Science

Physics

Mechanics, thermodynamics, electromagnetism, relativity, and quantum theory.

A study reference, not a substitute for primary sources. Updated 2026-06-02.

Classical Mechanics

Kepler’s Laws of Planetary Motion

First law (ellipses) — each planet orbits the Sun in an ellipse with the Sun at one focus.
Second law (equal areas) — a line from the Sun to the planet sweeps equal areas in equal times; planets move fastest at perihelion (nearest point) and slowest at aphelion.
Third law (periods) — the square of a planet’s orbital period is proportional to the cube of the semi-major axis: T² ∝ a³; in SI units T²/a³ = 4π²/(GM).

Newton’s Laws of Motion

First law (inertia) — a body at rest or in uniform motion stays so unless acted on by a net external force; defines the concept of an inertial reference frame.
Second law — net force equals mass times acceleration: F = ma; in its more general form, F = dp/dt (rate of change of momentum). SI unit of force: newton (N = kg·m/s²).
Third law — for every action there is an equal and opposite reaction; forces always occur in pairs on different objects.
Superposition of forces — the net force on an object is the vector sum of all individual forces acting on it.

Kinematics

Displacement, velocity, acceleration — displacement Δx is change in position; velocity v = dx/dt; acceleration a = dv/dt.
Kinematic equations (constant acceleration) — v = v₀ + at; x = x₀ + v₀t + ½at²; v² = v₀² + 2aΔx.
Projectile motion — horizontal and vertical components are independent; vertical acceleration is g ≈ 9.81 m/s² downward; range is maximized at 45° launch angle (in a vacuum).
Uniform circular motion — speed is constant but direction changes; centripetal acceleration a = v²/r directed toward center; centripetal force F = mv²/r.
Angular kinematics — analogous to linear: angular velocity ω = dθ/dt; angular acceleration α = dω/dt; torque τ = Iα.

Energy and Work

Work — W = F·d·cos θ (force component along displacement); SI unit: joule (J = N·m).
Work-energy theorem — the net work done on an object equals its change in kinetic energy: W_net = ΔKE; unifies the concepts of force and energy.
Kinetic energy — KE = ½mv²; the work-energy theorem states net work equals change in KE.
Potential energy — gravitational PE = mgh near Earth’s surface; elastic (spring) PE = ½kx² (Hooke’s law, F = –kx).
Young’s modulus (E) — a measure of the stiffness (tensile elasticity) of a solid material; defined as the ratio of tensile stress (force per unit area, σ = F/A) to tensile strain (fractional elongation, ε = ΔL/L₀): E = σ/ε; SI unit: pascals (Pa, typically GPa for solids); steel ~200 GPa, bone ~20 GPa, rubber ~0.01–0.1 GPa; valid within the elastic (linear) region of the stress-strain curve below the proportionality limit; named after Thomas Young (1807); one of several elastic moduli (others include the bulk modulus K and shear modulus G).
Hooke’s law — restoring force of a spring is proportional to displacement and opposite in direction: F = –kx, where k is the spring constant (N/m); valid within the elastic limit; underlies simple harmonic motion and much of elasticity theory.
Conservation of mechanical energy — total KE + PE is constant in the absence of non-conservative forces (friction, drag).
Power — rate of doing work: P = W/t = Fv; SI unit: watt (W = J/s); in circuits: P = IV = I²R = V²/R.
Conservative vs non-conservative forces — gravity and spring forces are conservative (path-independent work); friction is non-conservative and converts mechanical energy to heat.

Momentum and Collisions

Linear momentum — p = mv; conserved when net external force is zero.
Impulse-momentum theorem — the impulse J = FΔt equals the change in momentum Δp; a large force acting over a short time produces the same momentum change as a small force acting over a longer time.
Impulse — J = FΔt = Δp; relates force over time to change in momentum.
Elastic collision — both momentum and kinetic energy are conserved.
Inelastic collision — momentum is conserved, kinetic energy is not; in a perfectly inelastic collision objects stick together.
Center of mass — weighted average position of a system; r_cm = Σmᵢrᵢ / Σmᵢ; net external force equals total mass times acceleration of the center of mass.

Rotation and Angular Momentum

Torque — τ = r × F; the rotational analog of force.
Moment of inertia — I = Σmᵢrᵢ²; depends on mass distribution; analogous to mass in linear motion.
Angular momentum — L = Iω; conserved when net torque is zero. Explains why a spinning skater speeds up when pulling in arms.
Rotational kinetic energy — KE_rot = ½Iω².
Parallel axis theorem — I = I_cm + Md², where d is the distance from the center of mass to the new axis.

Gravitation

Newton’s law of universal gravitation — F = Gm₁m₂/r²; attractive force between any two masses; G ≈ 6.674 × 10⁻¹¹ N·m²/kg².
Gravitational field — g = GM/r² at distance r from mass M; near Earth’s surface g ≈ 9.81 m/s².
Orbital mechanics — for a circular orbit: gravitational force provides centripetal force, giving v = √(GM/r); orbital period T = 2πr/v.
Escape velocity — v_esc = √(2GM/r); the minimum speed to escape a body’s gravitational pull from its surface; for Earth ≈ 11.2 km/s.
Gravitational potential energy — U = –Gm₁m₂/r; negative by convention, increasing toward zero at infinite separation.

Fluid Mechanics

Pascal’s principle — a pressure change applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the container walls; basis of hydraulic lifts (F₂/F₁ = A₂/A₁).
Archimedes’ principle — a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces; F_b = ρ_fluid · V_displaced · g.
Bernoulli’s principle — along a streamline in steady, incompressible, inviscid flow: P + ½ρv² + ρgh = constant; higher flow speed corresponds to lower pressure; explains lift on an airfoil and the Venturi effect.
Navier-Stokes equations — the governing partial differential equations for viscous fluid flow; extend Euler’s equations by adding viscous stress terms; notoriously difficult to solve analytically; whether smooth solutions always exist in 3D is one of the Millennium Prize Problems.
Poiseuille’s law — volumetric flow rate of a viscous fluid through a cylindrical pipe: Q = πr⁴ΔP / (8ηL); flow scales with the fourth power of radius, making it highly sensitive to vessel diameter.
Stokes’ law — drag force on a slowly moving sphere in a viscous fluid: F_drag = 6πηrv; used to define terminal velocity and underlies sedimentation analysis.
Reynolds number — dimensionless ratio Re = ρvL/η comparing inertial to viscous forces; low Re → laminar flow; high Re → turbulent flow; transition typically around Re ≈ 2,300 in a pipe.

Thermodynamics

Temperature and Heat

Temperature scales — Celsius (°C), Fahrenheit (°F), Kelvin (K); T(K) = T(°C) + 273.15. Kelvin is the SI base unit.
Thermal equilibrium / zeroth law — if A is in thermal equilibrium with B, and B with C, then A is in equilibrium with C; this defines temperature.
Heat vs temperature — heat (Q) is energy transferred due to a temperature difference; temperature is a measure of average kinetic energy per particle.
Specific heat capacity — Q = mcΔT; energy needed to raise 1 kg of a substance by 1 K. Water has a high specific heat (≈4,186 J/kg·K).
Latent heat — energy absorbed or released during a phase transition at constant temperature; latent heat of fusion (solid↔liquid) and vaporization (liquid↔gas).

Laws of Thermodynamics

First law — energy is conserved: ΔU = Q – W, where ΔU is change in internal energy, Q is heat added to the system, and W is work done by the system.
Second law — entropy of an isolated system never decreases; heat flows spontaneously from hot to cold; no heat engine is 100% efficient.
Third law — entropy of a perfect crystal approaches zero as temperature approaches absolute zero (0 K); absolute zero is unattainable.
Entropy (S) — a measure of disorder or number of microstates; ΔS = Q_rev / T for a reversible process.

Heat Engines and Cycles

Carnot engine — the idealized most-efficient heat engine operating between temperatures T_H and T_C; efficiency η = 1 – T_C/T_H; sets an upper bound for all real engines.
Carnot cycle — four reversible steps: isothermal expansion, adiabatic expansion, isothermal compression, adiabatic compression.
Refrigerators and heat pumps — the reverse of a heat engine; work is input to move heat against the temperature gradient. Coefficient of performance (COP) replaces efficiency.
Ideal gas law — PV = nRT = NkT; P pressure, V volume, n moles, R = 8.314 J/(mol·K), N number of molecules, k = Boltzmann constant ≈ 1.381 × 10⁻²³ J/K.
Internal energy of an ideal gas — U = (f/2)nRT, where f is degrees of freedom (3 for monatomic, 5 for diatomic at moderate temperatures).

Kinetic Theory

Maxwell-Boltzmann distribution — statistical distribution of molecular speeds in an ideal gas; most probable, mean, and rms speeds are distinct.
Root-mean-square speed — v_rms = √(3kT/m); relates temperature directly to molecular kinetic energy.
Equipartition theorem — each quadratic degree of freedom contributes ½kT to the average energy per molecule.
Avogadro’s number — N_A ≈ 6.022 × 10²³ mol⁻¹; number of particles in one mole.
Mean free path — average distance a molecule travels between collisions: λ = 1/(√2 · nπd²), where n is number density and d is molecular diameter; increases with lower pressure.
Dulong-Petit law — molar heat capacity of most solid elements at room temperature ≈ 3R ≈ 25 J/(mol·K); explained by the equipartition theorem (6 quadratic degrees of freedom per atom); breaks down at low temperatures (quantum effects).
Phase diagram — map of thermodynamic states (solid, liquid, gas) as a function of temperature and pressure; the triple point is where all three phases coexist; above the critical point (T_c, P_c), the liquid-gas distinction vanishes and the substance becomes a supercritical fluid.
Joule-Thomson effect — a gas forced through a porous plug at constant enthalpy expands and (for real gases below the inversion temperature) cools; basis of most gas liquefaction processes.

Waves and Optics

Mechanical Waves

Wave parameters — wavelength λ, frequency f, period T = 1/f, wave speed v = fλ.
Transverse vs longitudinal — transverse: displacement perpendicular to propagation (e.g., string waves, light); longitudinal: displacement parallel (e.g., sound).
Sound — a longitudinal pressure wave in a medium; speed in air ≈ 343 m/s at 20°C; cannot travel through a vacuum.
Intensity — power per unit area (W/m²); for a point source I ∝ 1/r² (inverse-square law). Decibel scale: β = 10 log₁₀(I/I₀), where I₀ = 10⁻¹² W/m².
Doppler effect — observed frequency shifts when source or observer moves: f_obs = f_s(v ± v_obs)/(v ∓ v_s); used in radar, sonar, medical ultrasound, and measuring galactic recession (redshift).
Superposition and interference — waves add algebraically; constructive (in phase) vs destructive (out of phase) interference.
Standing waves — superposition of two identical waves traveling in opposite directions; nodes (zero amplitude) and antinodes.
Resonance — a system oscillates with maximum amplitude when driven at its natural frequency.

Optics

Reflection — angle of incidence equals angle of reflection (measured from the normal).
Refraction — bending of light when it crosses a boundary between media; described by Snell’s law: n₁ sin θ₁ = n₂ sin θ₂.
Index of refraction — n = c/v, the ratio of the speed of light in vacuum to its speed in the medium; glass ≈ 1.5, water ≈ 1.33.
Total internal reflection — occurs when light in a denser medium hits the boundary at an angle greater than the critical angle; basis for fiber optics.
Thin lens equation — 1/f = 1/d_o + 1/d_i; magnification m = –d_i/d_o. Converging (convex) lenses have positive focal lengths; diverging (concave) have negative.
Diffraction — spreading of waves around obstacles or through apertures; significant when slit width ≈ λ.
Young’s double-slit experiment — Thomas Young (1801) demonstrated that light produces interference fringes, establishing its wave nature; fringe spacing Δy = λL/d.
Polarization — transverse waves (including light) can be polarized; Malus’s law: intensity I = I₀ cos²θ after a polarizer.
Dispersion — different wavelengths refract by different amounts; a prism spreads white light into a spectrum because n varies with λ.
Huygens’ principle — every point on a wavefront acts as a source of secondary spherical wavelets; the new wavefront is the envelope of these wavelets; explains diffraction and refraction geometrically.
Brewster’s angle — at θ_B = arctan(n₂/n₁), reflected light is completely polarized (s-polarized); transmitted light is partially polarized; used in polarizing filters and glare reduction.
Wien’s displacement law — the peak wavelength of blackbody radiation is inversely proportional to temperature: λ_max · T = b, where b ≈ 2.898 × 10⁻³ m·K; hotter objects radiate at shorter (bluer) wavelengths.
Stefan-Boltzmann law — total power radiated per unit area by a blackbody: j = σT⁴, where σ ≈ 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴; total emitted power scales steeply with temperature.
Rayleigh-Jeans law — classical prediction for blackbody spectral radiance: B(f) ∝ f²T; agrees with Planck’s law at low frequencies but diverges at high frequencies, producing the ultraviolet catastrophe.
Planck’s radiation law — B(f,T) = (2hf³/c²) · 1/(e^(hf/kT) – 1); correctly describes the full blackbody spectrum by quantizing energy; reduces to Rayleigh-Jeans at low f and Wien’s law at high f.
Thin-film interference — light reflecting from the top and bottom surfaces of a thin film interferes constructively or destructively depending on film thickness and wavelength; explains soap bubble colors; a half-wavelength phase shift occurs on reflection from a denser medium.
Bragg’s law — condition for constructive interference of X-rays scattered by crystal planes: 2d sin θ = nλ; basis of X-ray crystallography.
Rayleigh criterion — the angular resolution limit of a circular aperture: θ ≈ 1.22 λ/D (first minimum of the Airy disk); two point sources are just resolvable when the central maximum of one falls on the first minimum of the other.
Michelson interferometer — splits a beam into two perpendicular paths, then recombines them; path-length differences produce interference fringes; used to measure wavelengths and (with LIGO’s km-scale variant) detect gravitational waves.

Electromagnetism

Electrostatics

Coulomb’s law — F = kq₁q₂/r²; k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²; ε₀ ≈ 8.854 × 10⁻¹² C²/(N·m²) is the permittivity of free space.
Electric field — E = F/q; field lines run from positive to negative charges.
Electric potential — V = kq/r; work done per unit charge; SI unit: volt (V). The electron-volt (eV) is 1.602 × 10⁻¹⁹ J.
Gauss’s law (electric) — the total electric flux through a closed surface equals the enclosed charge divided by ε₀: Φ_E = Q_enc/ε₀ (integral form); differential form (Maxwell): ∇·E = ρ/ε₀.
Capacitance — C = Q/V; SI unit: farad (F). For a parallel-plate capacitor, C = ε₀A/d. Energy stored: U = ½CV².

Circuits

Ohm’s law — V = IR; resistance R in ohms (Ω); applies to ohmic (linear) conductors.
Resistivity — R = ρL/A; resistivity ρ is a material property.
Kirchhoff’s laws — (1) Junction rule: current in equals current out (charge conservation). (2) Loop rule: sum of voltage changes around any closed loop equals zero (energy conservation).
Series and parallel resistors — series: R_total = R₁ + R₂ + …; parallel: 1/R_total = 1/R₁ + 1/R₂ + …
RC circuit — charging/discharging through a resistor and capacitor; time constant τ = RC.
Alternating current (AC) — voltage and current sinusoidal; rms values used for power calculations: P_avg = V_rms·I_rms·cos φ.

Magnetism

Magnetic force on a moving charge — F = qv × B; the Lorentz force; perpendicular to both v and B.
Force on a current-carrying wire — F = IL × B.
Biot-Savart law — gives the magnetic field produced by a current element.
Ampère’s law — relates the line integral of B around a closed loop to the enclosed current: ∮B·dl = μ₀I_enc; μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space).
Faraday’s law of induction — an changing magnetic flux induces an EMF: ε = –dΦ_B/dt. The negative sign reflects Lenz’s law: the induced current opposes the change in flux.
Inductance — L in henries (H); V = L(dI/dt); energy stored in an inductor: U = ½LI².
Mutual inductance — when current in coil 1 changes, it induces an EMF in coil 2: ε₂ = –M(dI₁/dt); M is the mutual inductance in henries; basis of transformers.
Eddy currents — circulating currents induced in a bulk conductor by a changing magnetic flux (Faraday’s law); cause resistive heating (used in induction cooktops) and braking forces; minimized in transformer cores by lamination.
LC and RLC circuits — an LC circuit oscillates at natural frequency ω₀ = 1/√(LC); an RLC circuit is a damped oscillator; Q factor measures sharpness of resonance; in the underdamped regime ω = √(1/LC – R²/4L²).
AC reactance and impedance — inductive reactance X_L = ωL; capacitive reactance X_C = 1/(ωC); total impedance Z = √(R² + (X_L – X_C)²); phase angle φ = arctan((X_L – X_C)/R).

Maxwell’s Equations

Gauss’s law (magnetic) — ∇·B = 0; no magnetic monopoles.
Faraday’s law — ∇×E = –∂B/∂t.
Ampère-Maxwell law — ∇×B = μ₀J + μ₀ε₀(∂E/∂t); Maxwell added the displacement current term, enabling the prediction of electromagnetic waves traveling at c = 1/√(μ₀ε₀).
Electromagnetic waves — transverse waves; E and B are perpendicular to each other and to the direction of propagation; speed in vacuum c ≈ 2.998 × 10⁸ m/s (exactly 299,792,458 m/s).
Poynting vector — S = (1/μ₀)(E × B); represents the directional energy flux (power per unit area) of an EM field; its magnitude is the intensity; integrates to give total power transported by an EM wave.
Radiation pressure — EM waves carry momentum; radiation pressure on a perfect absorber is P = I/c; on a perfect reflector P = 2I/c; relevant in stellar interiors, solar sails, and laser trapping.

Electromagnetic Spectrum

Region	Wavelength range
Radio	> 1 mm
Microwave	~1 mm – 1 m
Infrared	~700 nm – 1 mm
Visible	~400 – 700 nm
Ultraviolet	~10 – 400 nm
X-ray	~0.01 – 10 nm
Gamma	< 0.01 nm

Notable Effects and Phenomena

Lorentz force — the total electromagnetic force on a charged particle: F = q(E + v × B); the electric part accelerates the charge; the magnetic part is always perpendicular to velocity and does no work.
Hall effect — when a current-carrying conductor is placed in a transverse magnetic field, a voltage (Hall voltage) develops perpendicular to both current and field; used to measure carrier density and sign in semiconductors and to sense magnetic fields.
Zeeman effect — spectral lines split in the presence of an external magnetic field due to lifting of degeneracy of magnetic sublevels; normal Zeeman effect (singlet lines split into 3) vs. anomalous (more complex splitting requiring electron spin).
Stark effect — splitting and shifting of spectral lines by an external electric field; analogous to the Zeeman effect; first-order Stark effect applies to hydrogen; higher-order in most atoms.
Cherenkov radiation — electromagnetic radiation emitted when a charged particle travels through a medium faster than the phase velocity of light in that medium (v > c/n); produces a characteristic blue glow in nuclear reactors; angle of the radiation cone: cos θ = c/(nv).
Bremsstrahlung — German for “braking radiation”; electromagnetic radiation emitted when a charged particle (usually an electron) is decelerated by the Coulomb field of an atomic nucleus; produces a continuous X-ray spectrum (as opposed to the discrete characteristic X-ray lines from electron transitions); the intensity increases with the square of the atomic number (Z²) of the target and is greater for higher-energy electrons; the endpoint frequency corresponds to the electron’s full kinetic energy being converted to a single photon (hf_max = eV); the dominant mechanism of X-ray production in X-ray tubes; also significant in astrophysics (e.g., hot gas in galaxy clusters emitting X-rays) and in stopping fast charged particles in matter.
Birefringence (double refraction) — the optical property of a material in which the refractive index depends on the polarization direction of the incident light; such anisotropic materials (e.g., calcite, quartz, ice) split an unpolarized beam into two polarized rays (ordinary ray obeying Snell’s law and extraordinary ray with a different velocity) that travel at different speeds and emerge with a phase difference; the two refractive indices are n_o and n_e and their difference Δn is the birefringence; exploited in polarizing microscopy (to identify crystalline minerals and biological structures like collagen and starch), wave plates/retarders, and liquid crystal displays (LCD); stress-induced birefringence (photoelasticity) is used to visualize stress distributions in transparent materials.
Memristor — a two-terminal passive circuit element (the “missing” fourth fundamental circuit element alongside the resistor, capacitor, and inductor) theorized by Leon Chua (1971) and first fabricated by HP Labs (Stanley Williams, 2008); its resistance depends on the history of current through it (it “remembers” past current); characterized by a nonlinear charge-flux relationship; implements a resistance that changes with the integral of voltage (flux linkage); proposed for neuromorphic computing (as a compact analog memory and synapse), non-volatile memory (ReRAM), and neural network hardware accelerators; behavior arises physically from ion migration in metal oxide thin films.
Debye length — the characteristic length scale over which the electrostatic potential of a charge in a plasma or electrolyte solution decays by a factor of 1/e due to screening by surrounding charges; λ_D = √(ε₀kT / ne²) for a plasma (n = carrier density) or an analogous expression for electrolyte solutions; at distances » λ_D, the medium is electrically neutral; at distances « λ_D, the Coulomb interaction is unscreened; the Debye-Hückel theory of electrolytes uses this concept; in semiconductor physics, the Debye length governs the extent of space-charge regions in p-n junctions; in plasmas, Debye shielding maintains quasi-neutrality.
Antiferromagnetism — a type of magnetic ordering in which neighboring magnetic moments (spins) align in an antiparallel pattern on alternating sublattices, resulting in zero net macroscopic magnetization at low temperatures; occurs below the Néel temperature (T_N) analogous to the Curie temperature for ferromagnets; above T_N the material becomes paramagnetic; examples include MnO, NiO, and chromium; antiferromagnets do not produce external magnetic fields and were originally difficult to detect (requiring neutron diffraction, which detects magnetic periodicity in the crystal structure); important in exchange bias (used in GMR read heads in hard drives) and spintronics; contrasts with ferromagnetism (parallel alignment) and ferrimagnetism (antiparallel but unequal moments, net magnetization retained, e.g., magnetite Fe₃O₄).
Isochronous — describing a system or process in which the period is independent of amplitude; a simple pendulum is isochronous for small amplitudes (T = 2π√(L/g), independent of θ for small θ); Galileo reputedly observed this using his pulse to time a chandelier; the isochronous property is the foundation of the pendulum clock (Huygens, 1656); a mass on a spring (simple harmonic oscillator) is perfectly isochronous; the property fails for large-amplitude pendulum swings (the period increases with amplitude) or in anharmonic oscillators.

Qubit — a quantum bit; the basic unit of quantum information; whereas a classical bit can be 0 or 1, a qubit can exist in a quantum superposition α

0⟩ + β

1⟩ (where

² +

² = 1) and collapses to 0 or 1 upon measurement; physical implementations include superconducting transmon qubits (IBM, Google), trapped ions (IonQ, Honeywell), photons, and spin qubits in silicon; two-qubit entanglement enables quantum parallelism; quantum gates (unitary transformations) manipulate qubit states; decoherence (interaction with the environment) destroys quantum superpositions and is the primary obstacle to practical quantum computing; quantum error correction (Shor code, surface code) is needed for fault-tolerant computation.

Hawking radiation — Stephen Hawking (1974) predicted that black holes emit thermal radiation due to quantum effects near the event horizon; pair production near the horizon allows one particle to escape while the other falls in; causes black holes to slowly evaporate.
Meissner effect and superconductivity — below a critical temperature T_c, certain materials become superconductors: electrical resistance drops to zero and magnetic flux is expelled from the bulk (Meissner effect); explained by BCS theory (Cooper pairs); type I vs type II superconductors differ in how they respond to strong fields.
BCS theory — Bardeen, Cooper, and Schrieffer (1957 Nobel 1972) explained conventional superconductivity: lattice-mediated phonon interactions bind electrons into Cooper pairs that condense into a macroscopic quantum state; energy gap opens at the Fermi level.
Josephson effect — a supercurrent (DC or AC) can tunnel across a thin insulating barrier between two superconductors with no applied voltage (DC) or at a frequency proportional to the voltage (AC: f = 2eV/h); basis of SQUIDs (ultra-sensitive magnetometers) and voltage standards.
Quantum Hall effect — in a 2D electron gas at low temperature and high magnetic field, the Hall conductance is quantized in integer multiples of e²/h (Klaus von Klitzing, 1980, Nobel 1985); the fractional quantum Hall effect (Tsui, Störmer, Laughlin, Nobel 1998) arises from strongly correlated electron behavior.
Bose-Einstein condensate — at temperatures near absolute zero, a collection of bosons (integer-spin particles) collapses into the ground quantum state, forming a new state of matter in which quantum effects are macroscopic; first realized in 1995 by Cornell, Wieman, and Ketterle (Nobel 2001) using laser-cooled rubidium atoms.
Casimir effect — quantum vacuum fluctuations produce an attractive force between two uncharged parallel conducting plates in vacuum; the force per unit area is F/A = –ℏcπ²/(240d⁴), where d is the plate separation; first measured precisely by Lamoreaux (1997).
Aharonov-Bohm effect — a charged particle is affected by an electromagnetic potential (vector potential A) even in regions where E and B are zero; demonstrates that potentials, not just fields, are physically real in quantum mechanics; interference pattern shifts with enclosed magnetic flux.
Band theory of solids — quantum mechanical treatment of electrons in a periodic crystal lattice produces allowed energy bands separated by forbidden band gaps; conductors have partially filled bands; insulators have large gaps; semiconductors have small gaps (≈1 eV) bridgeable by thermal energy or doping.
Laser / stimulated emission — Light Amplification by Stimulated Emission of Radiation; an incoming photon stimulates an excited atom to emit a second photon of identical frequency, phase, and direction (Einstein, 1917); a population inversion is required; the resulting beam is coherent, monochromatic, and highly collimated.
Casimir-Polder effect — verify: the retarded version of van der Waals forces between a polarizable atom and a surface, also rooted in quantum vacuum fluctuations; distinct from but related to the Casimir effect.

Special Relativity

Michelson-Morley experiment (1887) — Albert Michelson and Edward Morley used an interferometer to detect Earth’s motion through the luminiferous ether; the null result (no fringe shift) demolished the ether hypothesis and was a key empirical precursor to special relativity.
Lorentz transformations — the coordinate transformations between inertial frames in special relativity: t’ = γ(t – vx/c²); x’ = γ(x – vt); derived by Lorentz before Einstein but given their physical meaning by Einstein; replace the Galilean transformations.
Postulates (Einstein, 1905) — (1) the laws of physics are the same in all inertial frames; (2) the speed of light c is the same for all inertial observers regardless of source motion.
Time dilation — a moving clock runs slow: Δt = γΔt₀, where γ = 1/√(1 – v²/c²) is the Lorentz factor; Δt₀ is the proper time (measured in the rest frame).
Length contraction — lengths along the direction of motion are shortened: L = L₀/γ.
Simultaneity — events simultaneous in one frame are generally not simultaneous in another; simultaneity is frame-dependent.
Relativistic momentum — p = γmv; increases without bound as v → c.
Mass-energy equivalence — E = mc² (rest energy); full form: E² = (pc)² + (mc²)².
Relativistic kinetic energy — KE = (γ – 1)mc²; reduces to ½mv² for v ≪ c.
Velocity addition — u’ = (u – v)/(1 – uv/c²); no combination of sub-luminal speeds exceeds c.
Spacetime interval — s² = (cΔt)² – Δx²; invariant across all inertial frames.

General Relativity

Equivalence principle — a uniform gravitational field is locally indistinguishable from a uniformly accelerating frame; this is Einstein’s key insight linking gravity to geometry; the weak equivalence principle (inertial mass = gravitational mass) was tested to high precision by the Eötvös experiment.
Eötvös experiment — Roland von Eötvös (1880s–1909) used a torsion balance to show that gravitational and inertial mass are equal to one part in 10⁸; this equality (the weak equivalence principle) is a cornerstone of general relativity.
Perihelion precession of Mercury — the anomalous 43 arcseconds/century precession of Mercury’s orbit that Newtonian gravity could not explain; Einstein’s general relativity predicted it exactly, providing the first quantitative confirmation of GR.
Spacetime curvature — mass and energy curve spacetime; objects follow geodesics (straightest possible paths) in curved spacetime; what we call gravity is this curvature.
Einstein field equations — G_μν = (8πG/c⁴)T_μν; relate the geometry of spacetime (left side) to the distribution of mass-energy (right side).
Gravitational time dilation — clocks run slower deeper in a gravitational well; confirmed by atomic clocks at different altitudes; critical for GPS accuracy.
Gravitational lensing — light follows curved geodesics around massive objects; first confirmed during the 1919 solar eclipse (Eddington).
Gravitational waves — ripples in spacetime from accelerating masses; predicted by Einstein (1916); first directly detected by LIGO in 2015 (from merging black holes).
Black holes in GR — the Schwarzschild radius r_s = 2GM/c² defines the event horizon of a non-rotating black hole.
Big Bang cosmology — GR predicts an expanding universe; solutions to the Einstein equations gave the Friedmann equations underpinning modern cosmology.

Quantum Mechanics Foundations

Blackbody radiation / ultraviolet catastrophe — classical physics (Rayleigh-Jeans law) predicted infinite radiated power at high frequencies (the ultraviolet catastrophe). Planck (1900) resolved this by postulating energy is quantized in discrete packets: E = hf; this was the birth of quantum theory.
Planck’s constant — h ≈ 6.626 × 10⁻³⁴ J·s; reduced form ℏ = h/2π ≈ 1.055 × 10⁻³⁴ J·s.
de Broglie hypothesis — matter has wave-like properties; wavelength λ = h/p; confirmed by electron diffraction experiments.
Wave-particle duality — light and matter exhibit both wave and particle behavior depending on the experiment.

Wave function and probability —

² gives the probability density of finding a particle at a given location (Born interpretation).

Superposition — a quantum system can exist in a linear combination of states until measured; measurement collapses the wave function.
Quantized energy levels — particle in a box, harmonic oscillator, and hydrogen atom each have discrete allowed energies; ground state energy of the hydrogen atom is –13.6 eV.
Rydberg formula — gives the wavelengths of spectral lines of hydrogen: 1/λ = R_∞ (1/n₁² − 1/n₂²), where n₁ < n₂ are principal quantum numbers and R_∞ ≈ 1.097 × 10⁷ m⁻¹ is the Rydberg constant; series include Lyman (n₁=1, UV), Balmer (n₁=2, visible), and Paschen (n₁=3, IR); derived empirically by Johann Balmer (1885) and Johannes Rydberg (1888) before quantum mechanics; quantum mechanics explains it via the Bohr model energy levels E_n = −13.6 eV/n²; also used for highly excited Rydberg atoms (n ~ 100s) which have exaggerated properties including giant atomic radii and long lifetimes.

Wave function (ψ) — the complex-valued function that fully describes the quantum state of a particle or system;

² gives the probability density of finding the particle at a given position (Born rule); for a particle in a box or hydrogen atom, the wave function is a standing wave solution to the Schrödinger equation; the wave function must be normalizable (total probability = 1), continuous, and single-valued; superposition of wave functions corresponds to quantum superposition of states; in many-particle systems, fermionic wave functions must be antisymmetric under particle exchange (Pauli exclusion principle).

Spin — an intrinsic angular momentum of particles; electrons have spin-½; measured values are ±ℏ/2.
Pauli exclusion principle — no two identical fermions (half-integer spin particles) can occupy the same quantum state simultaneously; governs electron configurations and the structure of matter.
Bohr model — electrons occupy discrete circular orbits; angular momentum quantized as L = nℏ; predicts hydrogen spectrum correctly but is superseded by quantum mechanics.
Quantum tunneling — a particle can pass through a potential barrier it classically cannot surmount; basis of scanning tunneling microscopy and nuclear fusion in stars.
Millikan oil-drop experiment — Robert Millikan (1909–1913) balanced gravitational and electric forces on charged oil droplets to measure the elementary charge e ≈ 1.6 × 10⁻¹⁹ C; Nobel Prize 1923; also confirmed charge quantization.
Rutherford/Geiger-Marsden gold-foil experiment — Hans Geiger and Ernest Marsden, under Rutherford’s direction (1909–1911), fired alpha particles at gold foil; most passed through but some scattered at large angles; Rutherford concluded that most atomic mass is concentrated in a tiny, dense, positively charged nucleus.
Davisson-Germer experiment — Clinton Davisson and Lester Germer (1927) observed diffraction of electrons from a nickel crystal, confirming de Broglie’s wave hypothesis and establishing the wave nature of matter; Nobel Prize to Davisson 1937.
Stern-Gerlach experiment — Otto Stern and Walther Gerlach (1922) passed silver atoms through an inhomogeneous magnetic field and observed discrete deflections into two spots rather than a continuous smear; demonstrated the quantization of angular momentum and (retrospectively) electron spin.
Franck-Hertz experiment — James Franck and Gustav Hertz (1914) showed that electrons lose energy to mercury atoms only in discrete quantized amounts (4.9 eV), directly confirming Bohr’s quantized energy levels; Nobel Prize 1925.
Compton scattering — Arthur Compton (1923) observed that X-rays scattered by electrons shift to longer wavelengths; the shift Δλ = (h/m_e c)(1 – cos θ) confirmed that photons carry momentum p = h/λ; Nobel Prize 1927.
Photoelectric effect — Einstein (1905) explained that light ejects electrons from a metal only if photon energy E = hf exceeds the work function φ; maximum KE = hf – φ; independent of intensity; confirmed light’s particle nature; Nobel Prize 1921.
Double-slit experiment (electrons) — when electrons (or any quantum particle) pass through two slits without which-path detection, they produce an interference pattern; with detection, the pattern vanishes; the paradigmatic demonstration of wave-particle duality and the role of measurement in quantum mechanics.
Cavendish experiment — Henry Cavendish (1798) used a torsion balance to measure the gravitational attraction between lead spheres, yielding the first precise value of G and hence the mass of the Earth; the apparatus was designed by John Michell.
EPR paradox and Bell’s theorem — Einstein, Podolsky, and Rosen (1935) argued that quantum mechanics is incomplete because entangled particles seem to allow instant action at a distance; John Bell (1964) derived inequalities that must hold if local hidden variables are real; violations of Bell inequalities (Aspect experiment, 1982) confirm that quantum mechanics is genuinely nonlocal.
Aspect experiment — Alain Aspect (1982) measured correlations in polarizations of entangled photons and found clear violations of Bell’s inequalities, ruling out local hidden-variable theories; Nobel Prize (Aspect, Clauser, Zeilinger) 2022.
Wu experiment / parity violation — Chien-Shiung Wu (1956) studied beta decay of ⁶⁰Co in a magnetic field and found the emitted electrons preferred one direction, showing that the weak force violates parity symmetry (P-symmetry); confirmed the Lee-Yang prediction; Nobel Prize to Lee and Yang 1957 (Wu controversially not included).
Schrödinger equation — iℏ(∂ψ/∂t) = Ĥψ; governs time evolution of the quantum wave function; the time-independent form Ĥψ = Eψ yields quantized energy eigenvalues; Erwin Schrödinger published it in 1926.
Dirac equation — Paul Dirac (1928) wrote a relativistic wave equation for spin-½ particles: (iγ^μ∂_μ – mc)ψ = 0; it predicted antimatter (the positron, discovered by Anderson in 1932) and naturally incorporates electron spin.
Heisenberg uncertainty principle — ΔxΔp ≥ ℏ/2; ΔEΔt ≥ ℏ/2; not a measurement disturbance but a fundamental property of quantum states; implies zero-point energy (particles cannot be perfectly at rest).
Noether’s theorem — Emmy Noether (1915/1918) proved that every continuous symmetry of a physical system’s action corresponds to a conserved quantity; time symmetry → energy conservation; translational symmetry → momentum conservation; rotational symmetry → angular momentum conservation; fundamental to all of modern physics.
Copenhagen interpretation — Bohr and Heisenberg’s orthodox view that the wave function is a complete description of a quantum system; observables have no definite values until measured; the wave function collapses on measurement; contrasted with many-worlds and pilot-wave interpretations.
Schrödinger’s cat — thought experiment (1935) illustrating the paradox of applying superposition to macroscopic objects; a cat in a sealed box is in a superposition of alive and dead until observed; highlights the measurement problem in quantum mechanics.

SI Named Units in Physics

Newton (N) — SI unit of force = kg·m/s²; named after Isaac Newton.
Joule (J) — SI unit of energy = N·m = kg·m²/s²; named after James Prescott Joule, who quantified the mechanical equivalent of heat.
Watt (W) — SI unit of power = J/s; named after James Watt for his work on the steam engine.
Pascal (Pa) — SI unit of pressure = N/m²; named after Blaise Pascal for his work on fluid pressure.
Hertz (Hz) — SI unit of frequency = 1 cycle/s; named after Heinrich Hertz for confirming electromagnetic waves.
Tesla (T) — SI unit of magnetic flux density = kg/(A·s²); named after Nikola Tesla.
Weber (Wb) — SI unit of magnetic flux = V·s = kg·m²/(A·s²); named after Wilhelm Eduard Weber.
Henry (H) — SI unit of inductance = kg·m²/(A²·s²); named after Joseph Henry, who independently discovered electromagnetic induction.
Farad (F) — SI unit of capacitance = C/V; named after Michael Faraday.
Ohm (Ω) — SI unit of electrical resistance = V/A; named after Georg Simon Ohm.
Coulomb (C) — SI unit of electric charge = A·s; named after Charles-Augustin de Coulomb.
Volt (V) — SI unit of electric potential = J/C = W/A; named after Alessandro Volta, inventor of the voltaic pile.
Kelvin (K) — SI base unit of thermodynamic temperature; named after William Thomson (Lord Kelvin); zero Kelvin = absolute zero.
Becquerel (Bq) — SI unit of radioactivity = 1 nuclear decay per second; named after Henri Becquerel, discoverer of radioactivity.
Gray (Gy) and Sievert (Sv) — SI units of absorbed radiation dose (Gy = J/kg, physical energy deposited) and effective dose (Sv = J/kg weighted by biological effect); named after Louis Harold Gray and Rolf Sievert, respectively.

Fundamental Constants

Constant	Symbol	Value
Speed of light	c	299,792,458 m/s (exact)
Planck’s constant	h	6.626 × 10⁻³⁴ J·s
Reduced Planck	ℏ	1.055 × 10⁻³⁴ J·s
Gravitational constant	G	6.674 × 10⁻¹¹ N·m²/kg²
Boltzmann constant	k	1.381 × 10⁻²³ J/K
Avogadro’s number	N_A	6.022 × 10²³ mol⁻¹
Elementary charge	e	1.602 × 10⁻¹⁹ C
Electron mass	m_e	9.109 × 10⁻³¹ kg
Proton mass	m_p	1.673 × 10⁻²⁷ kg
Permittivity of free space	ε₀	8.854 × 10⁻¹² C²/(N·m²)
Permeability of free space	μ₀	4π × 10⁻⁷ T·m/A
Gas constant	R	8.314 J/(mol·K)

Key Figures

Isaac Newton — Principia Mathematica (1687); laws of motion, universal gravitation, invented calculus (independently of Leibniz); reflecting telescope.
Galileo Galilei — free-fall experiments (uniform acceleration); projectile motion; telescopic observations; championed the heliocentric model.
James Clerk Maxwell — unified electricity, magnetism, and optics into electromagnetic theory; derived that light is an EM wave; Maxwell’s equations (1865).
Michael Faraday — discovered electromagnetic induction (1831); invented the electric motor concept; introduced the concept of field lines.
Ludwig Boltzmann — statistical mechanics; connected thermodynamic entropy to microstates: S = k ln W; Boltzmann’s constant named for him.
Albert Einstein — special relativity (1905); photoelectric effect explaining photons (1905, Nobel 1921); general relativity (1915); Brownian motion; stimulated emission.
Max Planck — quantized energy hypothesis to explain blackbody radiation (1900); initiated quantum theory; Nobel Prize 1918.
Niels Bohr — planetary model of the atom with quantized orbits (1913); explained hydrogen spectrum; articulated the complementarity principle and the correspondence principle; major architect of the Copenhagen interpretation; his Institute for Theoretical Physics in Copenhagen became the world center of quantum mechanics development.
Werner Heisenberg — uncertainty principle (1927); matrix mechanics formulation of quantum mechanics; Nobel Prize 1932.
Erwin Schrödinger — wave mechanics formulation of quantum mechanics (1926); Schrödinger equation; Schrödinger’s cat thought experiment.
Paul Dirac — combined quantum mechanics with special relativity; Dirac equation predicted antimatter; Nobel Prize 1933.
Marie Curie (Maria Skłodowska-Curie) — born Maria Skłodowska in Warsaw, Poland (1867); pioneering research on radioactivity; discovered polonium (named after her homeland) and radium; only person to win Nobel Prizes in two sciences (Physics 1903, Chemistry 1911); first woman to win a Nobel Prize; her birth name Skłodowska is referenced in quizbowl questions emphasizing her Polish origin; worked at the École Normale Supérieure; died of aplastic anemia from long-term radiation exposure.
Ernest Rutherford — gold foil experiment revealed the nuclear model of the atom (1911); first artificial nuclear transmutation; Nobel Prize in Chemistry 1908.
William Thomson (Lord Kelvin) — absolute temperature scale (Kelvin); contributions to thermodynamics and electromagnetism.
Heinrich Hertz — first experimental production and detection of radio waves (1887), confirming Maxwell’s predictions; also demonstrated the photoelectric effect.
Hendrik Lorentz — Lorentz transformations relating space and time in different inertial frames; paved the way for special relativity; Nobel Prize 1902.
Michael Faraday / André-Marie Ampère — Ampère established the relationship between current and magnetic fields (Ampère’s law); Faraday discovered induction.
Richard Feynman — path integral formulation of quantum mechanics; quantum electrodynamics (QED) with Schwinger and Tomonaga; Nobel Prize 1965; Feynman diagrams.
Enrico Fermi — first controlled nuclear chain reaction (Chicago Pile-1, 1942); theory of beta decay; invented the concept of statistical mechanics for particles obeying the exclusion principle (Fermi-Dirac statistics); Nobel Prize 1938; transuranium element fermium named for him.
Emmy Noether — proved Noether’s theorem (1915/1918) connecting symmetries to conservation laws; foundational to theoretical physics and modern algebra; worked at Göttingen despite official exclusion of women.
Pierre and Marie Curie — Marie Curie (with Pierre and Becquerel) won the 1903 Physics Nobel for radioactivity research; Pierre Curie also discovered piezoelectricity (with his brother Jacques) and the Curie temperature (above which ferromagnetic materials lose their magnetism).
J. J. Thomson — discovered the electron (1897) using cathode ray experiments; measured the charge-to-mass ratio e/m; Nobel Prize 1906; proposed the “plum pudding” model of the atom (later superseded by Rutherford’s nuclear model).
Lise Meitner — Austrian-Swedish physicist who co-discovered nuclear fission (with Otto Hahn and Fritz Strassmann, 1938) and provided the theoretical explanation using liquid-drop model; controversially excluded from the Nobel Prize awarded to Hahn alone in 1944; element meitnerium named for her.
Murray Gell-Mann — proposed the quark model (1964, independently of George Zweig); introduced the concept of strangeness; Nobel Prize 1969; the Standard Model’s eight gluons and the “Eightfold Way” classification of hadrons.

Max Born — provided the probability interpretation of the wave function (Born rule: probability ∝

²); Nobel Prize 1954; also contributed the Born-Oppenheimer approximation in molecular physics.

Wolfgang Pauli — exclusion principle (1925); predicted the existence of the neutrino (1930) to conserve energy in beta decay; Nobel Prize 1945; Pauli matrices used in spin-½ quantum mechanics.