechanics


Introduction

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at small scales, typically at the atomic and subatomic level. Unlike classical physics, which can often be viewed through simple mechanical arrangements, quantum mechanics delves deep into the behavior of the smallest particles in the universe, where rules and predictions become highly probabilistic and less intuitive.

Historical background

Quantum mechanics has its roots in early 20th-century studies, when scientists began investigating the nature of light, atoms, and other tiny particles. The notion that energy is quantized — that is, it comes in discrete quantities — was first proposed by Max Planck in 1900. Later, in 1905, Albert Einstein took this further by explaining the photoelectric effect using the idea of quanta, which he called photons.

The development of quantum mechanics continued with contributions from scientists such as Niels Bohr, who proposed the Bohr model of the atom, and Louis de Broglie, who suggested wave-particle duality. From these foundations, quantum theory evolved, with significant advances made through the formulation of Schrödinger's wave equation and Heisenberg's matrix mechanics.

Key concepts

Wave–particle duality

One of the main principles of quantum mechanics is wave-particle duality. This principle states that every particle or quantum entity can be partially described not only in terms of particles, but also in terms of waves. This is visually represented as follows:

particles <=> waves

For example, light can behave as both a wave (interfering and diffracting) and a particle (a stream of photons, as explained by the photoelectric effect). This duality is important for understanding quantum phenomena.

Uncertainty principle

Heisenberg's uncertainty principle is another cornerstone of quantum mechanics. It states that there is a fundamental limit to the accuracy of knowing certain pairs of physical properties, such as position and momentum, simultaneously. Mathematically, it is expressed as:

∆x ≥ Δp ≥ ħ / 2

Here, Δx denotes the uncertainty in position, Δp denotes the uncertainty in momentum, and ħ is the reduced Planck constant. This principle implies that, at quantum levels, there is a limit to how much we can know about a system, which affects our ability to predict the behavior of particles with great accuracy.

Quantum superposition

Quantum superposition refers to the concept that particles can exist in multiple states simultaneously, until they are observed or measured. A simple example of this is Schrödinger's cat, a thought experiment in which a cat locked in a box is both alive and dead until the box is opened and the state is resolved.

cat position

Quantum entanglement

Quantum entanglement is a phenomenon in which particles become linked together in such a way that the state of one particle directly affects the state of another particle, no matter the distance between them. If two particles are entangled, actions performed on one instantly affect the other, a concept that Einstein called "spooky actions at a distance." It is important to the fields of quantum computing and quantum cryptography.

Probability wave function

In quantum mechanics, particles are described by wave functions, which are mathematical functions that provide probabilities for all possible outcomes of a measurement. The wave function, usually denoted by Ψ(Psi), contains all the information about a system. The probability of finding a particle in a particular state or position is given by the square of the absolute value of its wave function:

p(x) = |Ψ(x)|²

This means that prediction of the behavior of a particle is inherently probabilistic and not deterministic.

Important equations

Schrödinger's equation

Schrödinger's equation is the fundamental equation of quantum mechanics and provides a way to calculate the wave function of a system. It can be written in its time-dependent form as:

∂Ψ/∂t = ĤΨ

Here, i is the imaginary unit, ħ is the reduced Planck constant, Ψ is the wave function, and Ĥ is the Hamiltonian operator, which represents the total energy of the system. The solution of this equation provides the future behavior of the quantum system.

The wave function Ψ

Born rules

The Born rule is a fundamental rule in quantum mechanics that describes how to calculate the probability of an outcome from a quantum experiment. It relates the wave function to probabilities using:

P = |Ψ|²

where P is the probability and Ψ is the wave function associated with the position being measured.

Applications in modern technology

Quantum mechanics is not just theoretical; it has important applications in modern technology that have changed the world.

Quantum computing

Unlike classical computers, which use bits as the smallest unit of data, quantum computers use quantum bits, or qubits. These can exist in multiple states at once due to superposition, allowing quantum computers to solve certain problems much faster than classical computers.

Quantum cryptography

Quantum cryptography uses the principles of quantum mechanics to create secure communication systems. The most famous concept is quantum key distribution (QKD), which enables two parties to share a secret key with security guaranteed by the rules of quantum mechanics.

Quantum sensors

Quantum sensors take advantage of quantum correlations such as entanglement to achieve higher sensitivity and accuracy in measurements than classical sensors. Its applications range from medical imaging to geophysical exploration.

Conclusion

Quantum mechanics is an interesting and complex field that revolutionizes our understanding of the universe at the most fundamental level. Despite its paradoxical and probabilistic nature, it has laid the groundwork for significant technological advances and remains a rich field of research. As we continue to explore this quantum world, we better understand the peculiar rules that govern the smallest scales of existence and develop more advanced applications that use these principles.


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Quantum mechanics

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