Physics
From subatomic particles to the cosmos. Every concept visualized with interactive 3D animations.
AC Impedance and Reactance
Impedance Z = R + jX is the complex generalization of resistance for AC circuits. Capacitive reactance is 1/(ωC), inductive reactance is ωL; the phase
ElectromagnetismAC vs DC Current · alternating
3D comparison. DC: electrons flow one direction steadily (battery). AC: electrons oscillate back and forth 60 times per second (wall outlet). Show wav
ElectromagnetismAbrikosov Vortex Lattice · How Type-II Superconductors Let Magnetic Flux In
The Abrikosov vortex lattice explained: how type-II superconductors admit quantized magnetic flux as a triangular grid of vortices, with equations, nu
Condensed MatterAcoustic Impedance · Why sound reflects at a boundary
Acoustic impedance Z = ρc is a medium's resistance to sound. A mismatch at a boundary reflects sound; matching transmits it. Why air-water reflects 99
Waves & OscillationsAcoustic Levitation · A standing sound wave traps small objects at its pressure nodes, suspending them in mid-air
Acoustic levitation traps small objects at the pressure nodes of an intense standing sound wave, holding them in mid-air against gravity. The trapping
Waves & OscillationsAction-Angle Variables · Turning Orbits into Straight-Line Flow
Action-angle variables explained: how a canonical transformation turns curved orbits into straight-line flow on a torus, with the ∮p dq action integra
Classical MechanicsAdiabatic Invariant · The quantity that refuses to change when you change everything slowly enough
An adiabatic invariant is a quantity — the action J = ∮ p dq — that stays constant when a system parameter changes slowly. Shorten a pendulum's string
Classical MechanicsAharonov-Bohm Effect · A charged particle picks up a phase from a magnetic field it never touches — A is real, not just B
A charged particle moving through a region of zero magnetic field still picks up a phase shift if the vector potential A is non-zero. The 1959 Aharono
Quantum MechanicsAiry Disk · Diffraction pattern from a circular aperture — central peak + rings, first dark at 1.22 λ/D
Light through a circular aperture forms an Airy disk: bright central peak plus concentric rings. First dark ring at θ ≈ 1.22 λ/D — the Rayleigh resolu
Wave OpticsAlfvén Wave · Pluck a magnetic field line and it twangs like a guitar string — the music of magnetised plasma
An Alfvén wave is a transverse MHD wave that travels along magnetic field lines as if they were vibrating strings under tension. Speed v_A = B/√(μ₀ρ).
Plasma PhysicsAlpha Decay & Quantum Tunneling · How a trapped helium nucleus escapes a wall it can't climb — and why that makes half-lives span 24 orders of magnitude
Alpha decay explained: an alpha particle escapes a nucleus by tunneling through the Coulomb barrier. Gamow theory and the Geiger-Nuttall law explain h
Nuclear PhysicsAmpère's Law · ∮ B·dℓ = μ₀ I_enc — line integral of B around a closed loop
Ampère's law (André-Marie Ampère, 1826): the line integral of magnetic field B around any closed loop equals μ₀ times the total current passing throug
ElectromagnetismAnderson Localization · When a medium gets random enough, interference traps the wave in place — and a conductor becomes an insulator
Anderson localization is the trapping of waves and electrons by disorder: when a medium is random enough, interference between scattered paths makes t
Condensed MatterAndreev Reflection · How an Electron Turns Into a Hole at a Superconductor
Andreev reflection explained: how an electron becomes a hole at a superconductor interface, transferring 2e charge. BTK theory, conductance doubling,
Condensed MatterAngular Momentum · spinning
3D ice skater spinning. Arms out: slow spin. Pull arms in: spin faster. Angular momentum L=Iω stays constant — decrease moment of inertia, increase an
Classical MechanicsAntenna Radiation · Accelerating charges launch electromagnetic waves — and a half-wave dipole presents 73 Ω to its feed line
Accelerating charges radiate EM waves. A dipole antenna's pattern follows sin²θ — strongest broadside, zero off the ends. Half-wave dipole input imped
Antennas & WaveguidesAntimatter · positron
3D electron meets its antiparticle (positron). They annihilate in a flash, converting entirely to two gamma ray photons. Reverse: high-energy photon c
Particle PhysicsAsymptotic Freedom · In QCD, the strong coupling α_s decreases at high energies — quarks act free at short distances
Asymptotic freedom: in quantum chromodynamics (QCD), the strong coupling α_s decreases logarithmically with increasing energy: α_s(Q²
Quantum ChromodynamicsAtomic Structure · protons
3D atom with a dense nucleus of protons (red) and neutrons (blue) surrounded by electron cloud (green orbiting dots). Zoom into nucleus to show strong
Quantum PhysicsAtwood Machine · Two masses, one pulley, and a gentler gravity
An Atwood machine is two masses hung over a pulley by a string; the heavier mass falls with acceleration a = (m1 − m2)g/(m1 + m2), a slowed-down gravi
Classical MechanicsBCS Theory · How electrons pair up through the lattice, condense as one, and switch off all electrical resistance
BCS theory explains superconductivity: electrons bind into Cooper pairs via phonon exchange and condense into a single quantum state, opening an energ
Condensed MatterBallistic Pendulum · Measuring a bullet’s speed with a swing
A ballistic pendulum measures a bullet's speed by firing it into a hanging block and measuring the swing height — momentum conservation in the collisi
Classical MechanicsBand Gap · The energy range with no electron states — separates filled valence from empty conduction
The band gap E_g is the energy range in a solid where there are no allowed electron states, separating the filled valence band from the empty conducti
Solid State PhysicsBeat Frequency · two close frequencies
3D two tuning forks with slightly different frequencies. Combined sound oscillates between loud (constructive) and quiet (destructive). Beat frequency
Waves & OscillationsBell's Inequalities · |S| ≤ 2 (CHSH) for any local realistic theory; QM violates with |S| ≤ 2√2
Bell's theorem (John Bell, 1964): any local hidden variable theory must satisfy certain inequalities; quantum mechanics violates them. The most-te
Quantum FoundationsBernoulli's Principle · fast flow = low pressure
3D fluid flowing through a pipe that narrows. In the narrow section: velocity increases, pressure decreases. Apply to airplane wing: faster air on top
Fluid DynamicsBerry Phase · A quantum memory of the path it took
The Berry phase is a geometric phase a quantum state acquires when its parameters are cycled adiabatically around a closed loop — it depends only on t
Quantum MechanicsBertrand's Theorem · Why Only 1/r and r² Forces Give Closed Orbits
Bertrand's theorem explained: why only the inverse-square (1/r²) and Hooke's-law (r) central forces produce closed bound orbits. Derivation, apsidal a
Classical MechanicsBeta Decay · A neutron turning into a proton, electron, and ghost
Beta decay is radioactive decay where a neutron turns into a proton (or vice versa), emitting a fast electron and a near-invisible antineutrino via th
Nuclear PhysicsBifurcation · When a tiny nudge to one knob rewrites a system's entire future — the mathematics of tipping points
A bifurcation is a qualitative change in dynamics as a parameter crosses a threshold — saddle-node, transcritical, pitchfork, and Hopf. The mathematic
Nonlinear DynamicsBiot-Savart Law · dB = (μ₀/4π) (I dℓ × r̂)/r² — magnetic field of an infinitesimal current segment
The Biot-Savart law (Jean-Baptiste Biot and Félix Savart, 1820) gives the magnetic field dB at point P due to an infinitesimal current element I dℓ at
ElectromagnetismBirefringence · Anisotropic crystals split a single beam into two orthogonally polarized rays
Anisotropic crystals like calcite have two refractive indices — ordinary and extraordinary — that split unpolarized light into two orthogonally polari
Wave OpticsBlack Body Radiation · thermal emission
3D heated object glowing through spectrum: red at low temp, orange, yellow, white hot. Classical prediction (UV catastrophe) diverges to infinity. Pla
Quantum PhysicsBloch Sphere · Every qubit is a point on a sphere — and every quantum gate is a rotation of that point
The Bloch sphere maps every pure single-qubit state to a point on a unit sphere: |0⟩ at the north pole, |1⟩ at the south, superpositions on the equato
Quantum MechanicsBloch's Theorem · Why an electron in a crystal is a plane wave wearing the lattice as a mask — and how that single fact gives rise to energy bands
Bloch's theorem explained: in a periodic crystal potential, electron wavefunctions are plane waves modulated by a lattice-periodic function, ψ_nk(r) =
Condensed MatterBohr Model of the Atom · electron orbits
3D atom with nucleus and electrons in discrete circular orbits. Electron jumps down an energy level and emits a photon (glowing particle). Jumps up by
Quantum PhysicsBoltzmann Distribution · Probability of a state with energy E in thermal equilibrium scales as exp(−E/kT)
The Boltzmann distribution is the fundamental probability law of statistical mechanics: in thermal equilibrium at temperature T, the probability of a
Statistical MechanicsBose-Einstein Condensate · Cool bosons below a critical temperature and they collapse into a single quantum wave
Below a critical temperature T_c that depends only on density and mass, a dilute gas of bosons abruptly piles a macroscopic fraction of its atoms into
Quantum PhysicsBose-Einstein Statistics · The quantum counting rule that lets an unlimited crowd of identical particles pile into one state
Bose-Einstein statistics governs indistinguishable integer-spin particles: the mean occupation is n = 1/(e^((E−μ)/kT) − 1). Unlimited bosons can share
Statistical MechanicsBoundary Layer · Prandtl's thin film of slowed fluid that sets nearly all real-world drag
The boundary layer is the thin film of slowed fluid right next to a solid surface. Ludwig Prandtl introduced it in 1904 and it transformed fluid dynam
Fluid DynamicsBra-Ket Notation · Dirac's language: kets |ψ⟩ are states, bras ⟨ψ| are their conjugates, ⟨φ|ψ⟩ is amplitude
Bra-ket notation: Dirac's compact language for quantum mechanics. Kets |ψ⟩ are column vectors, bras ⟨ψ| are row vectors, inner product ⟨ψ|φ⟩ is a scal
Quantum MechanicsBrachistochrone Curve · The fastest slide between two points isn’t a straight line
The brachistochrone curve is the path of fastest descent between two points under gravity — an inverted cycloid, not a straight line. See the equation
Classical MechanicsBrayton Cycle · The gas-turbine and jet-engine cycle — two adiabats plus two isobars, set by pressure ratio
The Brayton cycle is the gas-turbine and jet-engine cycle — two adiabats plus two isobars. Efficiency depends on pressure ratio P₂/P₁. Modern combined
ThermodynamicsBrazil Nut Effect · Shake a mix of grains and the largest rise to the top — size segregation in vibrated granular matter
The Brazil nut effect is the size segregation seen when a mix of granular particles is shaken: the largest grains rise to the top. Driven by void-fill
Statistical MechanicsBremsstrahlung · German for "braking radiation" — a continuous X-ray spectrum from a charged particle deflected by a nucleus
Bremsstrahlung is the electromagnetic radiation emitted when a charged particle is decelerated by another charge (usually a nucleus). The spectrum is
ElectromagnetismBrewster's Angle · At a special incidence angle, reflected light is 100% polarized perpendicular to the plane of incidence
Brewster's angle θ_B is the angle of incidence at which light reflected from a surface is completely polarized perpendicular to the plane of incidence
OpticsBrillouin Zone · Wigner-Seitz cell of the reciprocal lattice — k-space domain where Bloch states live
The Brillouin zone is the Wigner-Seitz primitive cell of the reciprocal lattice of a crystal. In real space, atoms repeat on a Bravais lattice {R}; in
Solid State PhysicsBrownian Motion · The jittery random walk of a grain bombarded by unseen molecules — Einstein's proof that atoms are real
Brownian motion is the jittery random walk of a microscopic grain bombarded from all sides by unseen molecules. Einstein's 1905 prediction — ⟨x²⟩ = 2D
Statistical MechanicsBuoyancy · Archimedes
3D object submerged in water with upward buoyant force arrow and downward weight arrow. If buoyancy > weight: object floats. Show Archimedes principle
Fluid DynamicsCKM Matrix · Three angles, one phase — and every cross-generation quark decay in the Standard Model
The Cabibbo-Kobayashi-Maskawa matrix is the 3x3 unitary matrix encoding how weak interactions mix quark flavors. Three mixing angles plus one complex
Particle PhysicsCMB Anisotropy · Tiny ΔT/T ~ 10⁻⁵ temperature fluctuations encode acoustic oscillations of pre-recombination plasma
The cosmic microwave background (CMB) is the relic radiation from when the universe became transparent to photons at z ≈ 1100 (cosmic age 380,000 year
CosmologyCabibbo Angle · The 13-Degree Rotation That Mixes Down and Strange Quarks
The Cabibbo angle explained: how a 13.02° rotation (sin θ_C ≈ 0.225) mixes down and strange quarks, suppresses strange-particle decays, and seeds the
Particle PhysicsCanonical Transformations and Generating Functions in Hamiltonian Mechanics
Canonical transformations and generating functions explained: the four F1–F4 types, the symplectic condition, worked harmonic-oscillator example, and
Classical MechanicsCapacitors · storing charge
3D parallel plate capacitor charging up. Electrons accumulate on one plate, creating an electric field between plates. Disconnect battery and capacito
ElectromagnetismCapillary Action
Capillary action is the rise (or fall) of a liquid in a narrow tube driven by surface tension and wetting. Jurin's law: h = 2γcos(θ)/(ρgr). Explains w
Fluid DynamicsCarnot Engine · maximum efficiency
3D heat engine cycle: absorb heat from hot reservoir, do work (piston expands), exhaust waste heat to cold reservoir. Efficiency = 1 - Tc/Th. No real
ThermodynamicsCatenary Curve · The exact shape a hanging chain takes
The catenary curve is the exact shape a uniform chain takes hanging under its own weight: y = a·cosh(x/a). It is not a parabola, and flipped it makes
Classical MechanicsCavitation · Low pressure boils a liquid into bubbles that collapse violently — eroding propellers and arming a shrimp's claw
Cavitation is the formation and violent collapse of vapor bubbles in a liquid when local pressure drops below the vapor pressure — the same temperatur
Fluid DynamicsCenter of Mass · balance point
3D irregularly shaped object balanced on a point — the center of mass. Two masses connected by a rod: center of mass is closer to the heavier one. Sys
Classical MechanicsCentripetal vs Centrifugal · real inward force
3D car turning in a circle. Centripetal: real force from friction pulling car inward. Centrifugal: fictitious force felt by passenger pushed outward.
Classical MechanicsChain Fountain (Mould Effect) · A bead chain that doesn't just pour out of a pot — it leaps above the rim first
The chain fountain (Mould effect) is a self-siphoning bead chain that arcs above its beaker before falling. The rising chain isn't just pulled by grav
Classical MechanicsChandrasekhar Limit · M_Ch ≈ 1.4 M_sun — maximum mass a white dwarf can have before collapse
The Chandrasekhar limit M_Ch ≈ 1.4 solar masses is the maximum mass a white dwarf can have before electron degeneracy pressure can no longer support i
AstrophysicsCharge-Density Wave · The Periodic Lattice Distortion That Gaps a Fermi Surface
Charge-density wave explained: how a 2k_F periodic lattice distortion (Peierls transition) gaps a Fermi surface, with real numbers from NbSe3, TaS2, a
Condensed MatterChemical Potential · The pressure that drives particles to move
Chemical potential is the energy cost of adding one particle to a system. Particles flow from high to low chemical potential until equilibrium equaliz
Statistical MechanicsCherenkov Radiation · Faster Than Light
Cherenkov radiation explained in 3D — watch a charged particle outrun light in water and produce an eerie blue glow. Interactive animation on Unseel.
PhysicsChiral Symmetry · Left and right-handed quarks transform independently — until the QCD vacuum picks a side
Chiral symmetry is the freedom to rotate left-handed and right-handed fermions independently. Spontaneously broken in QCD by the quark condensate, it
Particle PhysicsChladni Patterns · Sand on a vibrating plate draws the still lines — a map of the plate's resonant mode shape
Chladni patterns are the star-and-grid figures that sand draws on a vibrating plate: the grains slide off the moving antinodes and pile up along the s
Waves & OscillationsCircular Motion · centripetal force
3D ball on a string spinning in a circle. Velocity arrow tangent to circle, centripetal force arrow pointing inward. Cut the string and ball flies off
Classical MechanicsClausius Inequality · For any cyclic process: ∮ dQ_rev/T = 0 (reversible) or ∮ dQ/T ≤ 0 (irreversible)
The Clausius inequality (Rudolf Clausius, 1854) states that for any thermodynamic cycle, ∮ dQ/T ≤ 0, where Q is heat absorbed by the system and T is t
ThermodynamicsCoanda Effect · Why a fluid jet clings to and bends around a nearby curved surface
The Coanda effect is the tendency of a fluid jet to stay attached to and bend around a nearby convex surface. Entrainment lowers the pressure between
Fluid DynamicsCoherent States
A coherent state |α⟩ is the eigenstate of the annihilation operator, â|α⟩ = α|α⟩ — the minimum-uncertainty state that oscillates like a classical part
Quantum MechanicsColor and Light · RGB
3D white light hitting a prism and splitting into rainbow spectrum. Then show a red object: absorbs all colors except red which it reflects to your ey
OpticsCompton Scattering · Photons hitting electrons lose energy and gain wavelength — Compton's 1923 proof that light has momentum
When a high-energy photon strikes a free electron it bounces off carrying less energy and a longer wavelength, with the shift Δλ depending only on the
Quantum PhysicsConservation of Energy · kinetic
3D pendulum swinging. At top: max potential energy (gold glow), at bottom: max kinetic energy (cyan glow). Energy bar shows PE and KE trading back and
Classical MechanicsConservation of Momentum · p = mv
3D billiard balls colliding. Total momentum before equals total after. Show elastic collision (balls bounce) and inelastic (balls stick together). Mom
Classical MechanicsContinuity Equation · ∂ρ/∂t + ∇·(ρu) = 0 — what flows in must flow out (or accumulate)
The continuity equation states that mass is conserved in a fluid: ∂ρ/∂t + ∇·(ρu) = 0, where ρ is density and u velocity. The first term is local accum
Fluid DynamicsConvex and Concave Lenses · focal point
3D parallel light rays passing through a convex lens converging to a focal point. Then concave lens diverging rays outward. Show real vs virtual image
OpticsCooper Pairs · At low T, phonon-mediated attraction binds two opposite-spin electrons into a boson
A Cooper pair is a bound state of two electrons of opposite spin and momentum, held together by a weak attractive interaction mediated by lattice vibr
Condensed MatterCoriolis Effect · Rotating Frames
The Coriolis effect explained in 3D — watch projectiles curve in rotating frames, hurricanes spiral, and Foucault pendulums precess. Interactive anima
PhysicsCosmic Inflation · In the first 10⁻³² s after Big Bang, the universe expanded by ~10²⁶
Cosmic inflation, proposed by Alan Guth (1980), Andrei Linde (1982), Steinhardt and Albrecht (1982), is the theory that the universe underwent an expo
CosmologyCosmic Rays · high-energy particles
3D high-energy proton from space hitting upper atmosphere. Creates shower of secondary particles cascading downward: pions, muons, electrons, photons.
Particle PhysicsCosmological Constant Λ · Einstein's biggest blunder reborn as dark energy
The cosmological constant Λ is a constant added to Einstein's field equations: G_μν + Λ g_μν = 8πG T_μν / c⁴. Einstein introduced Λ in 1917 to allow a
CosmologyCoulomb's Law · F = kq₁q₂/r²
3D two charged particles with force arrows. Same charges repel, opposite attract. Double the distance: force drops to 1/4. Double a charge: force doub
ElectromagnetismCoupled Oscillators · Two pendulums joined by a spring — normal modes, beats, and energy transferring back and forth
Two pendulums or masses joined by a spring oscillate as a sum of two normal modes — a symmetric in-phase mode and an antisymmetric out-of-phase mode.
OscillationsCritical Exponents · The universal power laws that govern how matter teeters on the edge of a phase transition
Critical exponents are the power laws that govern continuous phase transitions: magnetization ~ (Tc-T)^β, susceptibility ~ |T-Tc|^-γ, correlation leng
Statistical MechanicsCrooks Fluctuation Theorem · The Forward vs Reverse Work Ratio
The Crooks Fluctuation Theorem explained: how the ratio of forward to reverse work distributions, P_F(W)/P_R(−W) = exp[(W−ΔF)/kT], recovers exact free
Statistical MechanicsCrow Instability · Why Aircraft Wingtip Vortices Pinch Into Rings
The Crow instability explained: how counter-rotating aircraft wingtip vortices buckle at an 8.6b wavelength and reconnect into vortex rings, with the
Fluid DynamicsCyclotron · A magnetic field bends, an alternating voltage kicks — and charged particles spiral outward to high energy
A cyclotron uses a steady magnetic field plus an alternating voltage to spiral charged particles outward to high energy. The field bends them in circl
ElectromagnetismCyclotron Resonance · Tune a microwave to the speed a particle spins in a magnetic field, and you can pump its orbit wide open
Cyclotron resonance: a charged particle gyrates in a magnetic field at omega_c = qB/m and resonantly absorbs EM energy at that frequency. It measures
Plasma PhysicsD'Alembert's Principle
D'Alembert's Principle recasts Newtonian dynamics as statics: adding the inertial force −ma to the applied forces makes every particle in equilibrium,
Classical MechanicsDamped Oscillation · Underdamped, critical, and overdamped regimes determined by b² − 4mk
A damped oscillator follows mẍ + bẋ + kx = 0, where m is mass, b is the damping coefficient, and k is the spring constant. Solutions split into three
Classical MechanicsDark Energy · 68% of the universe driving accelerated expansion — equation of state w ≈ −1
Dark energy is the unknown component making up ~68% of the energy density of the universe (Planck 2018), responsible for the accelerated expansion of
CosmologyDark Matter · Invisible Mass
Dark matter explained — invisible mass that holds galaxies together, bends light, and makes up 27% of the universe.
AstrophysicsDebye Model of Heat Capacity · Treats solid as a continuum of phonon modes up to a cutoff frequency ω_D — gives C_V ∝ T³ at low T
The Debye model (Peter Debye, 1912) treats a solid's vibrational modes as a continuum of phonons (quantized lattice vibrations) with frequencies up to
Solid State PhysicsDebye Shielding · How a plasma hides a charge — a screening cloud that turns a long-range field into an exponentially damped whisper
Debye shielding is how a plasma screens any stray charge — a cloud of opposite charge rearranges to cancel the field over the Debye length λ_D = √(ε₀k
Plasma PhysicsDeep Inelastic Scattering
Deep inelastic scattering (DIS) fires high-energy electrons at protons and — from the way they bounce off pointlike partons — revealed that the proton
Particle PhysicsDensity of States · How many quantum seats sit at each energy — the curve that quietly sets a material's heat capacity, color, and conductivity
Density of states g(E) counts how many quantum states sit in each sliver of energy. In 3D it grows as √E; in 2D it is flat; in 1D it diverges as E^(-1
Condensed MatterDetailed Balance
Detailed balance is the equilibrium condition P_i·W(i→j) = P_j·W(j→i) — every microscopic transition is exactly balanced by its reverse, so no net pro
Statistical MechanicsDiamagnetism & Paramagnetism · Materials that flee or follow a magnet
Diamagnetism and paramagnetism are the two weak magnetic responses of ordinary matter: diamagnets are repelled by a magnet (χ < 0), paramagnets are we
ElectromagnetismDielectric Polarization · How an insulator fights back against a field
Dielectric polarization is the field-induced alignment of bound charge inside an insulator, creating an opposing field that lowers the net field and b
ElectromagnetismDiffraction · wave bending around obstacles
3D wave passing through a narrow slit and spreading out on the other side. Show single slit diffraction pattern with central bright band and dimmer si
OpticsDiffraction Grating · N slits at spacing d sharpen interference into λ-resolving spectra
A diffraction grating is a periodic array of N slits (or grooves) at spacing d that produces sharp principal maxima at angles satisfying d sin θ_m = m
OpticsDirac Equation · The one equation that married quantum mechanics to special relativity — and demanded antimatter exist
The Dirac equation (iγ^μ∂_μ − m)ψ = 0 is the relativistic wave equation for spin-1/2 particles. It predicts antimatter, intrinsic spin, and g = 2.
Quantum Field TheoryDispersion Relation · How a wave's frequency depends on its wavelength
A dispersion relation is the function ω(k) linking a wave's frequency to its wavenumber. Its slope is phase velocity, its tangent group velocity, its
Waves & OscillationsDisplacement Current · Maxwell's missing term — the changing electric field that sources magnetic field and makes electromagnetic waves possible
Displacement current ε₀·∂E/∂t is Maxwell's 1861 correction to Ampère's law — the term that makes the capacitor magnetic field consistent, conserves ch
ElectromagnetismDoppler Effect · moving source
3D ambulance moving with circular wave fronts. Waves compress ahead (higher pitch, blue) and stretch behind (lower pitch, red). Stationary observer he
Waves & OscillationsDouble Pendulum Chaos · No randomness anywhere in the equations — and yet two near-identical pendulums end up doing completely different things
The double pendulum is fully deterministic yet chaotic — two near-identical starts diverge exponentially. The canonical desktop demo of deterministic
Nonlinear DynamicsDouble-Slit Experiment · Two slits, fringes spaced λL/d — and the same pattern emerges photon by photon
Light through two slits produces fringes spaced λL/d. With a 633 nm laser, d = 0.5 mm slits, L = 2 m screen: bands every 2.5 mm. Same pattern emerges
Wave OpticsDrag Coefficient · One number that captures how much air fights you
The drag coefficient (Cd) is a dimensionless number that captures how much a shape resists motion through a fluid. A sphere is ~0.47, a teardrop ~0.04
Fluid DynamicsDriven Oscillator and Resonance · Amplitude peaks when driving frequency ω matches natural ω₀
A driven damped oscillator obeys mẍ + bẋ + kx = F₀ cos(ωt). The steady-state response is x(t) = A(ω) cos(ωt − φ) with amplitude A(ω) = F₀ / √((k − mω²
Classical MechanicsDrude Model · A metal is a gas of free electrons bouncing off ions — and that single picture gives you Ohm's law
The Drude model treats a metal as a gas of free electrons that scatter off ions every relaxation time tau, deriving Ohm's law and the conductivity sig
Condensed MatterDuffing Oscillator · Add one cubic term to a spring and you get a bent resonance peak, a sudden amplitude jump, and — eventually — chaos
The Duffing oscillator x'' + δx' + αx + βx³ = γcos(ωt) is a driven nonlinear spring whose cubic restoring term produces a bent resonance curve, hyster
Nonlinear DynamicsE × B Drift · Charge-Independent Sideways Motion in Crossed Fields
E × B drift explained: why charged particles in crossed electric and magnetic fields drift sideways at v = E/B, independent of charge, mass, or energy
Plasma PhysicsEddy Currents · Closed loops of induced current that brake trains, heat induction cookers, and cause transformer losses
Drop a strong magnet through an aluminium tube and it falls in slow motion. Place a steel pan on an induction cooktop and the pan heats while the surf
ElectromagnetismEffective Mass · Why an electron in a crystal acts heavier or lighter
Effective mass is the apparent mass an electron acts as if it has inside a crystal, set by how sharply the energy band curves. It can be lighter, heav
Condensed MatterEffective Potential · Folding orbital motion into a 1D energy hill
The effective potential folds a central-force orbit's angular momentum into a single 1D energy curve, so radial motion reduces to a particle sliding o
Classical MechanicsEhrenfest Theorem · When Quantum Averages Obey Newton's Laws
The Ehrenfest theorem explained: how quantum expectation values of position and momentum obey Newton's laws, the derivation, when it holds exactly, an
Quantum MechanicsEinstein Field Equations · Ten coupled nonlinear equations that fuse gravity, space, and time into a single geometric law
The Einstein field equations explained: matter curves spacetime and curvature dictates motion. G_μν + Λg_μν = 8πG/c⁴ T_μν — ten coupled nonlinear PDEs
General RelativityElastic Potential Energy · stored in deformation
3D bow and arrow. Pull the string back: elastic PE stored (energy bar fills gold). Release: PE converts to KE as arrow flies. PE = ½kx² shown with spr
Classical MechanicsElastic vs Inelastic Collision · Momentum & Energy
All collisions conserve momentum. Elastic collisions also conserve kinetic energy; inelastic collisions lose it to heat and deformation.
Classical MechanicsElasticity Tensor · σᵢⱼ = Cᵢⱼₖₗ εₖₗ — Hooke's law generalized to 3D, with up to 21 independent constants
The elasticity tensor Cᵢⱼₖₗ generalizes Hooke's law to anisotropic 3D materials: σᵢⱼ = Cᵢⱼₖₗ εₖₗ. Symmetry reduces 81 components to at most 21 indepen
Continuum MechanicsElectric Current · electron flow
3D wire cross-section showing electrons drifting slowly through a lattice of positive ions. Current direction is conventional (opposite to electron fl
ElectromagnetismElectric Field · charges
3D positive and negative charges with electric field lines flowing from positive to negative. Bring charges closer and field lines intensify. Show rep
ElectromagnetismElectric Potential and Voltage
Electric potential V is the electric potential energy per unit charge at a point — V = U/q, measured in volts (1 V = 1 J/C). For a point charge V = kq
ElectromagnetismElectrical Power · P = IV
3D light bulb in a circuit. Power = current × voltage. Higher wattage = brighter glow. Show how your electricity bill relates to kilowatt-hours: power
ElectromagnetismElectromagnetic Induction · Faraday
3D magnet moving through a coil of wire. As the magnet enters, current flows one direction (meter deflects). As it exits, current reverses. Faster mov
ElectromagnetismElectromagnetic Spectrum · radio to gamma
3D spectrum bar from long radio waves to short gamma rays. Each region lights up with its characteristic color/representation. Show the inverse relati
ElectromagnetismElectromagnetic Waves · E and B fields
3D wave showing oscillating electric field (vertical) and magnetic field (horizontal) perpendicular to each other and to the direction of travel. Trav
ElectromagnetismElectroweak Unification · Weak and electromagnetic forces are two faces of one gauge symmetry — broken by the Higgs
Electroweak unification: the Weinberg–Salam model unifies the weak and EM forces above ~250 GeV via SU(2)×U(1). Higgs mechanism breaks the symmetry. N
Particle PhysicsEnthalpy
Enthalpy H is the total heat content of a system at constant pressure — H = U + PV. It equals the heat exchanged in a constant-pressure process (ΔH =
ThermodynamicsEntropy · disorder
3D box of gas particles. Start ordered in one corner: low entropy, few microstates. Particles spread randomly: high entropy, many microstates. This de
ThermodynamicsEquipartition Theorem · Each quadratic degree of freedom contributes ½kT to the mean energy of a system in thermal equilibrium
The equipartition theorem says each quadratic degree of freedom contributes ½kT to the mean energy of a system at temperature T. Predicts gas heat cap
Statistical MechanicsEquivalence Principle · A free-falling lab is locally indistinguishable from floating in space
The equivalence principle, Einstein's foundation for general relativity (1907), states that gravitational and inertial mass are identical — and conseq
General RelativityErgosphere · The region where spacetime itself is dragged
The ergosphere is the region outside a rotating black hole's event horizon where spacetime is dragged so violently that nothing can stay still — it mu
General RelativityEscape Velocity · overcoming gravity
3D rocket launching. Too slow: arcs back to Earth. At escape velocity (11.2 km/s): barely escapes gravity. Faster: escapes easily. Show how escape vel
Classical MechanicsEuler's Disk · A spinning disk's wobble accelerates into a whirring finale as energy drains and the contact point races around the rim
Euler's disk is a spinning, rolling disk whose wobble (precession) speeds up without limit as it settles — the contact point races around the rim at h
Classical MechanicsEvanescent Wave · The exponentially decaying field that leaks past a totally reflecting boundary
Evanescent waves are exponentially decaying fields that leak past a totally reflecting boundary. Penetration depth is fractions of a wavelength — the
OpticsEvent Horizon · The boundary at r = r_s = 2GM/c² beyond which nothing — including light — can escape
The event horizon of a black hole is the boundary at which the escape velocity equals c — beyond it, no causal signal (matter, photon, information) ca
Black HolesExciton · A bound electron–hole pair carrying energy, not charge
An exciton is a bound electron–hole pair created when a semiconductor absorbs a photon. It carries energy but no net charge, and recombines to emit li
Condensed MatterFabry-Perot Cavity · Two parallel mirrors, multiple bounces — razor-sharp resonances at λ = 2nL/m
A Fabry-Perot cavity uses two parallel partial mirrors to build sharp transmission resonances at λ = 2nL/m. The basis of laser cavities, optical filte
OpticsFaraday Rotation · A magnetic field that twists light's polarization
Faraday rotation is the rotation of light's polarization plane as it travels through a medium along a magnetic field. The angle is β = V·B·L, set by t
OpticsFaraday Waves · Parametric surface waves on a shaken fluid — the surface answers at half the drive frequency
Faraday waves are standing ripples that erupt on the surface of a vertically vibrated fluid once the shaking acceleration crosses a threshold. The sur
Fluid DynamicsFaraday's Law of Induction · EMF = −dΦ_B/dt — changing magnetic flux induces EMF
Faraday's law of induction: the EMF (electromotive force) induced in any closed loop equals the negative rate of change of magnetic flux through any s
ElectromagnetismFermat's Principle · Light takes the path that extremizes optical path length — Snell, reflection, geodesics all follow
Fermat's principle: light follows the path that extremizes optical path length. Implies Snell's law of refraction, the law of reflection, and geodesic
OpticsFermi Surface · In a metal, the constant-energy surface E(k) = E_F that determines transport, magnetism, superconductivity
The Fermi surface is the constant-energy surface in momentum (k-) space corresponding to the Fermi energy E_F — the highest occupied electron state at
Condensed MatterFermi-Dirac Statistics · Why electrons stack instead of pile — the law of the Fermi sea and its razor-sharp surface
Fermi-Dirac statistics govern indistinguishable half-integer-spin particles obeying Pauli exclusion. Mean occupation n = 1/(e^((E-μ)/kT) + 1) — a shar
Statistical MechanicsFerromagnetic Domains · Why iron can be magnetized and demagnetized
Ferromagnetic domains are microscopic regions of uniformly aligned atomic spins in iron, nickel and cobalt. An external field grows aligned domains un
ElectromagnetismFeynman Diagrams · Each diagram = one term in a perturbative expansion of the scattering amplitude
Feynman diagrams (Richard Feynman, 1948) are graphical representations of terms in the perturbative expansion of scattering amplitudes in QED, QCD, an
Quantum Field TheoryFine Structure · Why a single spectral line, looked at closely, splits into a tightly spaced doublet
Fine structure is the tiny splitting of atomic spectral lines caused by spin-orbit coupling and relativistic corrections, at the scale of alpha² (α =
Atomic PhysicsFirst Law of Thermodynamics · energy conservation
3D gas in a piston. Heat added increases internal energy. Piston expands doing work. ΔU = Q - W shown with energy bars. Energy is neither created nor
ThermodynamicsFokker-Planck Equation · The equation that turns a single unpredictable jiggle into a forecastable cloud of probability
The Fokker-Planck equation is the PDE for how a probability distribution evolves under drift plus diffusion — governing Brownian motion, diffusion, an
Statistical MechanicsForce on Current-Carrying Wire · F = BIL
3D wire carrying current placed in a magnetic field. Force pushes wire perpendicular to both current and field (right-hand rule). This is how electric
ElectromagnetismFoucault Pendulum · A long pendulum whose swing plane rotates as the Earth turns beneath it — Paris precession period 31.8 h
A long pendulum's swing plane rotates relative to the ground as the Earth turns beneath it. The precession period equals 24 h divided by sin(latitude)
Classical MechanicsFour-Vector · One time component, three space components, one invariant length — the natural variable of relativity
A four-vector groups one time component and three spatial components into a single object that transforms covariantly under Lorentz boosts. Its invari
Special RelativityFourier Optics
Fourier optics treats a lens as an analog Fourier-transform engine: the field in the back focal plane is the 2-D Fourier transform of the field at the
OpticsFourier's Law & the Heat Equation · Heat flows down temperature gradients — q = −k∇T — and the heat equation ∂T/∂t = α∇²T predicts how any body cools
Fourier's law says heat flux q = −k∇T — heat flows down the temperature gradient, fastest where temperature changes sharpest. Combine it with energy c
ThermodynamicsFrame Dragging · A spinning mass doesn't just curve spacetime — it grabs it and twists it around with the rotation
Frame dragging is the twisting of spacetime by a rotating mass — the Lense-Thirring effect. Gravity Probe B measured it at 37.2 mas/yr; Kerr black hol
General RelativityFraunhofer Diffraction · In the far field, the diffraction pattern is the spatial Fourier transform of the aperture
Fraunhofer diffraction (Joseph von Fraunhofer, 1820s) is the regime of wave diffraction where the source and observation point are effectively at infi
OpticsFresnel Diffraction
Fresnel diffraction is near-field diffraction, where the curved wavefronts and a finite Fresnel number N_F = a²/(λL) ≳ 1 make the pattern change with
OpticsFresnel Lens · Collapse a thick lens into concentric ridges that bend light the same way with a fraction of the glass
A Fresnel lens keeps only the curved surface of a thick lens, collapsing it into concentric ridges that refract light to the same focus with a fractio
OpticsFriction · static
3D block on a surface. Push force increases until static friction breaks (block starts moving). Then kinetic friction takes over at lower value. Show
Classical MechanicsGIM Mechanism · How the Charm Quark Suppresses Flavor-Changing Neutral
The GIM mechanism explained: how the charm quark and CKM unitarity suppress flavor-changing neutral currents like K_L → μμ to one part in 10⁸, and pre
Particle PhysicsGamma Decay · An excited nucleus shedding a high-energy photon
Gamma decay is when an excited atomic nucleus drops to a lower energy level by emitting a high-energy photon — a gamma ray — leaving the same element
Nuclear PhysicsGauss's Law · ∮ E·dA = Q_enc/ε₀ — electric flux through a closed surface
Gauss's law: the total electric flux through any closed surface equals the total enclosed charge divided by the permittivity of free space ε₀ = 8.854
ElectromagnetismGaussian Beam · Why a laser pinches to a waist, stays sharp for a while, then fans out into a cone — and the three numbers that predict all of it
A Gaussian beam is a laser beam with a bell-shaped intensity profile. Learn the waist w0, Rayleigh range z_R = π·w0²/λ, far-field divergence, and the
OpticsGeiger-Nuttall Law · Why Alpha Half-Lives Span 30 Orders of Magnitude
The Geiger-Nuttall law explained: how a factor-of-three change in alpha energy makes alpha-decay half-lives span 30+ orders of magnitude, via quantum
Nuclear PhysicsGeneral Relativity · Curved Spacetime
Einstein 1915: mass curves spacetime, and matter follows geodesics through that curvature. Gravity is geometry.
Modern PhysicsGeodesic Equation · Why a falling apple, an orbiting planet, and a bent ray of starlight are all just coasting along the straightest path geometry allows
The geodesic equation says free-falling objects follow the straightest possible paths through curved spacetime. Gravity is geometry, not a force — and
General RelativityGibbs Free Energy
Gibbs free energy G = H − TS is the thermodynamic potential that predicts spontaneity at constant temperature and pressure: a process runs when ΔG < 0
ThermodynamicsGibbs Phase Rule · Counting Degrees of Freedom in Phase Diagrams
The Gibbs phase rule F = C - P + 2 explained: how to count degrees of freedom in phase diagrams, its derivation, worked examples, and why the water tr
ThermodynamicsGinzburg-Landau Theory · The Order Parameter That Predicts Superconducting Coherence
Ginzburg-Landau theory explained: the complex order parameter ψ, the free-energy functional, coherence length ξ, penetration depth λ, and the κ = 1/√2
Condensed MatterGoos-Hänchen Shift · The Lateral Beam Displacement at Total Internal Reflection
The Goos-Hänchen shift is the lateral displacement of a light beam at total internal reflection. Learn the Artmann formula, evanescent-wave mechanism,
OpticsGrad-B Drift · Why Charged Particles Curve Across Magnetic Gradients
Grad-B drift explained: why charged particles drift perpendicular to a magnetic field gradient, the velocity formula, the ring current, and radiation-
Plasma PhysicsGrand Canonical Ensemble · Open the box to particles too — fix the temperature and the chemical potential, and let nature decide how many particles to keep
The grand canonical ensemble describes a system that exchanges both energy and particles with a reservoir at fixed temperature T and chemical potentia
Statistical MechanicsGravitational Fields · curved spacetime
3D rubber sheet analogy. Place a heavy mass: sheet curves creating a well. Smaller objects roll toward the well following curved paths. Shows how mass
Classical MechanicsGravitational Lensing · Mass bending light into rings and mirages
Gravitational lensing is the bending of light by mass-warped spacetime, producing Einstein rings, arcs, and multiple images. Deflection is 4GM/(c²b)..
General RelativityGravitational Redshift · Photons lose energy climbing out of gravity — verified by Pound-Rebka 1959 and GPS
Gravitational redshift: photons emitted from a region of strong gravitational potential are observed at lower frequency (longer wavelength) by an obse
General RelativityGravitational Waves · Spacetime Ripples
Ripples in spacetime from merging black holes, detected by LIGO in 2015 and earning the 2017 Nobel Prize.
AstrophysicsGravity · 9.8 m/s²
3D objects of different masses falling at the same rate in vacuum. Then show with air resistance where a feather falls slowly. Earth pulls objects wit
Classical MechanicsGravity Assist (Slingshot) · How a spacecraft steals a sliver of a planet's orbital momentum to gain speed without burning fuel
A gravity assist (slingshot) lets a spacecraft gain speed for free by flying past a moving planet. The planet's gravity bends the craft's path, and be
AstrophysicsGroup Velocity vs Phase Velocity · v_phase = ω/k vs v_group = dω/dk — envelope/info vs carrier
A monochromatic wave cos(kx − ωt) has phase velocity v_p = ω/k. A wave packet (sum of nearby frequencies) propagates with group velocity v_g = dω/dk —
Wave PhysicsGörtler Vortices · Centrifugal Streamwise Rolls on Concave Walls
Görtler vortices explained: counter-rotating streamwise rolls from centrifugal instability on concave walls, the Görtler number G, critical value, and
Fluid DynamicsHalf-Life · exponential decay
3D sample of radioactive atoms. Every half-life period, half the atoms decay (flash and disappear). Start with 1000, then 500, 250, 125... Exponential
Nuclear PhysicsHall Effect · When current crosses a magnetic field, charges pile sideways — turning a slab of conductor into a magnetometer
Send a current along a thin metal strip and stand a magnet across it. Within nanoseconds, charges deflect to one edge and a tiny voltage appears acros
ElectromagnetismHamilton-Jacobi Equation · All of mechanics squeezed into one equation for the action — and the secret doorway from Newton to Schrödinger
The Hamilton-Jacobi equation compresses all of mechanics into one PDE for the action S: ∂S/∂t + H(q, ∂S/∂q) = 0. Its characteristics are classical tra
Classical MechanicsHamiltonian Mechanics · Phase-space formulation where position and momentum are the protagonists, not force
Hamiltonian mechanics replaces force with energy and trades two second-order equations for 2n first-order ones in phase space. The Hamiltonian H(q, p)
Classical MechanicsHawking Radiation · Black Hole Evaporation
Black holes slowly evaporate as virtual particle pairs split at the event horizon. Stephen Hawking, 1974.
AstrophysicsHeat Transfer Methods · conduction
3D three panels side by side. Conduction: vibrating molecules pass energy along a metal bar. Convection: heated fluid rises, cool fluid sinks in a loo
ThermodynamicsHeisenberg Uncertainty Principle · position × momentum ≥ ℏ/2
3D particle with position shown as a fuzzy cloud. Measure position precisely (cloud shrinks) and momentum arrow becomes wildly uncertain. Measure mome
Quantum PhysicsHelmholtz Free Energy
Helmholtz free energy F = U − TS is the thermodynamic potential for a system at constant temperature and volume. It equals the maximum work extractabl
ThermodynamicsHelmholtz Resonance · Air in a cavity with a neck springs like a mass on a spring — the note you get blowing across a bottle
Helmholtz resonance is the low hum you get blowing across a bottle: the slug of air in the neck bounces like a mass on a spring against the springy ai
Waves & OscillationsHiggs Boson · Mass Giver
The particle associated with the Higgs field that gives other particles their mass. Discovered at CERN in 2012.
Particle PhysicsHohmann Transfer Orbit · The fuel-cheapest two-burn ellipse for moving between two circular orbits
The Hohmann transfer orbit is the fuel-cheapest two-burn elliptical path between two coplanar circular orbits: one prograde burn raises apoapsis to th
AstrophysicsHolography · Recording a 3D image in an interference pattern
Holography is recording a 3D image as an interference pattern between an object beam and a reference beam, then re-illuminating the pattern to rebuild
OpticsHomopolar Motor · A wire, a battery, and a magnet — the simplest possible motor, spun by a continuous Lorentz force
A homopolar motor is the simplest electric motor: a wire, a battery, and a magnet. A radial current I crossing an axial field B feels a Lorentz force
ElectromagnetismHong-Ou-Mandel Effect · Two Photons That Refuse to Split
The Hong-Ou-Mandel effect explained: two indistinguishable photons on a beam splitter bunch together, canceling coincidences. Physics, math, and the 1
Quantum PhysicsHooke's Law · F = -kx
3D spring with mass attached. Pull and release: oscillates. Double the displacement: double the restoring force. Stiffer spring (higher k): stronger f
Classical MechanicsHow Lasers Work · stimulated emission
3D atoms in a gain medium. Pump energy excites atoms to higher state (population inversion). One photon triggers stimulated emission: identical photon
OpticsHubble's Law · expanding universe
3D galaxies on an expanding grid. Farther galaxies move away faster (longer red arrows). Light from distant galaxies stretched to red (redshift). Show
AstrophysicsHuygens' Principle · Every point on a wavefront radiates a wavelet — the envelope of all wavelets is the next wavefront
Huygens' principle says every point on a wavefront acts as a source of secondary spherical wavelets; the new wavefront is their envelope. Pre-Maxwell
Wave OpticsHydrogen Atom Spectrum · Discrete energy levels E_n = −13.6 eV / n² produce sharp emission lines — the Rydberg formula
Hydrogen atom spectrum: discrete emission lines at E_n = -13.6 eV/n². Lyman series in UV (n→1), Balmer in visible (n→2), Paschen in IR. Foundation of
Quantum MechanicsHyperfine Structure · Nuclear Spin Splitting and the 21 cm Line
Hyperfine structure explained: how nuclear spin splits atomic levels, the physics of hydrogen's 21 cm line at 1420 MHz, worked numbers, and why it map
Atomic PhysicsIdeal Gas Law · PV = nRT
3D gas particles bouncing inside a container. Increase temperature: particles move faster, pressure increases. Decrease volume: particles hit walls mo
ThermodynamicsImpulse and Momentum · J = FΔt
3D egg dropping onto hard surface (short time, big force = crack) vs soft pillow (long time, small force = safe). Same impulse changes momentum by sam
Classical MechanicsInclined Plane · Trading force for distance — the original simple machine, and the foundation of every ramp, slope and slide in physics
An inclined plane tilts at angle θ. Decompose gravity into parallel mg·sin θ and perpendicular mg·cos θ components. Foundation for friction, mechanica
Classical MechanicsIsing Model · A grid of arrows that can't make up its mind — until it suddenly does, all at once
The Ising model is a lattice of spins ±1 with nearest-neighbour coupling J — the canonical model of ferromagnetism and phase transitions. In 2D, Onsag
Statistical MechanicsIsospin
Isospin is an approximate SU(2) symmetry that treats the proton and neutron as two states of a single nucleon (I = 1/2). Introduced by Heisenberg in 1
Particle PhysicsJarzynski Equality · Free Energy From Non-Equilibrium Work
The Jarzynski equality explained: how ⟨e^(−W/kT)⟩ = e^(−ΔF/kT) extracts equilibrium free energy from irreversible non-equilibrium work, with derivatio
Statistical MechanicsJaynes-Cummings Model · One Atom Trading a Photon with a Cavity
The Jaynes-Cummings model explained: how one two-level atom exchanges a single photon with a cavity mode, with the Hamiltonian, dressed states, vacuum
Quantum PhysicsJohnson-Nyquist Noise · Why Every Resistor Hisses at Temperature T
Johnson-Nyquist (thermal) noise explained: the 4kTR equation, its derivation from the fluctuation-dissipation theorem, worked numbers, quantum limits,
ElectromagnetismJosephson Junction · Two superconductors, one nanometre-thin gap — where a phase difference becomes a current and a voltage becomes a precise frequency
A Josephson junction is two superconductors split by a thin barrier where Cooper pairs tunnel: I = I_c·sin(Δφ). The AC effect ties voltage to frequenc
Condensed MatterJoule-Thomson Effect · A real gas cools or warms when forced through a porous plug at constant enthalpy — the engine behind every cryogenic plant
The Joule-Thomson effect: a real gas changes temperature when forced through a porous plug at constant enthalpy. Inversion temperature determines whet
ThermodynamicsKelvin's Circulation Theorem
Kelvin's circulation theorem states that in an inviscid, barotropic flow with conservative body forces, the circulation Γ = ∮ v·dl around a closed mat
Fluid DynamicsKelvin-Helmholtz Instability · When two fluid layers slide past each other, the interface buckles into rolling waves
The Kelvin-Helmholtz instability arises at the interface between two fluid layers in shear flow — when one layer moves over another with different vel
Fluid DynamicsKepler's Laws · elliptical orbits
3D planet orbiting a sun in an ellipse. Moves faster near the sun, slower far away (equal areas in equal times). Third law: farther planets have longe
Classical MechanicsKerr Effect · A field that changes how glass bends light
The Kerr effect is a change in a material's refractive index proportional to the square of an applied electric field — the basis of fast optical switc
OpticsKerr-Lens Mode-Locking · How Lasers Make Femtosecond Pulses
Kerr-lens mode-locking explained: how self-focusing in a Ti:sapphire crystal locks laser modes in phase to generate 5-femtosecond pulses, with the phy
OpticsKirchhoff's Laws · Current & Voltage
KCL: charge conserved at junctions. KVL: voltage sums to zero around loops. Foundation of all circuit analysis.
ElectromagnetismKutta Condition & Lift · Why circulation around a wing makes it fly
The Kutta condition is the rule that flow leaves a sharp trailing edge smoothly, fixing the circulation around an airfoil — and that circulation is wh
Fluid DynamicsKármán Vortex Street · The staggered double row of alternating vortices a fluid sheds behind a blunt body — and the f = St·U/d clock it ticks to
A Kármán vortex street is the staggered double row of alternating, counter-rotating vortices a fluid sheds behind a blunt body when the Reynolds numbe
Fluid DynamicsLC Circuit · Capacitor and inductor swap energy at a fixed frequency — the simplest electrical oscillator
Connect a charged capacitor to an inductor and the energy doesn't sit still. The capacitor discharges through the inductor, which winds the energy int
ElectromagnetismLadder Operators · Climbing quantum energy levels one rung at a time
Ladder operators are the raising (a†) and lowering (a) operators that step a quantum system between adjacent energy eigenstates — a† adds one quantum,
Quantum MechanicsLagrange Points · Five places in a two-body system where a third small body can sit still relative to the larger pair
Lagrange points are five locations in any two-body gravitational system where a small third body can sit nearly stationary relative to the two large o
AstrophysicsLagrangian Mechanics · Energy-difference reformulation that turns Newton's laws into a path-integral problem
Lagrangian mechanics reformulates Newton's laws as a single scalar equation: a system follows the path that makes the time-integral of L = T − V stati
Classical MechanicsLamb Shift · The 1057 MHz gap that should not exist — where the empty vacuum reached into a hydrogen atom and nudged an electron
The Lamb shift is a tiny 1057 MHz splitting between hydrogen's 2s and 2p levels that Dirac theory predicts should be degenerate. It is caused by QED v
Atomic PhysicsLandau Damping · How a plasma wave fades to nothing without a single collision
Landau damping explained: a plasma wave decays with no collisions, transferring energy to particles moving near its phase velocity. The damping rate t
Plasma PhysicsLandau Theory of Phase Transitions · One free-energy curve, one sign change — and a single valley splits into two as the world chooses a side
Landau theory expands the free energy in the order parameter: F = F0 + a(T-Tc)m² + bm⁴. When a(T) changes sign at Tc, a single minimum splits into two
Statistical MechanicsLangevin Equation · Newton's second law plus a coin flip — how a single noisy line of physics turns molecular chaos into Brownian motion
The Langevin equation m dv/dt = -γv + ξ(t) is the microscopic model of Brownian motion. The fluctuation-dissipation theorem ties the random force to f
Statistical MechanicsLarmor Precession · A spin wobbling around a magnetic field
Larmor precession is the wobble of a magnetic moment around an external magnetic field at angular frequency ω = γB. It underpins NMR, MRI, and electro
ElectromagnetismLaser Cooling · Six crossed laser beams that turn light into friction — and drag a gas of atoms to a few millionths of a degree above absolute zero
Laser cooling explained: counter-propagating red-detuned beams make atoms preferentially absorb photons opposing their motion, dragging gases down to
Atomic PhysicsLatent Heat · heat without temperature change
3D temperature graph with plateaus. Ice absorbs heat: temperature rises. At 0°C: temperature stops rising while ice melts (latent heat of fusion). Sam
ThermodynamicsLeidenfrost Effect · Why a water drop floats and skitters on a screaming-hot pan instead of boiling away
The Leidenfrost effect: above ~193 °C a water droplet stops touching the surface and floats on a thin layer of its own vapor, so it skitters around an
ThermodynamicsLength Contraction · Moving rulers shrink along the direction of motion — equal and opposite for each frame
Length contraction: moving objects appear shorter along the direction of motion. L = L₀√(1−v²/c²). Symmetric — each frame sees the other contracted. C
Special RelativityLenz's Law · induced current opposes change
3D magnet dropping through a copper tube. Induced eddy currents create opposing magnetic field that slows the magnet's fall. The magnet floats down sl
ElectromagnetismLight Cone · The causal boundary of any event — past inside the lower cone, future inside the upper, nothing outside
The light cone at an event P is the set of points that a photon emitted at P can reach (future cone) or whose photons could reach P (past cone). It pa
Special RelativityLiouville's Theorem · The phase-space blob can stretch and fold forever — but its area never changes
Liouville's theorem says phase-space density is conserved along Hamiltonian flow: dρ/dt = 0. Phase volume is incompressible — the foundation of statis
Classical MechanicsLiquid Drop Model · The nucleus as a charged drop of fluid — and the five-term formula that predicts binding energy and fission
The liquid drop model treats the nucleus as a charged liquid drop. Its semi-empirical mass formula B = a_V A − a_S A^(2/3) − a_C Z²/A^(1/3) − … predic
Nuclear PhysicsLogistic Map · One line of arithmetic, one knob to turn — and the simplest known doorway from order into chaos
The logistic map x → r·x(1−x) is the simplest equation that becomes chaotic. Watch a fixed point split into a 2-cycle, 4-cycle, then chaos past r = 3.
Nonlinear DynamicsLorentz Force · F = q(E + v × B) — total electromagnetic force on a charge
The Lorentz force is the total electromagnetic force on a point charge: F = q(E + v × B). The electric component qE acts along the field; the magnetic
ElectromagnetismLorentz Transformation · t' = γ(t − vx/c²), x' = γ(x − vt) — coordinates between inertial frames
The Lorentz transformation describes how spacetime coordinates of an event change between two inertial frames moving at relative velocity v: t' = γ(t
Special RelativityLyapunov Exponent · The number that turns "a butterfly flaps its wings" into a hard calendar deadline for prediction
The Lyapunov exponent λ measures how fast nearby trajectories diverge: separation grows as δ(t) = δ₀e^(λt). A positive λ means chaos and sets the pred
Nonlinear DynamicsMach-Zehnder Interferometer · Two beam splitters that reveal quantum phase — used in LIGO and quantum optics
The Mach-Zehnder interferometer (Ludwig Mach 1891, Ludwig Zehnder 1892) splits a beam into two paths via a 50:50 beam splitter, reflects each with mir
Quantum OpticsMagnetic Field · north
3D bar magnet with field lines looping from north to south pole. Place a compass nearby and watch the needle align with the field. Show iron filings p
ElectromagnetismMagnetic Hysteresis · Why a magnet remembers its past
Magnetic hysteresis is the lag of a ferromagnet's magnetization behind the applied field, so M traces a loop, not a line — leaving remanence at zero f
ElectromagnetismMagnetic Levitation · Meissner Effect
Magnetic levitation explained in 3D — see the Meissner effect expel magnetic flux, Cooper pairs form, and a superconductor float. Interactive animatio
PhysicsMagnetic Mirror · How a Field Pinch Reflects Charged Particles
The magnetic mirror effect explained: how converging field lines reflect charged particles via the conserved magnetic moment, the loss cone, mirror ra
Plasma PhysicsMagnetic Monopole · A hypothetical isolated north or south pole — the particle that would force charge quantization on the universe
A magnetic monopole would be a hypothetical isolated north or south magnetic charge. Dirac (1931) proved that if even one exists, every electric charg
ElectromagnetismMagnetic Reconnection · Field lines snapping and releasing stored energy
Magnetic reconnection is when oppositely directed field lines in a plasma break and rejoin at an X-point, converting stored magnetic energy into heat
Plasma PhysicsMagneto-Optical Trap · Six Beams and a Quadrupole Field
Magneto-optical trap explained: how six red-detuned laser beams plus a magnetic quadrupole field cool and confine neutral atoms to ~100 μK. Physics, e
Atomic PhysicsMagnetohydrodynamics (MHD) · Coupled Navier-Stokes + Maxwell — used for plasmas, solar wind, fusion
Magnetohydrodynamics (MHD) describes the dynamics of electrically conducting fluids — plasmas, liquid metals, salt water — under the combined effect o
Plasma PhysicsMagnus Effect · Spinning Flight
The Magnus effect explained in 3D — watch spinning balls curve through air, see pressure differences and airflow streamlines. Interactive animation on
PhysicsMalus's Law · I = I₀·cos²θ — how polarizer angle controls transmitted intensity
Malus's law: linearly polarized light through a polarizer at angle θ transmits intensity I = I₀·cos²θ. Aligned 100%, 45° = 50%, crossed = 0%. The cosi
OpticsMass Spectrometer · Ionize, accelerate, bend in a field — heavier atoms curve less, so the machine sorts them by mass
A mass spectrometer ionizes a sample, accelerates the ions through a voltage, then bends them in a magnetic field. Lighter ions curve tightly, heavier
ElectromagnetismMass-Energy Equivalence · E = mc²
3D equation E=mc² with a small mass converting into an enormous burst of energy. Show that c² is a huge number (9×10¹⁶), so even tiny mass creates mas
Modern PhysicsMaxwell Construction · The Equal-Area Rule That Fixes Van der Waals Isotherms
The Maxwell construction fixes the unphysical loop in van der Waals isotherms with an equal-area rule, setting the vapor pressure where liquid and gas
ThermodynamicsMaxwell Relations · Cross partial derivatives in thermodynamics — four identities from U, H, F, G
Maxwell's relations are four identities relating partial derivatives of thermodynamic state functions, derived from the equality of mixed second parti
ThermodynamicsMaxwell's Demon · A 150-year-old paradox that connects entropy, information, and the thermodynamic cost of computation
Maxwell's demon is a hypothetical entity that sorts molecules by speed to violate the second law. Resolved by Landauer's principle: erasing one bit of
ThermodynamicsMaxwell's Equations · Four equations that make the entire universe of light, radio, and electric current run
Maxwell's four equations bind electric and magnetic fields to charge, current, and each other — and predict that light is an electromagnetic wave trav
ElectromagnetismMaxwell-Boltzmann Distribution · The distribution of molecular speeds in a gas at thermal equilibrium
A jar of nitrogen at room temperature has 10²² molecules zipping around with no two moving at the same speed. The Maxwell-Boltzmann distribution descr
ThermodynamicsMean Free Path
The mean free path λ is the average distance a molecule travels between collisions: λ = 1/(√2·n·σ), set by number density n and collision cross-sectio
Statistical MechanicsMeissner Effect · Why a superconductor doesn't just stop resisting — it throws the magnetic field out of itself
The Meissner effect is the active expulsion of magnetic field from a superconductor below its critical temperature — perfect diamagnetism, not just ze
Condensed MatterMermin-Wagner Theorem · Why 2D Systems Can't Spontaneously Order
The Mermin-Wagner theorem explained: why continuous symmetries can't spontaneously break in 2D, the Goldstone-mode mechanism, BKT loophole, and real e
Statistical MechanicsMetamaterials and Negative Refractive Index
A metamaterial is an engineered sub-wavelength composite whose effective permittivity ε and permeability μ can both be negative, giving a negative ref
OpticsMichelson Interferometer · Split, bounce, recombine — every λ/2 of arm change is one full fringe shift
Michelson interferometer splits a beam down two arms, recombines them, and reads tiny path-length changes from the fringe shift. The same architecture
InterferometryMie Scattering · When a particle is the same size as the light hitting it — why clouds, fog, and milk are white instead of blue
Mie scattering is light scattering by particles comparable in size to the wavelength. It is only weakly wavelength-dependent and strongly forward-peak
OpticsMinkowski Spacetime · 4D space with metric ds² = −c²dt² + dx² + dy² + dz² — invariant interval
Minkowski spacetime is the 4D arena of special relativity, with one time and three space dimensions, equipped with the Minkowski metric ds² = −c²dt² +
Special RelativityMirage (Gradient-Index Refraction) · A temperature gradient bends light through air of varying refractive index, lifting a false reflection of the sky off hot ground
A mirage is gradient-index refraction: a steep temperature gradient near hot ground lowers the air's refractive index there, so light rays curve conti
OpticsMoment of Inertia · resistance to rotation
3D two wheels of same mass: solid disk vs ring. Ring has more moment of inertia (mass far from axis). Race them down a ramp: disk wins because it's ea
Classical MechanicsMoment of Inertia Tensor · 3×3 symmetric tensor with eigenvectors as principal axes
For a rigid body, the moment of inertia is generally a 3×3 symmetric tensor I, not a single scalar. Components: I_ij = ∫ ρ(r) (δ_ij r² − x_i x_j) dV —
Classical MechanicsMott Insulator · A material that should conduct but doesn't
A Mott insulator is a material that band theory predicts should be a metal but which insulates because strong electron-electron repulsion freezes elec
Condensed MatterMössbauer Effect · Recoil-free gamma emission so sharp it weighs photons
The Mössbauer effect is recoil-free emission and absorption of gamma rays by nuclei locked in a crystal lattice, giving lines so sharp they resolve pa
Nuclear PhysicsNavier-Stokes Equations · ρ(∂u/∂t + u·∇u) = −∇p + μ∇²u + f — Newton's second law for fluids
The Navier-Stokes equations are the master equations of viscous fluid flow: ρ(∂u/∂t + (u·∇)u) = −∇p + μ∇²u + f, where u is velocity, ρ density, p pres
Fluid DynamicsNegative Temperature · Hotter than infinity, below absolute zero
Negative temperature is a state where adding energy lowers entropy, achieved by population inversion in a bounded-energy system. It is hotter than any
Statistical MechanicsNeutrino Oscillations · Flavor superpositions of mass eigenstates — proves neutrinos have mass
Neutrinos come in three flavors (electron, muon, tau) and three mass eigenstates (ν₁, ν₂, ν₃) — these bases differ
Particle PhysicsNeutron Moderation · Slowing neutrons so a reactor can sustain fission
Neutron moderation is the slowing of fast fission neutrons by elastic collisions with light nuclei so they reach thermal energies that efficiently spl
Nuclear PhysicsNewton's First Law · inertia
3D object at rest stays at rest until a force acts on it. Then an object in motion continues at constant velocity until a force stops it. Show both ca
Classical MechanicsNewton's Second Law · F = ma
3D block with arrows showing force applied. Double the force, double the acceleration. Double the mass, half the acceleration. Animate F=ma with chang
Classical MechanicsNewton's Third Law · action
3D two objects pushing against each other with equal and opposite force arrows. Skater pushes wall and moves backward. Rocket exhaust pushes down, roc
Classical MechanicsNoether's Theorem · Every continuous symmetry of the action gives you a conserved quantity — the deepest theorem in physics
Noether's theorem says every continuous symmetry of the action gives a conserved quantity. Time-translation gives energy, space-translation gives mome
Classical MechanicsNormal Force · The surface pushes back — a constraint force perpendicular to whatever you stand on
Normal force is the contact-surface force perpendicular to the contact plane that prevents interpenetration. On flat ground N = mg; on incline angle θ
Classical MechanicsNormal Modes · How a tangle of coupled oscillators secretly contains a set of perfectly independent, single-frequency dances
Normal modes are the independent vibration patterns of coupled oscillators, each ringing at one frequency given by eigenvalues of K/M. N masses give N
Classical MechanicsNuclear Binding Energy
Nuclear binding energy is the energy holding a nucleus together — equal to the mass defect times c², E_B = Δm·c². Binding energy per nucleon peaks at
Nuclear PhysicsNuclear Fission · splitting atoms
3D uranium nucleus hit by a neutron, splitting into two smaller nuclei plus 2-3 neutrons. Those neutrons hit more uranium atoms creating a chain react
Nuclear PhysicsNuclear Fusion · combining nuclei
3D two hydrogen nuclei overcoming repulsion at extreme temperature and fusing into helium. Mass converts to energy (E=mc²). Show this is how the sun p
Nuclear PhysicsNuclear Magic Numbers · The 2, 8, 20, 28, 50, 82, and 126 Shell Closures
Nuclear magic numbers (2, 8, 20, 28, 50, 82, 126) explained: shell closures, the spin-orbit coupling that predicts them, doubly-magic nuclei, and how
Nuclear PhysicsNuclear Shell Model · Protons and neutrons fill quantized shells like atomic electrons — and closed shells at the magic numbers make a nucleus almost unbreakable
The nuclear shell model: protons and neutrons fill quantized energy shells like atomic electrons. Closed shells at the magic numbers 2, 8, 20, 28, 50,
Nuclear PhysicsNumerical Aperture · The cone of light that sets resolution
Numerical aperture is NA = n·sin θ, the size of the light cone a lens collects. It sets the diffraction-limited resolution: bigger cone, finer detail.
OpticsNutation
Nutation is the small, fast nodding wobble of a spinning top's axis superimposed on its slow steady precession. It arises from the Euler equations at
Rotational DynamicsOhm's Law · V = IR
3D simple circuit with battery, resistor, and ammeter. Increase voltage: current increases. Increase resistance: current decreases. Show electron flow
ElectromagnetismOptical Aberrations · Why a simple lens never makes a perfect image
Optical aberrations are imperfections that stop a lens or mirror from focusing all rays to one sharp point — spherical, chromatic, coma, astigmatism,
OpticsOptical Caustics · The bright cusped curves where a curved surface focuses light — the shimmer on a pool floor
Optical caustics are the bright cusped curves where a curved reflecting or refracting surface focuses light — the shimmering net on a pool floor, the
OpticsOptical Coherence · How long a light wave stays in step with itself
Optical coherence is how long a light wave stays in step with itself, set by coherence length and time. A laser stays coherent for kilometers; a bulb,
OpticsOptical Fiber · Trapping light in a glass thread by total reflection
An optical fiber is a hair-thin glass thread that guides light by total internal reflection in its high-index core, carrying terabits over hundreds of
OpticsOptical Tweezers · Grabbing a single cell with nothing but a beam of light — and measuring the force a molecular motor pulls with
Optical tweezers use a tightly focused laser to trap dielectric beads and living cells via the gradient force, exerting piconewton forces. Arthur Ashk
OpticsOptical Vortex (Orbital Angular Momentum) · A screw of light — the wavefront twists into a helix, leaving a thread of pure darkness on the axis
An optical vortex is a light beam whose wavefront is a helix (a screw of constant phase), forcing a dark core on the axis where the phase is undefined
OpticsOtto Cycle · The idealized four-stroke gasoline engine — efficiency η = 1 − r^(1−γ), set entirely by compression ratio
The Otto cycle is the idealized four-stroke gasoline engine — adiabatic compression, isochoric heat in, adiabatic expansion, isochoric heat out. Ideal
ThermodynamicsPair Production · Turning a photon into matter and antimatter
Pair production is when a high-energy photon converts into an electron and a positron near a nucleus. It needs at least 1.022 MeV — twice the electron
Particle PhysicsParallel Axis Theorem · I = I_cm + Md² — moment of inertia about a parallel axis
The parallel axis theorem (Steiner 1840s, but in earlier Euler form): for a rigid body of mass M, the moment of inertia I about an axis parallel to an
Classical MechanicsParametric Resonance · Pumping a swing by changing it, not pushing it
Parametric resonance is growth of oscillation when a system parameter is modulated at twice the natural frequency — like pumping a swing by standing a
Waves & OscillationsParity Violation · Wu experiment 1957: weak interactions distinguish left from right
Parity violation: in 1956 Lee and Yang theorized that the weak interaction might violate parity (P) symmetry — meaning a mirror-reflected weak i
Particle PhysicsParticle in a Box · Confine a particle to 0 < x < L with infinite walls — energy quantizes as E_n = n²π²ℏ²/(2mL²)
Particle in a 1D infinite square well: E_n = n²π²ℏ²/(2mL²), ψ_n(x) = √(2/L)·sin(nπx/L). The cleanest demonstration that boundary conditions create qua
Quantum MechanicsPartition Function · Z = Σ exp(−βEᵢ) — the Rosetta stone that maps microscopic energy levels to all of thermodynamics
The partition function Z = Σ exp(−βEᵢ) encodes all thermodynamic properties of a system at temperature T. Free energy F = −kT·ln Z. Average energy, en
Statistical MechanicsPascal's Principle · pressure transmitted equally
3D hydraulic press with small piston and large piston connected by fluid. Push small piston: pressure transmits equally. Large piston lifts heavy load
Fluid DynamicsPaschen's Law · The Voltage That Sparks a Gas Gap
Paschen's Law explained: how gas breakdown voltage depends on pressure times gap distance, the Paschen curve minimum (~327 V for air), the equation, a
ElectromagnetismPaul Trap (Ion Trap) · An oscillating quadrupole field spins a saddle so fast the ion can't fall off
A Paul trap (RF ion trap) confines a single charged ion using an oscillating quadrupole electric field. Earnshaw's theorem forbids a static electrosta
Atomic PhysicsPauli Exclusion Principle · no two identical fermions
3D atom with electron shells filling up. Each orbital holds max 2 electrons with opposite spins. Try to add a third: rejected. This principle builds t
Quantum PhysicsPeltier Effect · Run current across a junction of two materials and it pumps heat from one side to the other — solid-state cooling with no moving parts
The Peltier effect is the heating or cooling at the junction of two different conductors when current flows through it. Push electrons across the boun
ThermodynamicsPenrose Process · Stealing energy from a spinning black hole
The Penrose process is a way to extract rotational energy from a spinning black hole: a particle splits inside the ergosphere, one piece falls in with
General RelativityPercolation · Add random connections until a giant cluster suddenly spans the system — a razor-sharp connectivity threshold
Percolation is the sudden onset of system-spanning connectivity: add random links one at a time and, at a sharp critical fraction pc, a giant cluster
Statistical MechanicsPeriod-Doubling Route to Chaos · How Orbits Split 1-2-4-8 Into Turbulence
The period-doubling route to chaos explained: how stable orbits split 1-2-4-8 into chaos, the Feigenbaum constants δ≈4.6692 and α≈2.5029, and real exp
Nonlinear DynamicsPhase Changes · solid
3D molecules in three states. Solid: locked in lattice vibrating in place. Liquid: loosely bonded, flowing. Gas: free flying, filling container. Anima
ThermodynamicsPhonons · Bosonic quasi-particles representing collective vibration modes — energy ℏω, momentum ℏk
A phonon is the quantum-mechanical particle representation of a normal mode of lattice vibration in a crystal — analogous to a photon for the electrom
Solid State PhysicsPhotoelectric Effect · Einstein
3D metal surface hit by photons. Below threshold frequency: no electrons emitted regardless of intensity. Above threshold: electrons pop out. Brighter
Quantum PhysicsPhotovoltaic Effect · solar cells
3D solar cell cross-section. Photons hit semiconductor p-n junction, knock electrons free creating current. Show electron-hole pairs, depletion zone,
Modern PhysicsPiezoelectricity · Crystal Power
Piezoelectricity explained in 3D — squeeze a crystal and watch voltage appear. See ion displacement, electric fields, and the inverse effect. Interact
PhysicsPlasma · Fourth State of Matter
Ionized gas of free electrons and ions. Makes up 99% of visible matter in the universe, including stars and lightning.
ThermodynamicsPlasma Frequency · The natural ringing rate of an electron gas — and the cutoff that decides whether a wave passes through or bounces off
Plasma frequency ω_p = √(n·e²/ε₀·m_e) is the natural oscillation rate of electron density in a plasma. EM waves below it are reflected — that's the io
Plasma PhysicsPlasma Sheath · The charged skin a plasma grows against a wall
A plasma sheath is the thin, ion-rich layer a plasma forms against any wall, a few Debye lengths thick, where a strong field repels electrons and slam
Plasma PhysicsPlateau-Rayleigh Instability · Why a falling stream of water beads up and breaks into drops — surface tension amplifying its own pinches
The Plateau-Rayleigh instability is why a falling stream of water breaks into droplets: surface tension amplifies any pinch whose wavelength exceeds t
Fluid DynamicsPoincaré Section · Stop watching the whole orbit. Photograph it once per lap — and the dynamics fall out as dots.
A Poincaré section slices through phase space once per period, turning a continuous flow into a discrete map. Periodic motion lands on a fixed point,
Nonlinear DynamicsPoiseuille Flow · Steady viscous flow through a pipe — parabolic profile, quartic in radius
Poiseuille flow is the steady laminar motion of a viscous fluid through a cylindrical pipe. The velocity profile is a parabola; the flow rate Q = πr⁴Δ
Fluid DynamicsPoisson Brackets · The algebraic engine of Hamiltonian motion — and the classical skeleton that quantum mechanics inherited
The Poisson bracket {f,g} encodes Hamiltonian time evolution: df/dt = {f,H}. The fundamental bracket {q,p}=1 is the classical structure that becomes t
Classical MechanicsPoisson's Spot (Arago Spot) · The bright spot dead-center in a disk's shadow — the prediction meant to bury wave optics that instead proved it
Poisson's spot (the Arago spot) is the bright point of light that appears dead-center in the shadow of a circular disk — the wave-optics prediction me
Wave OpticsPolarization of Light · filter
3D unpolarized light waves oscillating in all directions pass through a polarizing filter. Only one orientation passes through. Add a second filter at
OpticsPotential Flow
Potential flow is idealized inviscid, irrotational fluid motion in which the velocity is the gradient of a scalar potential, u = ∇φ, and φ obeys Lapla
Fluid DynamicsPoynting Vector · S = E × H — how much electromagnetic energy is flowing where, in watts per square meter
The Poynting vector S = E × H is the energy flux density of the electromagnetic field, in watts per square meter. Its direction tells you which way en
ElectromagnetismPrecession (Gyroscope) · A spinning top refuses to fall — instead its axis sweeps out a cone at Ω = mgr/(Iω)
A spinning top under gravity precesses around the vertical axis at Ω = mgr/(Iω). Same physics powers bicycles, gyrocompasses, and MRI proton precessio
Rotational DynamicsPressure · P = F/A
3D same force applied to large area (low pressure, doesn't pop balloon) vs small area (high pressure, nail pops balloon). Show snowshoes vs heels on s
Fluid DynamicsPrinciple of Superposition · waves add together
3D two waves approaching each other. When they overlap: amplitudes add (constructive) or cancel (destructive). After passing through, they continue un
Waves & OscillationsPrism Dispersion · Splitting white light into a rainbow
Prism dispersion is the splitting of white light into its colors because a glass prism's refractive index changes with wavelength — violet bends more
OpticsProjectile Motion · parabolic path
3D ball launched at an angle tracing a parabolic arc. Decompose into horizontal (constant velocity) and vertical (accelerating) components shown as se
Classical MechanicsProper Time
Proper time τ is the time a clock reads along its own worldline — dτ² = dt² − dx²/c². It is a Lorentz invariant, the same in every inertial frame, and
Special RelativityPulley Systems · A wheel, a rope, and the trick that lets one person lift a piano
Pulley systems trade force for distance. Fixed pulley redirects force. Single movable pulley halves it. Compound systems with n movable pulleys give m
Classical MechanicsQuantum Decoherence · How the quantum world leaks into the classical
Quantum decoherence is the loss of phase coherence between a system's superposed branches as it entangles with its environment, making interference va
Quantum MechanicsQuantum Dot · A semiconductor nanocrystal that confines electrons like a particle in a box — so its color is tuned by size, not chemistry
A quantum dot is a semiconductor nanocrystal (~2–10 nm) that confines electrons like a particle in a box, so shrinking it widens the band gap and shif
Condensed MatterQuantum Entanglement · Spooky Action
Two particles linked so deeply that measuring one instantly affects the other, no matter the distance. Nobel Prize 2022.
Quantum PhysicsQuantum Eraser · Mark which path the particle took and the interference dies — erase that mark and the fringes come back
The quantum eraser shows that interference fringes vanish the moment which-path information becomes available — and reappear when you erase that infor
Quantum PhysicsQuantum Hall Effect · A staircase of perfect plateaus — where resistance becomes a number built from nothing but Planck's constant and the electron charge
The quantum Hall effect quantizes the Hall conductance of a 2D electron gas as σ_xy = ν·e²/h, with plateaus precise to 1 part in 10⁹. It defines the S
Condensed MatterQuantum Harmonic Oscillator · E_n = ℏω(n + ½) — evenly spaced energies, zero-point ½ℏω at the bottom
The quantum harmonic oscillator: E_n = ℏω(n+½), evenly spaced ladder. Ground state has E₀ = ½ℏω zero-point energy. Foundation of phonons, photons, and
Quantum MechanicsQuantum Perturbation Theory
Quantum perturbation theory approximates the energies and states of H = H₀ + λV when V is small. First-order shift E⁽¹⁾ₙ = ⟨n|V|n⟩; second-order sums
Quantum MechanicsQuantum Teleportation · Sending a qubit using entanglement and two bits
Quantum teleportation transfers an unknown qubit's state to a distant qubit using a shared entangled pair, a Bell measurement, and two classical bits.
Quantum MechanicsQuantum Tunneling · particles through barriers
3D particle approaching an energy barrier taller than its energy. Classically impossible to pass. But the quantum wavefunction leaks through, and the
Quantum PhysicsQuantum Zeno Effect · A watched quantum pot never boils — frequent measurements freeze evolution
A watched quantum state cannot decay. Repeated measurements project the system back to its initial state at every step, and the survival probability a
Quantum MechanicsQuarks & the Strong Force · Color Confinement
Protons and neutrons are made of three quarks bound by gluons exchanging color charge. The strong force gets stronger with distance — quarks are alway
Particle PhysicsRabi Oscillation · Shine a resonant field on a two-level atom and its quantum state flops — coherently, predictably, on demand
A two-level atom driven by a resonant field oscillates between ground and excited states at the Rabi frequency. P_excited = sin²(Ωt/2) — the basis of
Atomic PhysicsRadioactive Decay · alpha
3D unstable nucleus emitting three types of radiation: alpha particle (2p+2n cluster), beta particle (electron), gamma ray (photon wave). Show half-li
Nuclear PhysicsRailgun · A projectile launched by raw current — F = ½ L' I², no propellant required
A railgun fires a projectile by running a huge pulsed current up one rail, across a sliding armature, and back down the other rail. The current's own
ElectromagnetismRainbow Formation · One drop, three bends: refract in, reflect once, refract out — and the colors sort themselves into a 42° arc
A rainbow forms when sunlight refracts into a spherical raindrop, reflects once off the back surface, and refracts again on exit — emerging concentrat
OpticsRattleback · A spinning lump that picks a side — smooth one way, rattling backwards the other
A rattleback (celt or wobblestone) is a semi-ellipsoid that spins smoothly one way but stalls, rattles, and reverses when spun the other way. The asym
Rotational DynamicsRayleigh Scattering · The 1/λ⁴ law that paints the daytime sky blue and the setting Sun red
Rayleigh scattering is scattering by particles much smaller than the wavelength of light, with cross-section proportional to 1/λ⁴. Blue scatters ~9.4×
OpticsRayleigh-Bénard Convection · Heat a fluid from below and it organizes itself into rolling, hexagonal cells
Rayleigh-Bénard convection is the self-organized roll and hexagonal cell pattern a fluid forms when heated from below. Above the critical Rayleigh num
Fluid DynamicsRayleigh-Taylor Instability · Density inversion under gravity — heavy on top of light forms mushroom-cap plumes
The Rayleigh-Taylor instability occurs when a denser fluid sits above a less dense one in a gravitational field — any small perturbation grows exponen
Fluid DynamicsReduced Mass
Reduced mass μ = m₁m₂/(m₁+m₂) collapses the two-body problem into an equivalent one-body problem: a single particle of mass μ moving in the relative c
Classical MechanicsReflection · angle of incidence = reflection
3D light ray hitting a mirror surface. Angle of incidence equals angle of reflection. Show multiple rays reflecting to form an image. Normal line perp
OpticsRefraction · Snell's law
3D light ray entering water from air, bending toward the normal. Show the angle of incidence and angle of refraction. Demonstrate total internal refle
OpticsRelativistic Doppler Effect · Approach blue-shifts, recession red-shifts — and a transverse shift exists from pure time dilation
Relativistic Doppler: f' = f·√((1-β)/(1+β)) for recession. Unlike classical Doppler, a transverse shift exists from time dilation alone. Confirmed by
RelativityRelativistic Energy and Momentum
Relativistic energy is the total energy of a moving body, E = γmc², combining rest energy E₀ = mc² and kinetic energy (γ−1)mc². The invariant relation
Special RelativityRelativistic Momentum · p = γmv — diverges at light speed, conserved in every frame, equals E/c for photons
Relativistic momentum p = γmv diverges as v → c, conserved in every inertial frame. Reduces to Newtonian p = mv at low speeds. Photons: p = E/c.
Special RelativityRelativistic Velocity Addition
Relativistic velocity addition combines velocities so nothing exceeds light speed: u = (u′ + v) / (1 + u′v/c²). Derived from the Lorentz transformatio
Special RelativityRenormalization Group · Coarse-grain, rescale, repeat — and watch the laws of a system flow toward a fixed point
The renormalization group is systematic coarse-graining: integrate out short-wavelength fluctuations, rescale, and watch coupling constants flow towar
Statistical MechanicsResonance · natural frequency
3D swing being pushed at its natural frequency — amplitude grows dramatically. Push at wrong frequency — nothing happens. Show Tacoma Narrows bridge o
Waves & OscillationsReynolds Number · The dimensionless ratio that decides whether flow is laminar or turbulent
The Reynolds number Re = ρvL/μ is a dimensionless ratio of inertial to viscous forces. Below a critical value the flow is laminar — orderly, layered,
Fluid DynamicsRight-Hand Rule · current
3D hand with thumb (current direction), fingers (magnetic field direction), and palm (force direction). Apply to wire in magnetic field, motor, and so
ElectromagnetismRoche Limit · The orbit where a planet's tides win — and a moon is torn into a ring
The Roche limit is the distance below which a moon held together by its own gravity gets torn apart, because the planet's tidal force pulling on the n
AstrophysicsRolling Without Slipping · The contact-point-at-rest constraint that ties translation to rotation by v = ωR
Rolling without slipping enforces the constraint v = ωR. Kinetic energy splits into translation ½mv² + rotation ½Iω². For a solid sphere, total KE = 7
Rotational DynamicsRosensweig Instability (Ferrofluid Spikes) · When a magnetic field beats gravity and surface tension, a flat ferrofluid erupts into a hexagonal forest of peaks
The Rosensweig instability is the spontaneous breakup of a flat ferrofluid surface into a hexagonal array of peaks once a vertical magnetic field push
Fluid DynamicsRutherford Scattering
Rutherford scattering is the elastic Coulomb deflection of alpha particles by atomic nuclei — dσ/dΩ = (Z₁Z₂e²/4E)²/sin⁴(θ/2). The 1911 gold-foil resul
Classical MechanicsSQUID Magnetometer · Two Josephson junctions in a superconducting loop turn magnetic flux into a measurable voltage — the most sensitive magnetic detector ever built
A SQUID magnetometer is a superconducting loop with two Josephson junctions whose critical current is modulated by magnetic flux. Each flux quantum Φ₀
Condensed MatterSackur-Tetrode Equation · The Absolute Entropy of an Ideal Gas from Quantum Phase-Space
The Sackur-Tetrode equation explained: how quantum phase-space cells of size h³ give the absolute entropy of a monatomic ideal gas, matching measured
ThermodynamicsSagnac Effect · Counter-rotating light beams in a spinning loop return out of phase — and that phase measures absolute rotation
The Sagnac effect: send two light beams in opposite directions around a rotating loop and they return out of phase. The phase shift ΔΦ = 8πAΩ/(λc) is
Special RelativityScanning Tunneling Microscope · An atomically sharp tip reads the quantum tunneling current to map individual atoms — one row of pixels at a time
A scanning tunneling microscope (STM) drags an atomically sharp tip ~1 nm above a conducting surface and reads the quantum tunneling current, which fa
Quantum MechanicsSchrödinger's Cat · superposition
3D box containing a cat, a radioactive atom, and a Geiger counter. Before opening: cat is in superposition (ghostly alive+dead overlay). Upon observat
Quantum PhysicsSchwarzschild Radius · The radius at which an object's escape velocity equals the speed of light — a black hole's event horizon
Compress any mass M inside the radius r_s = 2GM/c² and you have a black hole. The Schwarzschild radius scales linearly with mass: 2.95 km per solar ma
RelativitySecond Law of Thermodynamics · entropy always increases
3D ordered particles in a box spontaneously spreading to fill the space. Entropy meter increases. Show that heat flows from hot to cold naturally, nev
ThermodynamicsSecond-Harmonic Generation · Two photons in, one photon of double the frequency out — the nonlinear trick that turns invisible infrared into green light
Second-harmonic generation (SHG) fuses two photons of frequency ω into one photon of 2ω inside a χ⁽²⁾ crystal — the trick that turns invisible 1064 nm
OpticsSeebeck Effect · Turning a temperature difference into voltage
The Seebeck effect is the generation of a voltage across a conductor when its two ends are held at different temperatures. It powers thermocouples and
Condensed MatterSelection Rules · Why atoms can only make certain jumps
Selection rules are conditions on quantum numbers that decide which atomic transitions are allowed. Electric-dipole jumps need Δl = ±1; violating them
Atomic PhysicsSelf-Focusing and Filamentation · When a Laser Beam Collapses on Itself
Optical Kerr self-focusing and filamentation explained: how an intense laser beam collapses on itself above the critical power, forms a plasma filamen
OpticsSelf-Organized Criticality (Sandpile) · A pile that tunes itself to the edge of collapse — power laws from one simple toppling rule
Self-organized criticality is how a slowly driven, dissipative system tunes itself to the brink of avalanches of every size, with no external paramete
Statistical MechanicsSemiconductors · silicon
3D silicon crystal lattice. Pure silicon: few free electrons. Add phosphorus (n-type): extra electrons. Add boron (p-type): electron holes. Put them t
Modern PhysicsSeries vs Parallel Circuits · current paths
3D series circuit (one path, current same everywhere) vs parallel circuit (multiple paths, voltage same across each). Remove a bulb in series: all go
ElectromagnetismShapiro Delay · Light passing near a massive body takes measurably longer — relativity's fourth classic test
The Shapiro delay is the extra travel time a light or radar signal accumulates when it passes near a massive body — relativity's fourth classic test.
General RelativityShock Wave · A thin discontinuity in pressure, density, and velocity propagating supersonically through a gas
A shock wave is a thin discontinuity where pressure, density, and velocity jump abruptly. Rankine-Hugoniot conditions relate the upstream and downstre
Fluid DynamicsSimple Harmonic Motion · spring
3D mass on a spring bouncing up and down. Position, velocity, and acceleration graphs trace sinusoidal curves in real-time. Show the relationship betw
Waves & OscillationsSimple Machines · lever
3D showcase of six simple machines. Lever: small force × long arm = big force × short arm. Inclined plane: less force over longer distance. All trade
Classical MechanicsSimple Pendulum · Harmonic Motion
A mass swinging from a pivot. Period T = 2π√(L/g) depends only on length and gravity, not mass or amplitude. Galileo 1583.
Classical MechanicsSkin Depth · δ = √(2/μσω) — how deep an AC field penetrates a conductor before falling to 1/e
Skin depth δ = √(2/μσω) is the characteristic depth to which an AC field penetrates a conductor. Copper at 60 Hz: δ ≈ 8.5 mm; at 1 GHz: δ ≈ 2.1 µm. Wh
ElectromagnetismSkin Effect · High-frequency current crowds toward a conductor's surface — and why power lines use stranded cable
The skin effect drives alternating current toward the outer skin of a conductor and starves the core. The characteristic penetration depth δ shrinks a
ElectromagnetismSmith Chart · Impedance Matching on the Reflection-Coefficient Plane
The Smith chart explained: how Phillip Smith mapped impedance onto the reflection-coefficient disk, with the Γ=(z−1)/(z+1) transform, VSWR circles, an
ElectromagnetismSnell's Law · n₁ sin θ₁ = n₂ sin θ₂ — refraction at an interface
Snell's law: when a wave (light, sound, water wave) passes from a medium of refractive index n₁ to one of n₂, the angles to the surface normal satisfy
OpticsSoliton · The wave that refuses to spread — where nonlinearity and dispersion fight to a perfect draw
A soliton is a self-reinforcing solitary wave in which nonlinearity exactly cancels dispersion, so the pulse keeps its shape forever and survives coll
Nonlinear DynamicsSonic Boom (Mach Cone) · An object faster than sound piles its wavefronts into a cone whose shock sweeps past as a boom
A sonic boom is the thunder-like shock that reaches you after an object flies faster than sound (Mach > 1). The aircraft outruns its own pressure wave
Waves & OscillationsSound Waves · longitudinal
3D speaker emitting longitudinal sound waves. Particles compress and rarefy in the direction of travel. Show how frequency changes pitch and amplitude
Waves & OscillationsSpecial Relativity · Einstein
3D spaceship approaching light speed. Clocks on the ship slow down (time dilation). The ship appears compressed in the direction of travel (length con
Modern PhysicsSpecific Heat Capacity · energy to raise temperature
3D comparison: same heat applied to water (high specific heat, slow temperature rise) vs metal (low specific heat, fast temperature rise). Temperature
ThermodynamicsSpin-Statistics Theorem · Integer-spin = bosons (BE); half-integer-spin = fermions (FD) — required by relativistic QFT
The spin-statistics theorem (Wolfgang Pauli, 1940): in relativistic QFT, particles of integer spin (0, 1, 2, …) must obey Bose-Einstein statistics — w
Quantum Field TheorySpin-½ · Two states, ℏ/2 each — and a 720° rotation to get home
Electrons, protons, neutrons, and quarks carry intrinsic angular momentum ℏ/2 with only two basis states |↑⟩ and |↓⟩. A 360° rotation flips the sign o
Quantum MechanicsSpinodal Decomposition · Uphill Diffusion and Spontaneous Phase Separation
Spinodal decomposition explained: how uphill diffusion and negative free-energy curvature drive spontaneous, barrier-free phase separation via the Cah
ThermodynamicsSpontaneous Symmetry Breaking
Spontaneous symmetry breaking is when the laws of a system are symmetric but its lowest-energy ground state is not — the system 'picks' one vacuum fro
Particle PhysicsStandard Model · quarks
3D periodic table of particles. Quarks (6 flavors in 3 colors), leptons (electron, muon, tau + neutrinos), force carriers (photon, gluon, W, Z, Higgs)
Particle PhysicsStanding Waves · nodes
3D string fixed at both ends vibrating in standing wave patterns. Show fundamental, 2nd harmonic, 3rd harmonic. Nodes stay still, antinodes oscillate
Waves & OscillationsStark Effect · Switch on an electric field and an atom's energy levels slide apart — the electric twin of the Zeeman effect
The Stark effect is the shifting and splitting of atomic energy levels in an electric field — linear (∝ E) for hydrogen, quadratic (∝ E²) for everythi
Atomic PhysicsStatic Electricity · charge transfer
3D balloon rubbed on hair, electrons transfer. Balloon becomes negative, hair positive. Balloon sticks to wall. Hair stands up from repulsion. Lightni
ElectromagnetismStefan-Boltzmann Law · Why a hot object's radiated power scales as T to the fourth — and why doubling temperature multiplies output by 16
The Stefan-Boltzmann law states that the power radiated per unit area by a black body is σT⁴, where σ = 5.67 × 10⁻⁸ W/m²K⁴. Doubling the absolute temp
ThermodynamicsStern-Gerlach Experiment · 1922 silver atoms split into two beams in a non-uniform magnetic field — the discovery of spin
Otto Stern and Walther Gerlach fired a beam of silver atoms through a steeply non-uniform magnetic field in 1922 and saw two sharp spots, not a smear.
Quantum PhysicsStirling Engine · A sealed gas shuttled between hot and cold spaces drives a piston — an external-combustion heat engine
A Stirling engine is a closed-cycle, external-combustion heat engine that converts a temperature difference into work by shuttling a fixed mass of gas
ThermodynamicsStokes Drag · Why tiny things fall slowly through fluid
Stokes drag is the viscous resistance F = 6πμrv on a small sphere moving slowly through fluid. It dominates at low Reynolds number and sets terminal v
Fluid DynamicsStrange Attractors · Lorenz's butterfly: a deterministic system that's still impossible to predict
A fractal-dimensional set in phase space toward which a chaotic system evolves — never repeats, never settles.
Chaos TheoryString Theory · Vibrating Strings
String theory visualized — tiny vibrating strings replace point particles, with different vibrations creating different particles across 10 dimensions
Theoretical PhysicsStructural Color · Color made of geometry, not dye — light interfering with nanoscale structure
Structural color is color produced by microscopic structures that interfere with light rather than by pigments that absorb it. Periodic layers, ridges
OpticsSuperconductivity · zero resistance
3D material cooling below critical temperature. Resistance drops to exactly zero (graph). Magnetic field expelled (Meissner effect): magnet levitates
Modern PhysicsSuperfluidity · Below 2.17 K, liquid helium becomes a single quantum wave with zero viscosity
A phase of matter where viscosity drops to zero. Helium-4 below 2.17 K and helium-3 below 2.5 mK enter this state.
Quantum FluidsSurface Plasmon · Light riding a wave of metal electrons
A surface plasmon is a collective oscillation of metal conduction electrons coupled to light, bound to a metal-dielectric interface and concentrating
Condensed MatterSurface Tension · Why water beads, soap films minimize, and small insects walk on liquid
Surface tension is the force per unit length that pulls a liquid surface into the smallest area possible. It comes from the unbalanced cohesion of mol
Fluid DynamicsTaylor-Couette Flow · Rotating-Cylinder Instability and Taylor Vortices
Taylor-Couette flow explained: how a viscous fluid between rotating cylinders forms Taylor vortices, the critical Taylor number ~1708, Rayleigh's crit
Fluid DynamicsTelegrapher's Equations · How Voltage Waves Travel Down a Transmission Line
The Telegrapher's Equations explained: how voltage and current waves travel down a transmission line, derived from distributed R, L, G, C, with impeda
ElectromagnetismTennis Racket Theorem · Spin a rigid body about its middle axis and it will flip — the intermediate axis is unstable
The tennis racket theorem says rotation about a rigid body's intermediate principal axis is unstable: flip a racket and it makes a half-twist mid-air.
Rotational DynamicsTerminal Velocity · drag equals gravity
3D skydiver falling. Initially accelerates (gravity > drag). As speed increases, drag force grows. When drag = gravity: acceleration stops, terminal v
Classical MechanicsTerrell-Penrose Rotation · Why a Fast Sphere Looks Rotated, Not Flattened
Terrell-Penrose rotation explained: why a relativistic sphere photographs as rotated, not Lorentz-flattened, with the derivation, angles, and worked n
Special RelativityThe Anharmonic Oscillator
An anharmonic oscillator is one whose restoring force deviates from Hooke's law F = -kx, gaining higher-order terms like -βx³ so its potential is
Waves & OscillationsThe Casimir Effect · Empty space pushes — measurable force from virtual particle modes
An attractive force between two uncharged conducting plates in vacuum, predicted by Casimir 1948 and measured by Lamoreaux 1997.
Quantum VacuumThe Clausius-Clapeyron Relation · Clausius-Clapeyron Relation
The Clausius-Clapeyron relation gives the slope of a phase-coexistence line: dP/dT = L / (T·ΔV). Integrated for a vapor, ln P = −L/(R·T) + const — a s
ThermodynamicsThe Density Matrix
The density matrix ρ is the operator that fully describes a quantum state — pure OR mixed — with ρ = Σ p_i |ψ_i⟩⟨ψ_i|, Tr(ρ)=1, expectations ⟨A⟩=Tr(ρA
Quantum MechanicsThe Ergodic Hypothesis
The ergodic hypothesis says a system's long-time average equals its ensemble average — because one trajectory eventually visits every accessible m
Statistical MechanicsThe Feigenbaum Constant · The Universal 4.669 Ratio Behind Period-Doubling Chaos
The Feigenbaum constant δ ≈ 4.669 explained: how period-doubling bifurcations accumulate into chaos, why it is universal, its derivation, α ≈ 2.5029,
Nonlinear DynamicsThe Fermi Energy · Fermi Energy
The Fermi energy E_F is the energy of the highest occupied electron state at absolute zero — set by Pauli exclusion filling a free-electron gas up to
Condensed MatterThe Fluctuation-Dissipation Theorem · Fluctuation-Dissipation Theorem
The fluctuation-dissipation theorem says a system's linear response to a small force is fixed by its spontaneous equilibrium fluctuations — same rando
Statistical MechanicsThe Franck-Condon Principle · Vertical Transitions on Molecular Potential Curves
The Franck-Condon principle explained: why electronic transitions are vertical on potential energy curves, Franck-Condon factors, vibronic band shapes
Atomic PhysicsThe Fresnel Equations
The Fresnel equations give the amplitude reflection and transmission coefficients of light at a planar interface as functions of incidence angle and p
OpticsThe Hellmann-Feynman Theorem · Forces from a Fixed Wavefunction
The Hellmann-Feynman theorem explained: how forces on atomic nuclei come from a fixed wavefunction via dE/dλ = ⟨ψ|∂H/∂λ|ψ⟩, with derivation, examples,
Quantum MechanicsThe Higgs Field · An invisible "ocean" of mass that every particle swims through — or doesn't
A scalar quantum field with non-zero vacuum expectation value (~246 GeV) that gives mass to particles via the Mexican Hat potential.
Quantum Field TheoryThe Kapitza Pendulum · Inverted Stability from Fast Vertical Vibration
The Kapitza pendulum explained: how fast vertical vibration of a pivot stabilizes an inverted pendulum, the effective potential, the (aν)² > 2gl condi
Classical MechanicsThe Kinetic Theory of Gases
The kinetic theory of gases explains pressure and temperature from molecular motion. Pressure PV = (1/3)Nm⟨v²⟩; temperature sets average kinetic energ
ThermodynamicsThe Kondo Effect
The Kondo effect is the logarithmic rise in a metal's electrical resistance below a temperature T_K, caused by conduction electrons collectively s
Condensed MatterThe Kosterlitz-Thouless Transition · Vortex Unbinding in Two Dimensions
The Kosterlitz-Thouless (BKT) transition explained: vortex unbinding in 2D superfluids and superconductors, the universal jump T_KT = (π/2)ρ_s, and th
Statistical MechanicsThe Larmor Radiation Formula · Larmor Formula
The Larmor formula gives the total power radiated by a non-relativistic accelerating point charge: P = q²a²/(6πε₀c³). Power scales with acceleration s
ElectromagnetismThe Magnetic Vector Potential · Magnetic Vector Potential
The magnetic vector potential A is a vector field whose curl gives the magnetic field: B = ∇ × A. It carries gauge freedom (A → A + ∇f), simplifies so
ElectromagnetismThe Method of Images
The method of images replaces a grounded conductor with a fictitious image charge that reproduces the same boundary condition (V = 0 on the surface).
ElectromagnetismThe Multipole Expansion
The multipole expansion writes the distant potential of a charge distribution as a series in powers of 1/r — monopole (1/r) + dipole (1/r²) + quadrupo
ElectromagnetismThe Path Integral Formulation
The path integral formulation is Feynman's reformulation of quantum mechanics: a particle's amplitude to go from A to B is a sum over ALL poss
Quantum MechanicsThe Physical Pendulum
A physical pendulum is any rigid body that swings about a fixed pivot under gravity, with period T = 2π√(I/(mgd)) — where I is the moment of inertia a
Classical MechanicsThe Quality Factor (Q)
The quality factor Q of an oscillator is 2π times the energy stored divided by the energy lost per cycle — equivalently Q = ω₀/Δω, the ratio of resona
Waves & OscillationsThe RC Circuit · RC Circuit
An RC circuit is a resistor and capacitor in series whose charging voltage follows V(t) = V₀(1 − e^(−t/RC)). The time constant τ = RC sets the pace: 6
ElectromagnetismThe Relativity of Simultaneity
The relativity of simultaneity is the result in special relativity that two events judged simultaneous in one inertial frame occur at different times
Special RelativityThe Scattering Cross Section
The scattering cross section σ is the effective target area (in barns, 1 b = 10⁻²⁸ m²) that quantifies how likely two particles are to interact. Event
Particle PhysicsThe Schrödinger Equation
The Schrödinger equation iℏ ∂ψ/∂t = Ĥψ is the foundational law of quantum mechanics, governing how the wavefunction ψ evolves in time. It sets the kin
Quantum MechanicsThe Semi-Empirical Mass Formula · The Five-Term Bethe-Weizsäcker Binding Energy Equation
The semi-empirical mass formula (Bethe-Weizsäcker equation) explained: its five terms, coefficient values in MeV, a worked example, and why it predict
Nuclear PhysicsThe Stress-Energy Tensor
The stress-energy tensor T^μν is the 16-component object that packages energy density, momentum density, pressure, and shear into the source of gravit
General RelativityThe Strong CP Problem · Why the QCD Theta Angle Is Smaller Than 10⁻¹⁰
The Strong CP problem explained: why QCD's theta angle θ̄ is smaller than 10⁻¹⁰, the neutron EDM bound, the Peccei-Quinn axion solution, and open ques
Particle PhysicsThe Talbot Effect · How a Grating Copies Itself With No Lens
The Talbot effect explained: how a periodic grating reproduces its own image without a lens, the Talbot length formula z_T = 2d²/λ, carpets, and appli
OpticsThe Tsiolkovsky Rocket Equation · Tsiolkovsky Rocket Equation
The Tsiolkovsky rocket equation gives a rocket's velocity change as Δv = v_e · ln(m₀/m_f) — exhaust velocity times the natural log of the mass rat
Classical MechanicsThe Variational Method
The variational method estimates a quantum system's ground-state energy: for ANY trial wavefunction, ⟨H⟩ ≥ E₀. Minimize ⟨H⟩ over parameters to squ
Quantum MechanicsThermal Expansion · heating → expanding
3D metal bar with atoms vibrating. Heat applied: atoms vibrate more, pushing apart. Bar visibly lengthens. Show expansion joints in bridges and railwa
ThermodynamicsThin Film Interference · soap bubbles
3D thin film (soap bubble) with light reflecting from top and bottom surfaces. Path difference causes constructive interference for some colors, destr
OpticsThird Law of Thermodynamics · As T → 0, the entropy of a perfect crystal → 0
The third law of thermodynamics (Walther Nernst, 1906; Max Planck refinement 1911) states: as the temperature of a system approaches absolute zero (T
ThermodynamicsThomas Precession · The Relativistic Half-Angle Twist of a Spinning Frame
Thomas precession explained: the relativistic frame rotation that halves spin-orbit coupling, its formula ω_T = γ²/(γ+1)·(a×v)/c², and its role in muo
Special RelativityThree-Body Problem · Three masses, no general solution, pure chaos
The three-body problem asks how three masses move under mutual gravity. There is no general closed-form solution — the motion is chaotic and sensitive
Classical MechanicsTidal Force · Gravity's difference across a body stretches it
The tidal force is the difference in gravity across a body — the near side is pulled harder than the far side, stretching the body along the line to t
Classical MechanicsTime Dilation · Relativity
Time dilation explained — see how speed slows time with light clocks, the twin paradox, and why GPS needs relativity corrections.
RelativityTippe Top · The spinning top that turns itself upside-down and stands on its stem — driven by friction, not magic
A tippe top is a spinning top that flips itself over to balance on its stem. Sliding friction at the contact point produces a torque that turns the sp
Rotational DynamicsTokamak Confinement · Toroidal field + plasma current → twisted helical field lines confine 100-million-degree fuel
A tokamak (Russian acronym, 1950s) is a toroidal magnetic confinement device for fusion plasma. Confinement is achieved by combining: (1) toroidal fie
Plasma PhysicsTollmien-Schlichting Waves · The First Step to Boundary-Layer Turbulence
Tollmien-Schlichting waves explained: the viscous boundary-layer instability that triggers turbulence. Critical Reynolds number, Orr-Sommerfeld equati
Fluid DynamicsTopological Insulators · An exotic state of matter where electrons flow lossless along the surface only
Materials whose interior is insulating but whose surface conducts electricity in scattering-immune edge states protected by topology.
Condensed MatterTorque · rotational force
3D wrench turning a bolt. Longer wrench (more lever arm) = more torque with same force. Show the perpendicular force component matters. Door hinge as
Classical MechanicsTotal Internal Reflection · critical angle
3D light ray inside glass hitting the surface at increasing angles. Below critical angle: some refracts out. At critical angle: ray skims along surfac
OpticsTransformer · step up
3D transformer with primary and secondary coils around an iron core. More turns on secondary: voltage steps up. Fewer turns: voltage steps down. Power
ElectromagnetismTurbulence and the Energy Cascade
Turbulence is chaotic fluid motion in which energy injected at large scales cascades through ever-smaller eddies until viscosity dissipates it as heat
Fluid DynamicsTwin Paradox · The travelling twin really does return younger — special relativity, not a contradiction
The twin paradox — one twin flies off at near-light speed and returns younger than the stay-at-home. Not a paradox: only the traveller changes frame.
Special RelativityUnruh Effect · Accelerate hard enough and empty space starts to feel warm — the vacuum is observer-dependent
The Unruh effect: an accelerating observer sees the empty quantum vacuum as a warm thermal bath of particles at temperature T = ℏa/(2πck_B). A referen
General RelativityVan der Pol Oscillator · Self-Sustained Limit-Cycle Relaxation Oscillations
The Van der Pol oscillator explained: the ẍ − μ(1−x²)ẋ + x = 0 equation, its self-sustained limit cycle, relaxation oscillations, amplitude ≈ 2, and p
Nonlinear DynamicsVirial Theorem · For a bound system: ⟨T⟩ = −(1/2)⟨V⟩ when V ∝ r^n with n = −1
The virial theorem (Clausius 1870) relates the time-averaged kinetic energy ⟨T⟩ and potential energy ⟨V⟩ of a bound system. For a power-law potential
Classical MechanicsViscosity · How fluids resist shear — Newtonian vs non-Newtonian, honey vs water
Viscosity is the property of a fluid that resists shear deformation. Newton's law μ = τ/(du/dy) defines it for ordinary liquids and gases; non-Newtoni
Fluid DynamicsViscous Fingering (Saffman-Taylor) · Push a thin fluid into a thick one and the flat front shatters into branching fingers
Viscous fingering, the Saffman-Taylor instability, happens when a low-viscosity fluid pushes into a high-viscosity one in a thin gap: the flat front b
Fluid DynamicsVortex Ring · A self-propelling doughnut of rotating fluid — smoke rings, bubble rings, and jellyfish propulsion
A vortex ring is a doughnut-shaped region of rotating fluid that carries itself forward — the smoke ring, the dolphin's bubble ring, the jellyfish's p
Fluid DynamicsVorticity · The local spin hidden inside a flow
Vorticity is the curl of the velocity field, ω = ∇ × u — twice the local angular velocity of a fluid parcel. It measures spin inside a flow, drives vo
Fluid DynamicsWKB Approximation · Semi-classical wave functions ψ ≈ A·exp(±i∫p(x)dx/ℏ) — connects classical action to QM amplitudes
WKB approximation: ψ(x) ≈ A·exp(±i∫p(x)dx/ℏ) where p(x) = √(2m(E−V)). Valid where wavelength varies slowly. Connects classical action to quantum ampli
Quantum MechanicsWave Function Collapse · |ψ⟩ → |k⟩ at random with probability |c_k|² when observed (Born rule)
Wave function collapse: in standard quantum mechanics, when a system in superposition |ψ⟩ = Σ c_n |n⟩ is measured in basis {|n⟩}, it "collapses&q
Quantum MechanicsWave Interference · constructive
3D two wave sources creating circular waves that overlap. Where crests meet crests: constructive interference (bright). Where crests meet troughs: des
Waves & OscillationsWave Packet · Many waves summing into a localized pulse
A wave packet is a group of waves of slightly different wavelengths that add up to one localized pulse. It travels at the group velocity and spreads v
Waves & OscillationsWave Properties · wavelength
3D transverse wave with labeled wavelength, amplitude, and frequency. Increase frequency to see more cycles. Increase amplitude to see taller waves. S
Waves & OscillationsWave Speed · v = fλ
3D wave traveling along a rope. Tighter rope: faster wave. Heavier rope: slower wave. Show v = fλ with frequency and wavelength labeled. Change one an
Waves & OscillationsWave-Particle Duality · light as wave and particle
3D light behaving as a wave (interference pattern through double slit) and as a particle (photon hitting a detector as a single dot). Same entity, two
Quantum PhysicsWaveguides · Hollow metallic tubes that confine and guide microwaves — above the cutoff frequency only
A waveguide is a hollow metallic tube that confines electromagnetic waves above a cutoff frequency. Rectangular guides support TE and TM modes; below
Antennas & WaveguidesWeak Force · Beta decay, massive W and Z bosons, and the only force that breaks mirror symmetry
The weak force mediates beta decay via massive W and Z bosons (80, 91 GeV). Range ~10⁻¹⁸ m. Violates parity and CP. Unified with EM in electroweak the
Particle PhysicsWien's Displacement Law · The peak emission wavelength shifts inversely with temperature — λ_max × T = 2.898 × 10⁻³ m·K
Heat anything to incandescence and it begins to glow. The colour of that glow is not arbitrary — Wien's displacement law fixes the wavelength at which
ThermodynamicsWigner Rotation · Why Two Boosts Don't Commute
Wigner rotation explained: why composing two non-collinear Lorentz boosts yields a boost plus a rotation, the Thomas precession formula, the fine-stru
Special RelativityWork and Energy · W = Fd
3D person pushing a box along a surface. Force arrow times distance equals work done. Energy transfers into the box as kinetic energy. Show the calcul
Classical MechanicsYoung's Modulus · stress vs strain
3D material bar being stretched. Stress-strain graph builds: linear elastic region (springs back), yield point (permanent deformation begins), ultimat
Classical MechanicsZeeman Effect · ΔE = μ_B g_J m_J B — atomic spectral lines splitting in magnetic fields
The Zeeman effect is the splitting of atomic spectral lines into multiple components when the atom is placed in an external magnetic field. Each line
Atomic Physicsde Broglie Wavelength · matter waves
3D electron shown as both a particle and a wave. Faster electron: shorter wavelength. Slower: longer wavelength. Demonstrate electron diffraction prov
Quantum Physics