Physics

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).
• Paramagnetism — a form of magnetism in which a material is weakly attracted into an applied magnetic field because unpaired electron spins (permanent atomic dipole moments) partially align with it, giving a small positive magnetic susceptibility; the alignment is destroyed by thermal agitation, so the susceptibility follows the Curie law (χ ∝ 1/T) or, for interacting moments, the Curie-Weiss law (χ ∝ 1/(T − θ)); ferromagnets become paramagnetic above their Curie temperature (iron above ~1043 K); the field-dependence of the magnetization is described by the Brillouin function (plotted against the ratio of field strength to temperature); Pauli paramagnetism is a weak, temperature-independent form arising from conduction electrons near the Fermi level in metals; oxygen’s diradical (triplet) ground state makes O₂ paramagnetic; superparamagnetism occurs in nanoparticles whose moments flip via Néel relaxation, averaging to zero with no net remanence; measurements often require SQUIDs because the effect is so weak; contrasts with diamagnetism (repulsion) and ferromagnetism (spontaneous alignment).
• Diamagnetism — a universal, weak form of magnetism in which an applied field induces atomic dipole moments that oppose the field (Lenz’s law at the atomic scale), so the material is weakly repelled and has a small negative magnetic susceptibility and a magnetic permeability less than μ₀; present in all matter but usually masked by stronger para- or ferromagnetism; the classical Langevin theory describes it in atoms and molecules, and the Bohr-van Leeuwen theorem shows classical statistical mechanics alone gives zero net magnetization, so diamagnetism is intrinsically quantum; the Meissner effect (flux expulsion by a superconductor) is perfect diamagnetism (χ = −1); strong enough diamagnetic repulsion can statically levitate materials (e.g., a frog in a strong field), circumventing Earnshaw’s theorem because field minima can exist in free space; Pascal’s constants give empirical molecular values; free-electron gases show the Landau type.
• 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

  α

  ² +

  β

  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.
• Superfluidity — a state of matter in which a fluid flows with exactly zero viscosity (and effectively infinite thermal conductivity), first demonstrated in liquid helium-4 cooled below its lambda point (~2.17 K), where it becomes helium II; Pyotr Kapitsa (and independently Allen and Misener) discovered it in 1937 (Kapitsa Nobel 1978); a superfluid creeps up and over the walls of its container as a thin Rollin film (the Onnes effect) and can fountain or leak through tiny pores; excitations are described by quasiparticles called phonons and rotons (Lev Landau’s two-fluid model; Feynman and Cohen refined the roton); heating a superfluid generates second sound, a wave of temperature/entropy rather than pressure; the non-rotation effect is an analogue of the Meissner effect; helium-3, a fermion, becomes superfluid only at far lower (millikelvin) temperatures by forming Cooper-like pairs (Lee, Osheroff, Richardson, Nobel 1996); superfluidity is also invoked for the cores of neutron stars (pulsar glitches) and is closely related to Bose-Einstein condensation.
• 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.
• Mössbauer effect — the recoil-free resonant emission and absorption of gamma rays by atomic nuclei bound in a solid: because the entire crystal lattice takes up the recoil momentum, essentially no energy is lost to phonons and the emission line stays sharp enough for resonant reabsorption (it occurs only when the recoil energy E_R is below about half the transition linewidth); discovered by Rudolf Mössbauer (1958; Nobel Prize in Physics 1961); the extraordinary energy resolution enables Mössbauer spectroscopy, whose parameters include the isomer (chemical) shift and quadrupole splitting (and magnetic sextet splitting that probes hyperfine fields, e.g., in iron compounds); most famously used in the Pound-Rebka experiment (1959) to measure the tiny gravitational redshift predicted by general relativity; the fraction of recoil-free events is the Lamb-Mössbauer factor.

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 —

  ψ

  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;

  ψ

  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 ∝

  ψ

  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.