Roadmap

A physical quantity is a number together with a unit; SI fixes seven base units that every other unit is built from.

Precision describes how repeatable a measurement is; accuracy describes how close it is to the true value — a measurement can have one without the other.

An atom has a tiny, dense, positively charged nucleus made of protons and neutrons, surrounded by electrons; atoms of the same element with different neutron counts are called isotopes.

Electric charge comes in two kinds, positive and negative, and an electric current is simply charge flowing through a conductor over time.

Light reflects off a surface at the same angle it arrives, and refracts — changes direction — when it passes from one transparent material into another.

Speed is the distance an object covers divided by the time it takes, and it tells you nothing about the direction the object is moving.

Temperature measures how hot or cold something is; heat is energy that flows between objects because they are at different temperatures.

A wave carries energy from place to place through repeating oscillations, without the medium itself travelling along with it.

Pressure in a fluid increases with depth because each layer must support the weight of everything above it, so the pressure at a depth h below the surface equals the surface pressure plus the fluid's density times gravitational acceleration times depth.

An object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces — whether the object floats or sinks depends on how that buoyant force compares to its own weight.

A scalar has only magnitude; a vector has both magnitude and direction, and the two must never be added together.

Significant figures record how precisely a measured or calculated value is known, and a result can never be more precise than its least precise input.

Uncertainty is the range within which the true value is expected to lie; error is the difference between a measured value and the true value, which is usually unknown.

An unstable nucleus can spontaneously decay, emitting radiation; its half-life is the time it takes for half of a sample of that isotope to decay.

Light delivers its energy in discrete packets called photons, each carrying an energy proportional to the light's frequency.

The electric force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them, acting along the line joining them.

Voltage drives current through a circuit, and resistance opposes it — for many materials the three are linked by Ohm's law: voltage equals current times resistance.

An electric charge creates an electric field around it, and any other charge placed in that field feels a force proportional to the field strength there.

Electric power dissipated in a resistor equals current times voltage, and since the two are linked by Ohm's law it can equally be written as I²R or V²/R — three faces of one quantity that give the same answer whichever is easiest to use.

Resistivity is a material property describing how strongly a substance opposes current, independent of the size of a particular sample, so a wire's actual resistance grows with its length and shrinks with its cross-sectional area.

Moving charges and electric currents create magnetic fields, and a magnetic field in turn exerts a force on any moving charge that crosses it.

Snell's law relates the angles of incidence and refraction to the refractive indices of the two materials: a ray bends more the greater the difference between them.

A converging lens bends parallel rays of light so that they meet at its focal point, and the focal length measures how far away that point is.

Displacement, velocity and acceleration are vectors: each has a direction as well as a size, unlike the distance and speed used at a lower level.

An object stays at rest or keeps moving at constant velocity unless a net force acts on it — this resistance to a change in motion is inertia.

A projectile's horizontal and vertical motions are independent of each other, so a ball thrown sideways and one simply dropped from the same height hit the ground at the same time, even though only one of them is also travelling forward.

An object moving at constant speed around a circle is still accelerating, because its velocity's direction keeps changing even though its magnitude does not — that acceleration always points toward the centre of the circle.

Force equals mass times acceleration.

Whenever one object exerts a force on a second object, the second exerts an equal and opposite force back on the first.

Centripetal force is not a new kind of force but a name for whatever net force actually points toward the centre of a circular path — tension, gravity, friction, or a combination — and it equals mass times the speed squared divided by the radius.

Friction opposes relative sliding between two surfaces in contact, with a maximum static value that must be overcome before motion starts and a slightly smaller kinetic value once sliding is underway.

Impulse is the product of a force and the time it acts for, and it equals the change in an object's momentum.

Work is done when a force moves an object through a distance, and it transfers energy from one form or object to another.

An object at rest and staying at rest needs both its net force and its net torque to be zero — force balance alone can leave it free to spin, so both conditions must be checked independently.

Heat moves by three distinct mechanisms — conduction through direct contact, convection by moving fluid, and radiation by electromagnetic waves that need no medium at all — and identifying which one dominates in a given situation is what makes a heat-transfer problem solvable.

Most materials expand when heated because their particles move faster and take up more space on average.

Specific heat capacity is the energy needed to raise the temperature of one kilogram of a substance by one degree, and it differs widely between materials.

During a phase transition such as melting or boiling, the energy added goes into breaking bonds between particles rather than raising the temperature, which stays constant until the transition is complete.

A wave's speed equals its frequency times its wavelength, so for a wave travelling at a fixed speed, a higher frequency always means a shorter wavelength.

Sound is a pressure wave that needs a material medium to travel through, and its intensity — power per unit area — falls off with the square of distance from a point source, which is why loudness drops fast as you move away.

In simple harmonic motion the restoring force is proportional to the displacement from equilibrium and always points back towards it, producing a smooth, repeating oscillation.

For an incompressible fluid flowing through a pipe, the same volume must pass every cross-section each second, so the fluid speeds up wherever the pipe narrows and slows down wherever it widens.

Pressure applied anywhere to an enclosed, incompressible fluid is transmitted equally throughout it, so a small force on a narrow piston can produce a much larger force on a wide one — the basis of hydraulic lifts and brakes.

Surface tension arises because molecules at a liquid's surface are pulled inward by unbalanced intermolecular attraction, making the surface behave like a stretched elastic membrane that resists an increase in area, and this same effect drives capillarity — the rise or fall of liquid in a narrow tube — depending on whether the liquid's attraction to the tube wall is stronger or weaker than its attraction to itself.

An object falling through a fluid experiences a drag force that grows with its speed, and as it accelerates the drag force eventually grows to equal the net downward force, at which point the acceleration reaches zero and the object continues falling at a constant terminal velocity rather than speeding up indefinitely.

Earth's interior is layered by density — a solid inner core, a liquid outer core, a solid mantle that flows over geological time, and a thin rigid crust broken into tectonic plates that ride on the slowly convecting mantle beneath — driven by the same buoyancy and pressure-gradient physics that governs any fluid: hotter, less dense material rises while cooler, denser material sinks, and the resulting plate motions build mountains, open ocean basins, and drive nearly all earthquakes and volcanism.

A planet in energy balance radiates away, as long-wavelength infrared, exactly as much power as it absorbs from its star, and gases in its atmosphere that are transparent to incoming visible sunlight but absorb and re-emit outgoing infrared raise the planet's surface temperature above what a bare-rock energy balance alone would predict — the greenhouse effect.

Earth's atmosphere and its oceans are both fluids held in hydrostatic balance by gravity, so pressure and density fall off roughly exponentially with altitude in the air and rise with depth in the water; because air obeys the ideal gas law, a rising parcel expands and does work against the lower pressure around it, cooling adiabatically even with no heat exchanged — which is why the troposphere gets colder with altitude — while in the ocean, temperature and salinity together set the water's density and organise it into stable layers rather than a single well-mixed temperature.

An unstable nucleus sheds excess energy through one of several characteristic decay modes — alpha decay ejects a helium nucleus and reduces both mass and atomic number, beta decay converts a neutron to a proton (or vice versa) via the weak force while emitting an electron or positron and a neutrino, and gamma decay releases a high-energy photon from an already-formed nucleus without changing its identity at all.

Different detectors exploit different physical effects of ionizing radiation — gas ionization in Geiger-Müller counters, light flashes in scintillators, charge collection in semiconductor detectors — and each has real limitations, such as a dead time after every count during which further events go unregistered. Dosimetry turns raw counts into a biologically meaningful quantity by tracking the energy actually absorbed per unit mass of tissue.

A nucleus's mass is always slightly less than the total mass of its separate protons and neutrons, and this mass defect, converted via E = mc², equals the binding energy that would be needed to pull the nucleus completely apart — the binding energy per nucleon peaks around iron, which is why fusion releases energy for light nuclei and fission releases energy for heavy ones.

Fission splits a heavy nucleus into lighter fragments while fusion combines light nuclei into a heavier one, and both processes release energy because they move the resulting nuclei toward the peak of the binding-energy-per-nucleon curve near iron; fission can sustain a self-propagating chain reaction via released neutrons, while fusion requires extreme temperature and pressure to overcome mutual electrostatic repulsion between nuclei.

Capacitance measures how much charge a component can store per volt across it, and a charged capacitor holds energy in the electric field between its plates.

Kirchhoff's current law says the currents into a junction must equal the currents out; his voltage law says the voltage changes around any closed loop must sum to zero — together they solve any network of resistors and sources.

Inserting a dielectric between a capacitor's plates increases its capacitance by weakening the field between them, and the energy a charged capacitor stores is one half its capacitance times the square of its voltage.

Any network of resistors and sources, however complicated, looks from its two output terminals exactly like one ideal voltage source in series with one resistor (its Thévenin equivalent) or one ideal current source in parallel with one resistor (its Norton equivalent) — letting the whole network be replaced by two numbers for any further analysis.

A changing magnetic field through a loop of wire induces an electromotive force in it — the faster the field changes, the larger the induced voltage.

A charged particle moving through a magnetic field feels a force perpendicular to both its velocity and the field, so the force never speeds the particle up or slows it down — it only bends its path into a circle or a helix.

Materials respond to an applied magnetic field in three qualitatively different ways — diamagnetic (weakly repelled), paramagnetic (weakly aligned) and ferromagnetic (strongly aligned and self-sustaining) — and a ferromagnet's magnetization lags behind the field it has already seen, tracing out a hysteresis loop whose enclosed area is the energy lost to heat on every cycle.

Piezoelectric materials generate a voltage when mechanically stressed, and deform slightly when a voltage is applied, because squeezing their asymmetric crystal structure shifts internal charge centres apart; ferroelectric materials go further still, holding a spontaneous electric polarization even with no field applied at all — the electrical counterpart of ferromagnetism, complete with domains, a hysteresis loop and a Curie temperature above which the effect vanishes.

A generator spins a coil in a magnetic field to convert mechanical motion into an alternating EMF through electromagnetic induction, a motor is the very same machine run in reverse to convert electrical energy into mechanical motion, and a thermocouple sidesteps magnetism altogether — converting a temperature difference directly into a voltage through the Seebeck effect.

A charged particle moving perpendicular to a uniform magnetic field travels in a circle whose radius grows with the particle's momentum and shrinks with its charge and the field strength — the exact principle a mass spectrometer uses to sort ions by mass and a cyclotron uses to accelerate them along an ever-widening spiral.

When a current-carrying conductor sits in a magnetic field perpendicular to the current, the moving charge carriers are pushed sideways by the magnetic force, building up a small transverse voltage — the Hall voltage — that reveals both the sign and the density of the carriers doing the carrying.

Heating a metal surface gives its mobile conduction electrons enough thermal energy to overcome the material's work function and escape into the surrounding vacuum — thermionic emission — the electron source at the heart of vacuum-tube devices like the diode, which lets current flow in only one direction, and the triode, whose small control-grid voltage governs a much larger current, making it an early electronic amplifier.

A transformer uses a changing magnetic flux, shared between two coils wound on the same core, to step an AC voltage up or down by the ratio of the coils' turns — and because power in equals power out, stepping the voltage up steps the current down, which is why power grids transmit electricity at very high voltage: less current means far smaller resistive losses over long distribution lines.

Electric potential is potential energy per unit charge — a property of a location in a field, independent of any particular test charge placed there — so the potential energy of a specific charge is its own charge times the potential at its position.

Millikan suspended charged oil droplets between two capacitor plates and adjusted the electric field until each one hung motionless — gravity exactly balanced by the electric force — letting him read off that droplet's charge; repeating this for many droplets, every measured charge turned out to be a whole-number multiple of the same smallest value, showing that electric charge itself comes in discrete lumps rather than a continuous range.

A material's refractive index depends slightly on wavelength, so different colours of light bend by different amounts at an interface — the effect that spreads white light into a rainbow through a prism and is also what limits a simple lens's sharpness.

Beyond a critical angle, light travelling into a less dense material reflects entirely back rather than refracting through, which is how optical fibres trap light inside them.

Light is a transverse wave whose electric field can oscillate in any direction perpendicular to its motion; polarizing it restricts that oscillation to a single direction.

Magnification compares the size of an image to the size of the object producing it, and a negative value signals an inverted image — combining lenses in instruments like microscopes and telescopes multiplies each stage's magnification to reach far higher total magnification than any single lens could give alone.

A diffraction grating is a surface ruled with thousands of closely spaced slits, and its many-slit interference pattern produces bright fringes far sharper and better separated than a simple double slit gives, which is what makes it the standard tool for measuring wavelengths precisely.

Light reflecting off the top and bottom surfaces of a thin film — a soap bubble, an oil slick, an anti-reflective coating — travels slightly different distances and can interfere constructively or destructively depending on the film's thickness and the light's wavelength, which is why such films show shifting rainbow colours rather than a single reflected colour.

A photomultiplier tube (PMT) turns single photons into a large, measurable pulse of current by knocking one electron loose via the photoelectric effect, then multiplying it through a cascade of dynodes that each release several secondary electrons per electron they absorb. A CCD instead uses the photoelectric effect inside a semiconductor, at millions of independent pixels at once, storing each pixel's accumulated charge and reading it out to build up an image.

Radiometric quantities (radiant flux, in watts) measure light as pure physical energy, unweighted; photometric quantities (luminous flux, in lumens) weight that same light by how sensitive the human eye is at each wavelength. Two sources with equal radiant flux can have wildly different luminous flux if one emits mostly green light, where the eye is most sensitive, and the other mostly deep red or infrared, where it barely responds at all.

Rayleigh scattering, off particles much smaller than the light's wavelength (air molecules), scales as 1/λ⁴ and favours blue light — why the sky is blue and sunsets are red. Mie scattering, off particles comparable to the wavelength (dust, water droplets), is nearly wavelength-independent — why clouds and haze look white.

In Raman scattering, most incident photons scatter elastically (Rayleigh scattering), but a small fraction exchange a little energy with the material's vibrations, coming out shifted to a different wavelength — the size of that shift reveals which vibrational modes are present, making it a fingerprint of molecular structure. Fluorescence is a different process entirely: a photon is fully absorbed, promoting an electron to a higher energy level, which then relaxes non-radiatively partway before emitting a new, lower-energy (longer-wavelength) photon — the Stokes shift.

When two waves of slightly different frequency overlap, they alternately reinforce and cancel, producing a slow throbbing in loudness called beats, whose rate equals the plain difference between the two frequencies.

When two waves overlap they add together point by point — interference — and a wave passing through a narrow opening spreads out into the region behind it — diffraction.

The energy carried by a mechanical wave is proportional to the square of its amplitude, so a wave twice as tall carries four times the energy — a rule that makes intensity fall off far more steeply than amplitude does.

When a wave source and observer are moving toward or away from each other, the observed frequency shifts up or down from the emitted frequency, because relative motion changes how often successive wave crests arrive — not because the wave itself has actually changed.

When a system is driven at its natural oscillation frequency, the amplitude of its response grows much larger than it would at any other driving frequency.

When quantities are added or subtracted their absolute uncertainties add; when they are multiplied or divided their relative uncertainties add.

Plotting experimental data as a straight line rather than a curve makes deviations, systematic errors, and the fit parameters themselves far easier to read by eye, so a nonlinear relationship is routinely linearised — by re-plotting a transformed combination of the variables, such as log(y) against x or y against x squared — chosen so the underlying physical law predicts a straight line.

Since 2019 every SI base unit is defined by fixing the numerical value of a constant of nature — the second from a caesium transition, the metre from the speed of light, the kilogram from the Planck constant — so the last physical artefact, the platinum-iridium kilogram in Paris, was retired and any laboratory with the right apparatus can realise a unit from first principles. Getting from that definition to a number on a bench instrument is what traceability means: an unbroken chain of calibrations, each with its own stated uncertainty, linking the instrument back to a realisation of the definition. A reading without that chain is a number without provenance, however many digits it shows.

When a photon scatters off a loosely bound or free electron, it transfers some of its energy and momentum to the electron and emerges with a longer wavelength, an effect that only makes sense if light is treated as particle-like photons carrying momentum, and that provided decisive evidence for the particle nature of light.

Light striking a metal surface ejects electrons only if a single photon carries enough energy to free one, no matter how bright the light is — evidence that light behaves as particles, not just waves.

Every particle, including electrons, has an associated wavelength, and every wave, including light, can transfer energy and momentum in discrete particle-like packets.

X-rays are produced when fast electrons strike a target and either decelerate abruptly, radiating a continuous spectrum of bremsstrahlung, or knock out an inner-shell electron from a target atom, producing sharp characteristic lines whose energies are set by the target element's own atomic energy levels.

Plotting stars by their surface temperature (or spectral colour) against their luminosity reveals that most stars fall along a single diagonal band, the main sequence, rather than being scattered randomly, with a star's position along that band set primarily by its mass — and stars evolving off the main sequence, into giants or down to white dwarfs, trace out the distinct evolutionary stages of stellar life.

Nearly every distant galaxy shows its light redshifted, and the redshift grows in direct proportion to distance (Hubble's law), a pattern that is not galaxies flying apart through static space but space itself expanding uniformly everywhere, stretching the wavelength of light in transit and carrying distant galaxies away from every observer faster the farther away they are.

Stars near the outer edges of spiral galaxies orbit far faster than Newtonian gravity applied to the galaxy's visible, luminous matter alone can account for — their rotation curves stay roughly flat with radius instead of falling off the way planetary orbits in the Solar System do — and the same mass discrepancy shows up independently in the dynamics of galaxy clusters and in gravitational lensing of light passing near them; the leading explanation is that galaxies are embedded in far larger haloes of non-luminous dark matter, which interacts gravitationally but, as far as every experiment to date can tell, not electromagnetically, and so neither emits, absorbs, nor scatters light, while outweighing all of the universe's ordinary matter roughly five to one.

The Solar System condensed roughly 4.6 billion years ago from a collapsing, slowly rotating cloud of gas and dust; conservation of angular momentum flattened the collapsing cloud into a spinning disk, gravity concentrated most of its mass into the young Sun at the centre, and the leftover disk material stuck together in stages, from dust grains to planetesimals to planets, with the resulting planets' nearly circular, coplanar orbits obeying Kepler's laws and Newtonian gravitation to good approximation.

A star's life is a running balance between gravity, which continuously pulls it inward, and the outward pressure produced by nuclear fusion in its core; as each fuel is exhausted the core contracts and heats until the next, heavier fuel ignites, driving the star along a path across the Hertzsprung-Russell diagram from the main sequence through giant phases, and when fusion can no longer supply enough pressure to hold the star up, gravity finally wins, leaving behind — depending on the star's initial mass — a white dwarf, a neutron star, or a black hole.

Every moving particle has an associated wavelength inversely proportional to its momentum, λ = h/p, which is negligibly small for everyday objects but large enough to produce measurable diffraction for electrons and other subatomic particles.

A quantum particle is described by a wavefunction whose squared magnitude gives the probability density of finding the particle at a given position — the particle has no definite location until it is measured.

Position and momentum cannot both be known to arbitrary precision at once — narrowing the uncertainty in one necessarily widens the uncertainty in the other, a limit built into the wave nature of matter, not a flaw in measuring instruments.

In a system with no external forces, the total momentum before an interaction equals the total momentum after it.

Torque measures how effectively a force twists an object about a pivot, equal to the force times the perpendicular lever arm; angular momentum is its rotational analogue of momentum, conserved when no net torque acts.

Real oscillators lose energy to friction or resistance, so their amplitude decays over time while the oscillation itself continues — and depending on how strong that damping is, the system can still ring through many cycles, creep back slowly with no ringing at all, or sit at the boundary between the two.

Energy cannot be created or destroyed, only converted between forms — the total energy of an isolated system stays constant.

The centre of mass of a system is the mass-weighted average position of all its parts, and it moves exactly as if the system's entire mass were concentrated there and only external forces acted on it — internal forces never shift it.

Moment of inertia measures an object's resistance to changes in rotational speed, playing the same role for rotation that mass plays for straight-line motion, so that torque equals moment of inertia times angular acceleration.

Momentum is conserved in every collision, but kinetic energy is conserved only in an elastic collision — in an inelastic one, some kinetic energy converts into heat, sound or deformation, with a perfectly inelastic collision losing the most because the objects stick together.

A rotating object stores kinetic energy in its spin, given by one half its moment of inertia times its angular velocity squared — the rotational counterpart of one half mv².

Kepler's three laws describe planetary orbits as ellipses that sweep out equal areas in equal times, with a fixed relationship between a planet's orbital period and the size of its orbit — laws found from observation alone, later shown by Newton to follow directly from an inverse-square gravitational force.

An object rolling without slipping has its centre-of-mass speed and its angular speed locked together by the radius, so it carries both translational and rotational kinetic energy at once, which is why a rolling ball starting from rest on a slope always arrives slower than a frictionless block sliding the same distance.

A gas's temperature is a measure of the average kinetic energy of its particles, which move randomly and collide elastically with each other and the container walls.

For an ideal gas, pressure, volume, temperature and the amount of substance are linked by a single equation: pV equals nRT.

The first law of thermodynamics is energy conservation applied to a system's internal energy: it changes only by heat added and work done, ΔU = Q − W, and internal energy itself depends only on the system's current state, not on how it got there.

Isothermal, adiabatic, isobaric and isochoric processes each hold one variable fixed while a gas changes state, and that constraint alone determines how heat, work and internal energy divide between each other for that particular process.

In a gas at thermal equilibrium, molecular speeds spread out over a characteristic distribution set by temperature and mass — most molecules cluster near a typical speed, with a tail of much faster and much slower ones.

When a capacitor charges or discharges through a resistor, its voltage approaches its final value exponentially rather than instantly, with a time constant set by the product of resistance and capacitance; an inductor produces the analogous exponential delay in current.

Ampère's law says the circulation of the magnetic field around any closed loop is proportional to the electric current passing through it, giving the magnetic-field analogue of Gauss's law and, once corrected by Maxwell, half of the electromagnetic wave equation.

The Biot–Savart law gives the magnetic field produced by a small segment of current-carrying wire, falling off with the square of distance like Coulomb's law but pointing perpendicular to both the current and the direction to the point, not along it.

In an alternating-current circuit, resistors, capacitors and inductors oppose current with an impedance that combines resistance with frequency-dependent reactance, so current and voltage can drift out of phase with each other.

A changing electric field creates a changing magnetic field and vice versa, and this self-sustaining pair propagates through space as an electromagnetic wave, including visible light.

An inductor opposes changes in the current through it, not the current itself, so in an RL circuit the current approaches its final value exponentially — the same shape as RC charging, but delaying current instead of voltage.

Four equations tie electric and magnetic fields to their sources and to each other, unifying electricity, magnetism and optics into one theory and predicting that changing fields propagate as electromagnetic waves at the speed of light.

Inside a material, an electromagnetic wave slows down and its wavelength shortens because the material's electrons respond to the oscillating field and re-radiate with a delay, an effect summarised in the refractive index — while the wave's frequency, set by the source, never changes.

The Poynting vector describes the direction and rate of electromagnetic energy flow at each point in space, built from the cross product of the electric and magnetic fields, and it accounts for how a light wave, a radio signal, or a power line actually delivers energy from source to destination.

A capacitor and inductor connected together exchange energy back and forth at a fixed resonant frequency, exactly as a mass and spring exchange kinetic and potential energy, and adding resistance drains that energy away just as friction damps a mechanical oscillator.

Gauss's law says the total electric flux out through any closed surface is proportional to the charge enclosed, which turns hard direct calculations of the field into easy ones whenever the charge distribution has enough symmetry to choose a convenient surface.

The Michelson interferometer splits a beam of light in two, sends each half down a different path to a mirror and back, then recombines them — any change in one path's length by even a fraction of a wavelength shifts the interference pattern, making it one of the most precise length-measuring instruments ever built and the device that first showed light's speed does not depend on Earth's motion through space.

A birefringent material has two different refractive indices depending on the light's polarisation direction, so the two polarisation components travel at different speeds and pick up a phase difference as they pass through. A wave plate is a slab of birefringent material cut to a thickness that introduces a specific phase difference — a quarter turn or a half turn — used to convert between polarisation states.

The Fresnel equations give the fraction of light reflected and transmitted at a boundary between two media, and that fraction depends on both the angle of incidence and the polarisation direction. At one specific angle — Brewster's angle — light polarised parallel to the plane of incidence is not reflected at all, so the reflected beam is purely polarised perpendicular to it.

Any optical instrument's aperture diffracts the light passing through it, spreading a point source into a small blurred disc rather than a true point, and the Rayleigh criterion sets the standard for how close two such discs can sit before they can no longer be told apart — a limit set by diffraction, not by imperfect engineering.

Passing light through two closely spaced slits produces an interference pattern of bright and dark fringes, direct evidence that light behaves as a wave.

At the intensities ordinary light produces, a material's polarisation response is proportional to the electric field, and light passing through doesn't change colour. At the very high intensities a focused laser can reach, that proportionality breaks down: the response also contains terms in the square (and higher powers) of the field, and one consequence is second-harmonic generation, where some of the light emerges at exactly twice the input frequency — half the wavelength.

Spectroscopy reads the light a substance emits or absorbs as a record of its energy levels, so a spectrum identifies what a sample is made of and how its atoms and molecules are bound.

An accelerating charge — in the simplest case, one oscillating back and forth like a dipole antenna — radiates electromagnetic waves whose intensity is not the same in every direction: it is zero along the dipole's own axis and greatest in the plane perpendicular to it, following a sin²θ pattern with θ measured from that axis.

A perfect absorber and emitter of radiation produces a spectrum that depends only on its temperature, and classical physics could not explain its shape at all until Planck showed that treating the emitted energy as coming in discrete packets fixed the prediction — the discovery that began quantum mechanics.

An electron bound in an atom can only occupy certain discrete energy levels, and it absorbs or emits a photon of exactly the right energy when it jumps between them.

A laser produces intense, coherent light by exploiting stimulated emission — a passing photon of exactly the right energy can trigger an already-excited atom to emit an identical, in-phase photon — which requires a population inversion (more atoms in the excited state than the ground state) to dominate over ordinary absorption and spontaneous emission.

For a general function of measured quantities, the uncertainty in the result is found from the function's partial derivatives with respect to each input.

Repeated measurements of the same quantity, subject to many small independent random error sources, tend to scatter according to the normal (Gaussian) distribution, whose spread is characterised by the standard deviation, and this predictable shape is what justifies reporting a single mean value plus an uncertainty and what underlies standard statistical tests for comparing measurements.

Linear regression finds the straight line through a set of data points that minimises the sum of the squared vertical distances between each point and the line — the least-squares criterion — giving objective, reproducible values for the slope and intercept along with their own uncertainties, rather than relying on a subjective by-eye best-fit line.

Any signal can be decomposed into sinusoidal components, and its frequency spectrum is as complete a description as its time trace — a duality that underlies filtering, modulation and the limits on how a signal may be sampled.

Newton's laws hold exactly only in inertial frames; an accelerating or rotating observer can still use them by adding a fictitious force that exists purely because of that frame's own acceleration and disappears entirely when viewed from an inertial frame.

When oscillators are linked or pushed by an external periodic force, their combined motion develops normal modes and, near resonance, an amplitude and phase lag that depend sharply on how close the driving frequency is to the natural frequency.

Every pair of masses attracts each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

A force that always points toward or away from a fixed centre conserves angular momentum about that centre regardless of its strength, which is why every orbit — circular, elliptical or otherwise — sweeps out area at a constant rate, and why orbital motion can be reduced to a one-dimensional problem in the distance from the centre alone.

Instead of tracking forces directly, Lagrangian mechanics derives the equations of motion from the difference between kinetic and potential energy, expressed in whichever coordinates most naturally describe the system's constraints.

Hamiltonian mechanics rewrites Lagrangian mechanics in terms of position and momentum instead of position and velocity, turning each second-order equation of motion into two coupled first-order ones and revealing energy as the quantity generating time evolution — the formulation that carries most directly into quantum mechanics.

Noether's theorem shows that every continuous symmetry of a system's Lagrangian — a change that leaves it unchanged — corresponds to a conserved quantity: time-translation symmetry gives energy conservation, spatial-translation symmetry gives momentum conservation, and rotational symmetry gives angular-momentum conservation.

A constraint that can be written as a relation between coordinates alone (holonomic) can simply be used to eliminate a coordinate before forming the Lagrangian, but a constraint on velocities that cannot be integrated into such a relation (nonholonomic, like rolling without slipping in general) cannot — it must instead be kept explicit and enforced with a Lagrange multiplier, which turns out to equal the constraint force itself.

A canonical transformation changes position and momentum coordinates while preserving Hamilton's equations' form; the most powerful choice makes the new Hamiltonian identically zero, so every new coordinate is constant in time — finding the generating function that does this is the Hamilton–Jacobi equation, and solving it is equivalent to solving the entire motion at once.

The laws of physics are identical in every inertial frame of reference, and the speed of light in vacuum is the same for every observer, regardless of the motion of the light source or the observer.

A clock moving relative to an observer ticks more slowly than an identical clock at rest in the observer's own frame — the closer the relative speed is to the speed of light, the larger the effect.

An object measured while moving past an observer is shorter along its direction of motion than when it is measured at rest — length in the direction of travel contracts by the same factor that stretches time.

Two events that are simultaneous in one inertial reference frame are generally not simultaneous when viewed from a different frame moving relative to the first, a direct consequence of the finite, constant speed of light — simultaneity is not an absolute, frame-independent fact but depends on the observer's state of motion.

The Lorentz transformations are the equations that convert an event's position and time coordinates between two inertial frames moving at constant relative velocity, replacing the Galilean transformations of classical mechanics and correctly encoding time dilation, length contraction, and the relativity of simultaneity all at once, while reducing to the familiar Galilean equations whenever the relative speed is much less than light speed.

As speed approaches the speed of light, momentum grows faster than the classical mv predicts, and even a mass at rest carries an intrinsic energy given by E = mc², the basis of nuclear energy release.

Velocities in special relativity do not add the way classical intuition suggests — the relativistic velocity-addition formula u' = (u + v)/(1 + uv/c²) replaces simple addition and guarantees that combining any two speeds each less than c always yields a result still less than c, with the speed of light itself unchanged by any addition.

Special relativity is expressed most naturally by treating space and time together as a single four-dimensional spacetime and packaging an event's or a particle's properties into four-vectors, whose spacetime interval or invariant length stays the same for every inertial observer even though the individual space and time components each observer measures differ.

The relativistic Doppler effect combines the classical frequency shift from relative motion with an additional shift from time dilation itself, producing a nonzero frequency change even for pure transverse motion — light from a source moving directly across the line of sight is still redshifted, a purely relativistic effect with no classical acoustic analogue.

A standing wave forms when two identical waves travel in opposite directions and interfere, producing fixed points of no displacement, called nodes, that never move.

The wave equation is a single differential equation satisfied by any disturbance travelling through a medium at fixed speed, and its solutions — the family of possible waveforms — are fixed entirely by how the medium's tension, stiffness or elasticity relates to its inertia.

Confining a quantum particle between rigid walls forces its wavefunction to fit a whole number of half-wavelengths, restricting its energy to a discrete ladder of allowed values rather than a continuous range.

The Schrödinger equation is the central equation of non-relativistic quantum mechanics, playing the same role for a wavefunction that Newton's second law plays for a classical trajectory — given the forces acting on a particle, it determines exactly how the wavefunction evolves in time.

A quantum particle has a nonzero probability of appearing on the far side of an energy barrier it classically could not cross, because its wavefunction decays but does not vanish inside the barrier.

Solving the quantum mechanics of an electron bound to a single proton reproduces the discrete energy levels of hydrogen and introduces the quantum numbers that organise every atom's electron structure.

A molecule's full Schrödinger equation couples every electron's motion to every nucleus's motion, which is intractable directly — the Born-Oppenheimer approximation exploits the huge mass difference between nuclei and electrons to solve them separately: electrons adjust to the nuclei's positions essentially instantly, giving a potential energy surface on which the much heavier, slower nuclei then vibrate and rotate, each motion quantized into its own ladder of discrete energy levels.

Every measurable physical quantity in quantum mechanics is represented by a mathematical operator that acts on the wavefunction, and the only values a measurement can ever return are the operator's eigenvalues — sharp, well-defined outcomes even though the wavefunction itself is generally spread out.

The quantum version of a mass on a spring has evenly spaced discrete energy levels, En = (n + 1/2)ħω, and even in its lowest possible state it retains a nonzero zero-point energy — it can never sit motionless at the bottom of the potential well the way a classical oscillator at rest could.

Solving the Schrödinger equation for a step or a finite-depth well shows that the wavefunction always leaks into a classically forbidden region as a decaying exponential rather than stopping dead at the boundary, and that a finite well binds only a limited number of discrete states instead of the infinite ladder an infinitely deep well would allow.

Electrons carry an intrinsic angular momentum called spin that has no classical analogue, comes only in discrete half-integer amounts, and — together with the Pauli exclusion principle — determines how electrons fill atomic orbitals.

For a system not in an eigenstate of the quantity being measured, quantum mechanics predicts only an expectation value — the probability-weighted average result over many identical measurements — while any single measurement returns one of the operator's eigenvalues chosen at random, with the wavefunction collapsing onto that outcome.

Solving the Schrödinger equation for an electron in a hydrogen-like atom produces wavefunctions — orbitals — labelled by a discrete set of quantum numbers (n, l, ml, ms) that together fix the electron's energy, its orbital angular momentum and shape, and its spin, replacing the classical picture of an electron on a definite orbit with a three-dimensional probability cloud.

The hydrogen energy levels predicted by treating the electron as a point charge orbiting the nucleus are only an approximation: the electron's own spin magnetic moment interacts with the magnetic field it experiences from its orbital motion around the nucleus, slightly splitting each orbital energy level according to how the electron's spin and orbital angular momentum combine.

An external magnetic field splits a single atomic spectral line into several closely spaced lines, because the field lifts the energy degeneracy between states that share the same orbital quantum number but differ in magnetic quantum number. The Stern-Gerlach experiment showed this splitting is real and quantized even in a beam of atoms with no orbital angular momentum at all, which is what first revealed electron spin.

The Schrödinger picture puts all of a quantum system's time dependence into the evolving state, with fixed operators; the Heisenberg picture is a physically equivalent bookkeeping choice that instead evolves the operators and leaves the state fixed. Time-dependent perturbation theory uses this same machinery to answer a different question — given a small time-dependent nudge to a system already sitting in an energy eigenstate, at what rate does it transition to another eigenstate — with Fermi's golden rule giving that rate to leading order.

No two electrons — or any other fermions — in the same quantum system can occupy an identical quantum state, a rule that forces electrons in an atom to stack into successively higher orbitals rather than all collapsing into the lowest one, and that is the fundamental reason ordinary matter has structure, volume, and resists being compressed.

General relativity replaces Newton's picture of gravity as a force acting between masses with geometry: mass and energy curve spacetime, and every object, including light, follows the straightest possible path (a geodesic) through that curved geometry — which is what is perceived as gravitational attraction, as clocks running slower deep in a gravitational well, and as light bending around massive objects; applied to the universe as a whole, Einstein's field equations combined with the observed expansion (Hubble's law) and the cosmic microwave background give the Big Bang model of a hot, dense, expanding early universe, refined further by an early inflationary phase and a present accelerating expansion attributed to dark energy.

A fixed point is a state at which every rate of change vanishes, so a system left exactly there stays there — and drawing the trajectories in the phase plane (position against velocity, say) turns the question of whether it stays there when nudged into geometry: linearise about the point, and the eigenvalues of the resulting 2×2 matrix decide whether nearby trajectories spiral in, are thrown out along a preferred direction, or circle indefinitely, which is why a pendulum hanging down and the same pendulum balanced upright are one equation with two completely different fixed points.

Turn a control parameter slowly — a load, a flow rate, a temperature difference — and for almost every setting nothing qualitative happens: the fixed points shift slightly and keep the stability they had. At isolated values they do not, and those values are bifurcations: a stable state and an unstable one collide and annihilate (saddle-node), a symmetric state loses stability to two new asymmetric ones (pitchfork), or a fixed point hands its stability to a growing oscillation (Hopf). This is where behaviour changes in kind rather than degree, and it is the same event behind a loaded column buckling, convection rolls appearing in a heated fluid, and a laser crossing its threshold.

A limit cycle is an isolated closed trajectory that neighbouring trajectories wind onto: unlike the nested orbits of an undamped pendulum, its amplitude is set by the system itself rather than by how hard it was started, which is what a heartbeat, a violin string under a bow and a laser above threshold all have in common — each loses energy and takes it back in over one cycle. Because a closed loop in phase space is awkward to read directly, the standard tool is the Poincaré section: pick a surface the flow crosses, record only the crossing points, and a limit cycle collapses to a single dot, a quasi-periodic motion to a closed curve, and chaos to a scatter of points with visible structure.

Chaos is what a fully deterministic system does when nearby states pull apart exponentially fast: no randomness anywhere in the equations, yet two starting conditions that differ in the tenth decimal place end up on entirely different trajectories. The Lyapunov exponent measures that rate — a small separation grows roughly as an exponential of the exponent times elapsed time, so its reciprocal sets the time scale on which prediction decays — and a positive exponent together with motion that stays bounded is what marks genuine chaos rather than a system simply running away, which is why weather has a forecast horizon of days no matter how good the instruments get.

Along a streamline of a flowing fluid, the sum of pressure, kinetic energy per volume and gravitational potential energy per volume stays constant — where the fluid speeds up, its pressure must drop.

Real fluids resist internal sliding between layers, an effect called viscosity; in a narrow pipe this drag makes flow either smooth and layered at low speed or chaotic and turbulent once a critical speed is crossed.

The Reynolds number is a dimensionless ratio of a flow's inertial forces to its viscous forces, and it predicts whether a given flow will be smooth and orderly (laminar, at low Reynolds number) or chaotic and eddying (turbulent, at high Reynolds number), letting the same criterion apply across wildly different fluids, speeds, and length scales.

A random walk is a sequence of steps whose directions are independent of one another, and its central result is that the walker's typical distance from the start grows as the square root of the elapsed time rather than in proportion to it: the average displacement is exactly zero, while the root-mean-square displacement is not. Molecules colliding in a fluid do exactly this, so diffusion inherits the same square-root law — which is why a cell a few micrometres across can rely on diffusion to move material about, why the same process would take hours to cross a room, and why almost all mixing you actually notice is convection instead.

A pollen grain in water jitters visibly under the microscope because it is struck by molecules from every side and the strikes never quite balance — Brownian motion, the visible end of a random walk driven by thermal energy. Einstein's insight in 1905 was that the same molecular collisions cause both the kicking and the drag, so the two must be linked: the Stokes–Einstein relation gives the diffusion coefficient as thermal energy divided by the drag coefficient of the particle, tying a random microscopic quantity to two things anyone can measure. It was the argument that made atoms hard to deny, and it is still how particle sizes are measured in a dynamic light scattering instrument.

A process is Markovian when the probabilities of what happens next depend only on the present state and not on the route that led to it — the whole history is already carried in where the system is now. That single assumption is what makes a stochastic model tractable: the dynamics reduce to a table of transition probabilities, repeated application drives most such chains towards a stationary distribution independent of where they started, and requiring detailed balance in that distribution is exactly how a Monte Carlo simulation is made to sample the Boltzmann distribution. Radioactive decay, a random walk and a thermally activated jump between two wells are all Markovian; a process with a memory, like fatigue in a material, is not.

The equipartition theorem says thermal energy distributes equally among every independent way a system can store energy, ½kT per quadratic degree of freedom, which is why molecules with more ways to move or vibrate carry more internal energy at the same temperature than simpler ones.

The first law of thermodynamics says energy is conserved: the change in a system's internal energy equals the heat added minus the work done by it.

The mean free path is the average distance a molecule travels between collisions, and since it is inversely proportional to pressure it stretches from about 70 nanometres in room air to kilometres in a good ultra-high vacuum. Comparing it with the size of the vessel is what defines the regimes: while it is much smaller than the chamber the gas behaves as a fluid and is pumped by pushing it along, and once it is much larger the molecules fly wall to wall without meeting each other, so the gas conducts heat less and less well as pressure falls and a thin-film source can reach the substrate without being scattered. That crossover is why a vacuum system needs two kinds of pump in series and why a thermos flask works at all.

The ideal gas law assumes molecules are points that never interact, and real gases depart from it because molecules have actual volume and attract each other weakly — the van der Waals equation corrects for both, and the departure grows largest at high pressure and low temperature, exactly where a gas approaches condensing into a liquid.

The second law of thermodynamics says the entropy of an isolated system never decreases, which is why heat flows from hot to cold and not the other way round.

Maxwell's relations are four equalities between partial derivatives of pressure, volume, temperature and entropy, derived from the fact that each thermodynamic potential's mixed second partial derivatives must agree — they let a hard-to-measure quantity such as (∂S/∂V)_T be replaced by an easily measured one such as (∂p/∂T)_V.

A crystalline solid's atoms are arranged in a repeating, periodic pattern described by a lattice — a mathematical scaffold of points related by translation symmetry — combined with a basis of one or more atoms attached to each lattice point, and this underlying periodicity is what makes concepts like energy bands and Bragg diffraction well defined.

When atoms bond into a crystal, their discrete atomic energy levels broaden into continuous allowed energy bands separated by forbidden gaps, and whether a material conducts, semiconducts, or insulates is determined entirely by how these bands are filled with electrons and how large the gap is between the highest filled band and the next empty one.

A crystal's planes of atoms are spaced at roughly the same scale as an X-ray wavelength, so waves reflected from successive planes interfere, and they reinforce only at angles where the extra path travelled by the deeper reflection is a whole number of wavelengths — Bragg's law. That turns a diffraction pattern into a measurement: each sharp spot or ring fixes one plane spacing, the set of spacings identifies the lattice, and their relative intensities carry where the atoms sit within the unit cell. It is how the structures of salt, metals, minerals, DNA and essentially every protein of known shape were determined.

A thin film is built atom by atom onto a substrate, either physically — evaporating or sputtering material that then flies through vacuum to the surface — or chemically, from a gas that reacts at the hot substrate and leaves a solid behind. Which method is available depends on the vacuum: physical deposition needs a mean free path longer than the source-to-substrate distance so the atoms arrive in a straight line. What happens after they land decides the film: arriving atoms diffuse across the surface and either wet it layer by layer, ball up into islands, or do one then the other, and that growth mode is what sets whether the result is a mirror, a rough grey coating or an epitaxial single crystal.

Doping a semiconductor with carefully chosen impurity atoms deliberately adds either extra mobile electrons (n-type, from donor atoms with a spare valence electron) or extra mobile holes (p-type, from acceptor atoms with one fewer valence electron), and controlling this doping is what makes semiconductors useful for building diodes, transistors, and every other solid-state electronic device.

Below a critical temperature, certain materials lose all electrical resistance entirely and also expel magnetic fields from their interior (the Meissner effect), a genuinely quantum phenomenon explained in conventional superconductors by electrons pairing into Cooper pairs that move through the lattice without scattering, rather than merely being an extreme case of ordinary good conductivity.

Shrink a piece of semiconductor until it is comparable to the size an electron-hole pair naturally occupies, and the continuous bands of the bulk crystal break up into discrete levels whose spacing grows as the crystal gets smaller — so the same material emits blue at two nanometres and red at six, with no change of chemistry. Restricting motion in one, two or three directions gives quantum wells, wires and dots, and each changes not just the number of available states but the shape of the density of states, which is why a quantum-well laser reaches threshold at a far lower current than a bulk one and why these structures are engineered rather than merely observed.

Protons and neutrons fill discrete energy shells inside the nucleus, much like electrons do in an atom, and nuclei with full shells at "magic numbers" of protons or neutrons are unusually stable. The model also predicts each nucleus's total spin and parity from which shells its outermost nucleons occupy.

A nuclear reaction that absorbs energy (a negative Q-value) cannot proceed at just any projectile energy — momentum conservation means some of the projectile's kinetic energy is unavoidably tied up in moving the products forward, so the projectile needs strictly more kinetic energy than |Q| to make the reaction happen at all.

The Standard Model classifies all known elementary particles into matter particles (quarks and leptons, both fermions) and force-carrying particles (bosons, one for each of the electromagnetic, weak, and strong interactions), explaining ordinary matter as built from just a handful of truly fundamental building blocks rather than the many hundreds of composite particles once thought equally fundamental.

Particle interactions must respect a set of conservation laws — energy, momentum, electric charge, and additional quantum numbers such as baryon number and lepton number — which act as strict bookkeeping rules that any proposed reaction must satisfy, and a reaction that violates even one of these laws simply cannot occur no matter how much energy is available.

Entropy measures the number of microscopic arrangements consistent with a system's observed macroscopic state — the second law's tendency toward higher entropy is simply the statistical tendency toward the overwhelmingly more probable arrangement.

No heat engine operating between two temperatures can be more efficient than the idealised Carnot cycle, whose efficiency depends only on the ratio of the cold and hot reservoir temperatures.

Unlike a first-order transition (boiling, melting), a continuous (second-order) phase transition has no latent heat and no jump in density — instead an order parameter shrinks smoothly to zero at a critical temperature, while quantities like heat capacity and susceptibility diverge there following power laws whose exponents are shared by whole classes of otherwise unrelated systems.

Gibbs free energy combines a system's enthalpy and entropy into a single quantity whose decrease predicts whether a process will occur spontaneously at constant temperature and pressure.

A macrostate is what can be observed about a system from outside; a microstate is one exact configuration of every particle consistent with it, and the multiplicity — the number of microstates sharing a macrostate — is what statistical entropy directly counts.

Liouville's theorem says that as a cloud of representative points in phase space (each one a whole system's position and momentum coordinates) evolves under Hamilton's equations, the density of points around any given one stays constant — the cloud's shape can stretch and fold arbitrarily, but its phase-space volume never compresses or expands, exactly the property that lets statistical mechanics treat probability as conserved along a trajectory rather than something that has to be tracked separately.

The partition function sums the Boltzmann factor over every microstate a system can occupy, and once known it generates every thermodynamic quantity — energy, entropy, free energy — by differentiation, condensing all of a system's statistical mechanics into one function of temperature.

Because the Pauli exclusion principle forbids identical fermions from sharing a quantum state while identical bosons face no such restriction, the two particle classes follow different statistical distributions — Fermi-Dirac and Bose-Einstein — which reduce to the classical Maxwell-Boltzmann distribution only when the states available vastly outnumber the particles occupying them.

An aerosol is solid or liquid particles suspended in air, and almost everything about one follows from its size: particles above a few micrometres settle out within minutes, those below a micrometre stay airborne for days and are carried by air currents and Brownian motion rather than gravity, and only the small ones reach deep into the lungs — which is why air quality limits are written for PM10 and PM2.5 rather than for mass alone. Measuring them means choosing which physical effect stands in for size: weighing what a filter collects, counting scattered-light pulses one particle at a time, or reading an attenuation, each of which answers a different question and can disagree with the others.