Reference Library

Special relativity rests on two postulates: the laws of physics are the same in every inertial reference frame, and the speed of light in vacuum, c, is the same for every observer, regardless of their own motion or that of the source. Everything else in special relativity — time dilation, length contraction, the relativity of simultaneity — follows from these two statements alone.

An observer watching a clock or a ruler in motion relative to them measures time running slow and length shrunk along the direction of motion.

The Lorentz transformation relates the space and time coordinates of an event as measured in two inertial frames moving relative to one another, replacing the Galilean transformation of Newtonian mechanics. It reduces to the ordinary Galilean transformation when v ≪ c, and is the underlying source of time dilation, length contraction, and the relativity of simultaneity.

Mass and energy are interconvertible; every object at rest carries an intrinsic energy proportional to its mass.

A hot object radiates across a continuous spectrum that shifts to shorter wavelengths as temperature rises, with total power growing steeply with temperature. Classical physics could not reproduce the observed spectrum — it predicted infinite radiated power at short wavelengths, the 'ultraviolet catastrophe'. Planck resolved this by proposing that radiation is emitted in discrete energy packets, the first appearance of quantization in physics.

Light striking a metal can eject electrons, but only above a threshold frequency — a result classical wave theory could not explain.

A photon scattering off a free electron shifts to a longer wavelength, exactly as expected if the photon carries momentum like a particle.

Just as light behaves as both wave and particle, so does matter — every moving particle has an associated wavelength.

The wave function ψ describes a quantum system completely; it has no direct physical meaning itself, but its squared magnitude |ψ|² gives the probability density of finding the particle at a given position. Unlike a classical particle's definite trajectory, a quantum particle's position is described only probabilistically until measured.

The equation governing how a quantum system's wave function evolves, playing the role for quantum mechanics that Newton's second law plays classically.

Confining a particle between two impenetrable walls restricts it to a discrete set of allowed energies.

Quantum mechanics allows a particle a nonzero probability of passing through a potential energy barrier even when its energy is less than the barrier's height — classically forbidden, but permitted because the wave function decays exponentially rather than dropping to zero inside the barrier. Tunnelling probability falls off exponentially with barrier width and height, which is why it is significant only on atomic and subatomic scales — radioactive alpha decay and the scanning tunnelling microscope both rely on it.

An electron bound in the Coulomb potential of a proton can only occupy certain discrete energy levels.

A hydrogen electron state is specified by four quantum numbers: principal n (energy), orbital ℓ (angular momentum magnitude, 0 to n−1), magnetic m_ℓ (angular momentum orientation, −ℓ to +ℓ), and spin m_s (±½). Together they label every distinct quantum state available to the electron.

Electrons are fermions with spin ½, obeying the Pauli exclusion principle: no two electrons in the same atom can occupy an identical quantum state — all four quantum numbers matching. This single rule explains the structure of the periodic table, as successive elements fill available quantum states in order, producing the recurring pattern of chemical properties.

Transitions between hydrogen's energy levels emit or absorb photons of specific wavelengths, producing its characteristic line spectrum. A laser exploits population inversion — more atoms in an excited state than the ground state — so a passing photon of the right energy triggers a cascade of identical, coherent photons through stimulated emission rather than being absorbed.

Atoms bond by sharing (covalent), transferring (ionic), or pooling (metallic) valence electrons to lower the total energy of the system relative to isolated atoms. The type of bonding — and the resulting arrangement of atoms — determines whether a solid is a crystal, a glass, or something in between.

In a solid, the discrete energy levels of individual atoms broaden into continuous bands as atoms are brought close together and their wave functions overlap. Whether a material conducts, insulates, or behaves as a semiconductor depends on how these bands are filled and how large the energy gap between them is.

A semiconductor has a small energy gap between its filled valence band and empty conduction band — small enough that thermal energy or added impurities (doping) can promote electrons across it. Doping with donor atoms (n-type) adds free electrons; doping with acceptor atoms (p-type) adds free 'holes' — missing electrons that behave as positive charge carriers. A junction between p-type and n-type material is the basis of the diode and the transistor.

Below a critical temperature, certain materials lose electrical resistance entirely — not merely a very small resistance, but exactly zero. The effect arises from electrons pairing up (Cooper pairs) in a way that lets them move through the lattice without scattering. A current started in a superconducting loop persists, in principle, indefinitely.

A nucleus is bound more tightly than the sum of its separate protons and neutrons, with the difference carried away as binding energy.

An unstable nucleus decays at a rate proportional to how many undecayed nuclei remain, giving a fixed half-life independent of sample size or age.

Fission splits a heavy nucleus into lighter fragments; fusion combines light nuclei into a heavier one. Both release energy because the products sit higher on the binding-energy-per-nucleon curve — closer to iron — than the reactants did. Fission powers current nuclear reactors; fusion powers stars and remains an active area of energy research on Earth.

The standard model classifies every known elementary particle into quarks and leptons (matter particles, obeying the Pauli exclusion principle) and gauge bosons (force carriers: photons for electromagnetism, gluons for the strong force, W and Z bosons for the weak force). Only gravity remains outside this framework, with no confirmed quantum description.