Newton's second law F ⃗ = m a ⃗ \vec F = m\vec a F = m a F ⃗ = d p ⃗ / d t \vec F = d\vec p/dt F = d p / d t Work-energy theorem Net work equals the change in kinetic energy: W n e t = Δ K E = 1 2 m v f 2 − 1 2 m v i 2 W_{net} = \Delta KE = \tfrac{1}{2}mv_f^2 - \tfrac{1}{2}mv_i^2 W n e t = Δ K E = 2 1 m v f 2 − 2 1 m v i 2 Kinetic energy K E = 1 2 m v 2 KE = \tfrac{1}{2}mv^2 K E = 2 1 m v 2 Gravitational potential energy (near Earth) U = m g h U = mgh U = m g h Momentum p ⃗ = m v ⃗ \vec p = m\vec v p = m v Impulse-momentum theorem J ⃗ = ∫ F ⃗ d t = Δ p ⃗ \vec J = \int \vec F\,dt = \Delta \vec p J = ∫ F d t = Δ p
Elastic collision (equal masses, one at rest) The velocities are exchanged: the incoming mass stops and the struck mass moves off at the original speed.
Perfectly inelastic collision Objects stick together; momentum is conserved (they move at the center-of-mass velocity) but kinetic energy is not. Centripetal acceleration a c = v 2 r a_c = \dfrac{v^2}{r} a c = r v 2 Centripetal force F c = m v 2 r = m ω 2 r F_c = \dfrac{mv^2}{r} = m\omega^2 r F c = r m v 2 = m ω 2 r Acceleration on a frictionless incline (angle θ) a = g sin θ a = g\sin\theta a = g sin θ Hooke's law F = − k x F = -kx F = − k x Angular frequency of a mass-spring system ω = k m \omega = \sqrt{\dfrac{k}{m}} ω = m k Period of a simple pendulum T = 2 π L g T = 2\pi\sqrt{\dfrac{L}{g}} T = 2 π g L Simple harmonic motion Motion under a restoring force proportional to displacement: x ( t ) = A cos ( ω t + ϕ ) x(t) = A\cos(\omega t + \phi) x ( t ) = A cos ( ω t + ϕ ) Torque τ ⃗ = r ⃗ × F ⃗ \vec\tau = \vec r\times\vec F τ = r × F τ = I α \tau = I\alpha τ = I α Moment of inertia I = ∑ m i r i 2 I = \sum m_i r_i^2 I = ∑ m i r i 2 ∫ r 2 d m \int r^2\,dm ∫ r 2 d m Parallel-axis theorem I = I c m + M d 2 I = I_{cm} + Md^2 I = I c m + M d 2 d d d Moment of inertia of a thin rod about its end I = 1 3 M L 2 I = \tfrac{1}{3}ML^2 I = 3 1 M L 2 1 12 M L 2 \tfrac{1}{12}ML^2 12 1 M L 2 Moment of inertia of a solid sphere (center) I = 2 5 M R 2 I = \tfrac{2}{5}MR^2 I = 5 2 M R 2 7 5 M R 2 \tfrac{7}{5}MR^2 5 7 M R 2 Angular momentum L ⃗ = I ω ⃗ \vec L = I\vec\omega L = I ω r ⃗ × p ⃗ \vec r\times\vec p r × p Rotational kinetic energy K E r o t = 1 2 I ω 2 KE_{rot} = \tfrac{1}{2}I\omega^2 K E r o t = 2 1 I ω 2 Why a spinning skater speeds up pulling in her arms Conservation of angular momentum I ω = I\omega = I ω = I I I ω \omega ω Newton's law of universal gravitation F = G m 1 m 2 r 2 F = \dfrac{Gm_1 m_2}{r^2} F = r 2 G m 1 m 2 Escape velocity v e s c = 2 G M R v_{esc} = \sqrt{\dfrac{2GM}{R}} v esc = R 2 GM 2 × \sqrt{2}\times 2 × Kepler's third law T 2 ∝ a 3 T^2 \propto a^3 T 2 ∝ a 3
Kepler's second law A planet sweeps equal areas in equal times — a consequence of angular-momentum conservation; it moves fastest at perihelion. Lagrangian L = T − V L = T - V L = T − V Euler-Lagrange equation d d t ∂ L ∂ q ˙ − ∂ L ∂ q = 0 \dfrac{d}{dt}\dfrac{\partial L}{\partial \dot q} - \dfrac{\partial L}{\partial q} = 0 d t d ∂ q ˙ ∂ L − ∂ q ∂ L = 0 Hamiltonian H = T + V H = T + V H = T + V q ˙ = ∂ H / ∂ p , p ˙ = − ∂ H / ∂ q \dot q = \partial H/\partial p,\ \dot p = -\partial H/\partial q q ˙ = ∂ H / ∂ p , p ˙ = − ∂ H / ∂ q Generalized (canonical) momentum p i = ∂ L ∂ q ˙ i p_i = \dfrac{\partial L}{\partial \dot q_i} p i = ∂ q ˙ i ∂ L L L L q i q_i q i Coriolis acceleration a ⃗ C o r = − 2 ω ⃗ × v ⃗ \vec a_{Cor} = -2\,\vec\omega\times\vec v a C or = − 2 ω × v Centrifugal acceleration (rotating frame) − ω ⃗ × ( ω ⃗ × r ⃗ ) -\vec\omega\times(\vec\omega\times\vec r) − ω × ( ω × r ) ω 2 r \omega^2 r ω 2 r Tsiolkovsky rocket equation Δ v = v e ln m i m f \Delta v = v_e \ln\!\dfrac{m_i}{m_f} Δ v = v e ln m f m i Reduced mass (two-body problem) μ = m 1 m 2 m 1 + m 2 \mu = \dfrac{m_1 m_2}{m_1 + m_2} μ = m 1 + m 2 m 1 m 2 Period of a physical pendulum T = 2 π I m g d T = 2\pi\sqrt{\dfrac{I}{mgd}} T = 2 π m g d I d d d Power P = d W d t = F ⃗ ⋅ v ⃗ P = \dfrac{dW}{dt} = \vec F\cdot\vec v P = d t d W = F ⋅ v Conservative force A force whose work is path-independent and equals minus a potential-energy change; F ⃗ = − ∇ U \vec F = -\nabla U F = − ∇ U Friction force (kinetic) f k = μ k N f_k = \mu_k N f k = μ k N f s ≤ μ s N f_s \le \mu_s N f s ≤ μ s N
Terminal velocity The constant speed where drag balances gravity, so net force and acceleration are zero. Coulomb's law F = 1 4 π ε 0 q 1 q 2 r 2 F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2} F = 4 π ε 0 1 r 2 q 1 q 2 Electric field of a point charge E = 1 4 π ε 0 q r 2 E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2} E = 4 π ε 0 1 r 2 q Gauss's law ∮ E ⃗ ⋅ d A ⃗ = Q e n c ε 0 \oint \vec E\cdot d\vec A = \dfrac{Q_{enc}}{\varepsilon_0} ∮ E ⋅ d A = ε 0 Q e n c ε 0 \varepsilon_0 ε 0
Electric field inside a conductor (electrostatic equilibrium) Zero; any excess charge resides on the conductor's surface. Electric potential energy / potential U = 1 4 π ε 0 q 1 q 2 r U = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r} U = 4 π ε 0 1 r q 1 q 2 V = 1 4 π ε 0 q r V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r} V = 4 π ε 0 1 r q Relation between E and V E ⃗ = − ∇ V \vec E = -\nabla V E = − ∇ V Capacitance of a parallel-plate capacitor C = κ ε 0 A d C = \dfrac{\kappa\varepsilon_0 A}{d} C = d κ ε 0 A κ \kappa κ Energy stored in a capacitor U = 1 2 C V 2 = Q 2 2 C = 1 2 Q V U = \tfrac{1}{2}CV^2 = \dfrac{Q^2}{2C} = \tfrac{1}{2}QV U = 2 1 C V 2 = 2 C Q 2 = 2 1 Q V Ohm's law V = I R V = IR V = I R Power dissipated in a resistor P = I V = I 2 R = V 2 R P = IV = I^2 R = \dfrac{V^2}{R} P = I V = I 2 R = R V 2 Resistors in series vs parallel Series: R = R 1 + R 2 + … R = R_1 + R_2 + \dots R = R 1 + R 2 + … 1 R = 1 R 1 + 1 R 2 + … \dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dots R 1 = R 1 1 + R 2 1 + … Capacitors in series vs parallel Parallel: C = C 1 + C 2 + … C = C_1 + C_2 + \dots C = C 1 + C 2 + … 1 C = 1 C 1 + 1 C 2 + … \dfrac{1}{C} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dots C 1 = C 1 1 + C 2 1 + … RC time constant τ = R C \tau = RC τ = R C e − t / τ e^{-t/\tau} e − t / τ RL time constant τ = L R \tau = \dfrac{L}{R} τ = R L
Kirchhoff's rules Junction rule: currents into a node sum to zero (charge conservation). Loop rule: voltages around a closed loop sum to zero (energy conservation). Lorentz force F ⃗ = q E ⃗ + q v ⃗ × B ⃗ \vec F = q\vec E + q\vec v\times\vec B F = q E + q v × B Magnetic field of a long solenoid B = μ 0 n I B = \mu_0 n I B = μ 0 n I n n n Ampere's law ∮ B ⃗ ⋅ d l ⃗ = μ 0 I e n c \oint \vec B\cdot d\vec l = \mu_0 I_{enc} ∮ B ⋅ d l = μ 0 I e n c Biot-Savart law d B ⃗ = μ 0 4 π I d l ⃗ × r ^ r 2 d\vec B = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec l\times\hat r}{r^2} d B = 4 π μ 0 r 2 I d l × r ^
Force between parallel currents Attractive when the currents are in the same direction, repulsive when opposite. Faraday's law of induction ε = − d Φ B d t \varepsilon = -\dfrac{d\Phi_B}{dt} ε = − d t d Φ B
Lenz's law The induced current flows so as to oppose the change in magnetic flux that produced it (the minus sign in Faraday's law). Self-inductance of a solenoid Proportional to the square of the number of turns, L ∝ N 2 L \propto N^2 L ∝ N 2 Energy stored in an inductor U = 1 2 L I 2 U = \tfrac{1}{2}LI^2 U = 2 1 L I 2 SI unit of magnetic flux The weber (Wb); 1 Wb = 1 T ⋅ m 2 1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2 1 Wb = 1 T ⋅ m 2 Maxwell's equations Gauss's law, Gauss's law for magnetism (∇ ⋅ B ⃗ = 0 \nabla\cdot\vec B = 0 ∇ ⋅ B = 0 Speed of an EM wave in vacuum c = 1 μ 0 ε 0 ≈ 3.0 × 10 8 c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3.0\times10^8 c = μ 0 ε 0 1 ≈ 3.0 × 1 0 8 Properties of electromagnetic waves Transverse, require no medium, travel at c c c E ⃗ \vec E E B ⃗ \vec B B Poynting vector S ⃗ = 1 μ 0 E ⃗ × B ⃗ \vec S = \dfrac{1}{\mu_0}\vec E\times\vec B S = μ 0 1 E × B Capacitive reactance X C = 1 ω C X_C = \dfrac{1}{\omega C} X C = ω C 1 Inductive reactance X L = ω L X_L = \omega L X L = ω L Resonance in an LC / RLC circuit ω 0 = 1 L C \omega_0 = \dfrac{1}{\sqrt{LC}} ω 0 = L C 1 X L = X C X_L = X_C X L = X C Magnetic flux Φ B = ∫ B ⃗ ⋅ d A ⃗ \Phi_B = \int \vec B\cdot d\vec A Φ B = ∫ B ⋅ d A Cyclotron frequency ω = q B m \omega = \dfrac{qB}{m} ω = m q B
Principle of superposition (fields) The net electric (or magnetic) field at a point is the vector sum of the fields from each source. Wave speed relation v = λ f v = \lambda f v = λ f
What stays constant when a wave changes medium? The frequency. The wavelength and speed change, but the frequency is fixed by the source. Snell's law n 1 sin θ 1 = n 2 sin θ 2 n_1\sin\theta_1 = n_2\sin\theta_2 n 1 sin θ 1 = n 2 sin θ 2 Index of refraction n = c v n = \dfrac{c}{v} n = v c v v v Total internal reflection Occurs when light in a denser medium hits the boundary beyond the critical angle θ c = sin − 1 ( n 2 / n 1 ) \theta_c = \sin^{-1}(n_2/n_1) θ c = sin − 1 ( n 2 / n 1 ) Thin-lens / mirror equation 1 f = 1 d o + 1 d i \dfrac{1}{f} = \dfrac{1}{d_o} + \dfrac{1}{d_i} f 1 = d o 1 + d i 1 m = − d i d o m = -\dfrac{d_i}{d_o} m = − d o d i Double-slit bright fringes d sin θ = m λ d\sin\theta = m\lambda d sin θ = mλ Single-slit diffraction minima a sin θ = m λ a\sin\theta = m\lambda a sin θ = mλ m = ± 1 , ± 2 , … m = \pm1, \pm2, \dots m = ± 1 , ± 2 , … a a a Double-slit fringe spacing Proportional to the wavelength and inversely proportional to the slit separation: Δ y = λ L d \Delta y = \dfrac{\lambda L}{d} Δ y = d λ L Brewster's angle θ B = tan − 1 ( n ) \theta_B = \tan^{-1}(n) θ B = tan − 1 ( n ) Malus's law I = I 0 cos 2 θ I = I_0\cos^2\theta I = I 0 cos 2 θ θ \theta θ Unpolarized light through a polarizer Exactly half the intensity, 1 2 I 0 \tfrac{1}{2}I_0 2 1 I 0
Doppler effect (light) Approaching source → blueshift (higher frequency); receding source → redshift (lower frequency).
Interference vs diffraction Interference = superposition of waves from multiple sources; diffraction = bending/spreading of a single wave around edges or apertures.
Which phenomenon needs the particle theory of light? The photoelectric effect. Interference, diffraction, and polarization are all explained by the wave theory. Diffraction-grating maxima d sin θ = m λ d\sin\theta = m\lambda d sin θ = mλ Rayleigh criterion (resolution) θ ≈ 1.22 λ D \theta \approx 1.22\dfrac{\lambda}{D} θ ≈ 1.22 D λ D D D Standing wave on a string fixed at both ends λ n = 2 L n , f n = n v 2 L \lambda_n = \dfrac{2L}{n},\ f_n = \dfrac{nv}{2L} λ n = n 2 L , f n = 2 L n v n = 1 , 2 , 3 , … n = 1, 2, 3, \dots n = 1 , 2 , 3 , …
Color with the shortest visible wavelength Violet (refracted/dispersed most by a prism); red has the longest visible wavelength.
Why does a prism disperse light? The index of refraction depends on wavelength (dispersion), so different colors bend by different amounts. First law of thermodynamics Δ U = Q − W \Delta U = Q - W Δ U = Q − W
Second law of thermodynamics The entropy of an isolated system never decreases; heat does not spontaneously flow from cold to hot.
Third law of thermodynamics The entropy of a perfect crystal approaches zero as the temperature approaches absolute zero. Ideal gas law P V = n R T PV = nRT P V = n R T P V = N k B T PV = Nk_B T P V = N k B T Isothermal process T T T Δ U = 0 \Delta U = 0 Δ U = 0 Q = W Q = W Q = W Isobaric process P P P W = P Δ V W = P\,\Delta V W = P Δ V Isochoric (isovolumetric) process V V V W = 0 W = 0 W = 0 Q = Δ U Q = \Delta U Q = Δ U Adiabatic process No heat exchange, Q = 0 Q = 0 Q = 0 Δ U = − W \Delta U = -W Δ U = − W P V γ = PV^\gamma = P V γ = Adiabatic free expansion (ideal gas) No work and no heat, so Δ U = 0 \Delta U = 0 Δ U = 0 U U U T T T Carnot efficiency η = 1 − T c T h \eta = 1 - \dfrac{T_c}{T_h} η = 1 − T h T c Entropy change (reversible) d S = d Q r e v T dS = \dfrac{dQ_{rev}}{T} d S = T d Q r e v Boltzmann entropy formula S = k B ln Ω S = k_B \ln \Omega S = k B ln Ω Ω \Omega Ω Equipartition theorem Each quadratic degree of freedom contributes 1 2 k B T \tfrac{1}{2}k_B T 2 1 k B T Average translational kinetic energy of a gas molecule ⟨ K E ⟩ = 3 2 k B T \langle KE\rangle = \tfrac{3}{2}k_B T ⟨ K E ⟩ = 2 3 k B T RMS speed of gas molecules v r m s = 3 k B T m v_{rms} = \sqrt{\dfrac{3k_B T}{m}} v r m s = m 3 k B T Maxwell-Boltzmann distribution The classical distribution of molecular speeds in an ideal gas; raising T T T Stefan-Boltzmann law P = σ A T 4 P = \sigma A T^4 P = σ A T 4 T 4 T^4 T 4 Wien's displacement law λ m a x T = \lambda_{max} T = λ ma x T = Boltzmann constant k B ≈ 1.38 × 10 − 23 k_B \approx 1.38\times10^{-23} k B ≈ 1.38 × 1 0 − 23 k B = R / N A k_B = R/N_A k B = R / N A Heat capacity / specific heat Q = m c Δ T Q = mc\,\Delta T Q = m c Δ T C P − C V = R C_P - C_V = R C P − C V = R
Heat pump / refrigerator Moves heat from cold to hot using external work — a consequence (and demonstration) of the second law.
Fermi-Dirac vs Bose-Einstein statistics Fermi-Dirac governs fermions (one per state, Pauli); Bose-Einstein governs bosons (many can share a state). Internal energy of an ideal gas Depends only on temperature; for a monatomic ideal gas U = 3 2 n R T U = \tfrac{3}{2}nRT U = 2 3 n R T Schrodinger equation (time-independent) H ^ ψ = E ψ \hat H\psi = E\psi H ^ ψ = E ψ ψ \psi ψ E E E Schrodinger equation (time-dependent) i ℏ ∂ ψ ∂ t = H ^ ψ i\hbar\dfrac{\partial \psi}{\partial t} = \hat H\psi i ℏ ∂ t ∂ ψ = H ^ ψ Born rule (probability density) ∣ ψ ( x ) ∣ 2 |\psi(x)|^2 ∣ ψ ( x ) ∣ 2 [ a , b ] [a,b] [ a , b ] ∫ a b ∣ ψ ∣ 2 d x \int_a^b |\psi|^2\,dx ∫ a b ∣ ψ ∣ 2 d x Normalization condition ∫ − ∞ ∞ ∣ ψ ∣ 2 d x = 1 \int_{-\infty}^{\infty} |\psi|^2\,dx = 1 ∫ − ∞ ∞ ∣ ψ ∣ 2 d x = 1 Infinite square well energies E n = n 2 h 2 8 m L 2 = n 2 π 2 ℏ 2 2 m L 2 E_n = \dfrac{n^2 h^2}{8mL^2} = \dfrac{n^2\pi^2\hbar^2}{2mL^2} E n = 8 m L 2 n 2 h 2 = 2 m L 2 n 2 π 2 ℏ 2 n = 1 , 2 , 3 , … n = 1, 2, 3, \dots n = 1 , 2 , 3 , … Infinite square well wave functions ψ n ( x ) = 2 L sin ( n π x L ) \psi_n(x) = \sqrt{\dfrac{2}{L}}\sin\!\left(\dfrac{n\pi x}{L}\right) ψ n ( x ) = L 2 sin ( L nπ x ) Quantum harmonic oscillator energies E n = ( n + 1 2 ) ℏ ω E_n = \left(n + \tfrac{1}{2}\right)\hbar\omega E n = ( n + 2 1 ) ℏ ω 1 2 ℏ ω \tfrac{1}{2}\hbar\omega 2 1 ℏ ω Canonical commutator [ x ^ , p ^ ] = i ℏ [\hat x,\hat p] = i\hbar [ x ^ , p ^ ] = i ℏ Heisenberg uncertainty principle Δ x Δ p ≥ ℏ 2 \Delta x\,\Delta p \ge \dfrac{\hbar}{2} Δ x Δ p ≥ 2 ℏ Δ E Δ t ≥ ℏ 2 \Delta E\,\Delta t \ge \dfrac{\hbar}{2} Δ E Δ t ≥ 2 ℏ Hamiltonian operator H ^ \hat H H ^ Momentum operator p ^ = − i ℏ ∂ ∂ x \hat p = -i\hbar\dfrac{\partial}{\partial x} p ^ = − i ℏ ∂ x ∂ Orthonormality of eigenstates Energy eigenstates are orthogonal and normalized: ∫ ψ m ∗ ψ n d x = δ m n \int \psi_m^* \psi_n\,dx = \delta_{mn} ∫ ψ m ∗ ψ n d x = δ mn Expectation value ⟨ A ⟩ = ∫ ψ ∗ A ^ ψ d x \langle A\rangle = \int \psi^* \hat A\,\psi\,dx ⟨ A ⟩ = ∫ ψ ∗ A ^ ψ d x A ^ \hat A A ^
Quantum tunneling A particle has a nonzero probability of passing through a potential barrier even when its energy is below the barrier height. Electron spin quantum number s = 1 2 s = \tfrac{1}{2} s = 2 1 m s = ± 1 2 m_s = \pm\tfrac{1}{2} m s = ± 2 1
Pauli exclusion principle No two identical fermions can occupy the same quantum state simultaneously. de Broglie wavelength λ = h p \lambda = \dfrac{h}{p} λ = p h Ehrenfest's theorem Quantum expectation values obey the classical equations of motion (m d ⟨ x ⟩ / d t = ⟨ p ⟩ m\,d\langle x\rangle/dt = \langle p\rangle m d ⟨ x ⟩ / d t = ⟨ p ⟩ Eigenvalue equation A ^ ψ = a ψ \hat A\psi = a\psi A ^ ψ = a ψ A ^ \hat A A ^ ψ \psi ψ a a a Angular-momentum quantization L 2 = l ( l + 1 ) ℏ 2 L^2 = l(l+1)\hbar^2 L 2 = l ( l + 1 ) ℏ 2 L z = m l ℏ L_z = m_l\hbar L z = m l ℏ Bohr model energy levels (hydrogen) E n = − 13.6 eV n 2 E_n = -\dfrac{13.6\,\text{eV}}{n^2} E n = − n 2 13.6 eV − 13.6 -13.6 − 13.6 Bohr quantization of angular momentum L = n ℏ L = n\hbar L = n ℏ ℏ \hbar ℏ Photon energy E = h f = h c λ E = hf = \dfrac{hc}{\lambda} E = h f = λ h c Rydberg formula 1 λ = R ( 1 n 1 2 − 1 n 2 2 ) \dfrac{1}{\lambda} = R\left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right) λ 1 = R ( n 1 2 1 − n 2 2 1 ) Lyman, Balmer, Paschen series Transitions ending at n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 n = 3 n = 3 n = 3 Longest-wavelength visible hydrogen line The n = 3 → 2 n = 3 \to 2 n = 3 → 2 Four atomic quantum numbers Principal n n n l l l m l m_l m l m s m_s m s Photoelectric effect K E m a x = h f − ϕ KE_{max} = hf - \phi K E ma x = h f − ϕ Work function ϕ \phi ϕ
Fine structure Small spectral-line splitting from spin-orbit coupling — the electron's spin interacting with its orbital motion.
Hyperfine structure Even smaller splitting from the interaction between the nuclear spin and the electron cloud.
Zeeman effect Splitting of atomic energy levels (and spectral lines) in an external magnetic field.
Stark effect Shifting and splitting of spectral lines in an external electric field (the electric analog of the Zeeman effect).
Stern-Gerlach experiment An atomic beam splits into discrete components in a non-uniform magnetic field — proof that angular momentum (spin) is quantized. Compton scattering Δ λ = h m e c ( 1 − cos θ ) \Delta\lambda = \dfrac{h}{m_e c}(1-\cos\theta) Δ λ = m e c h ( 1 − cos θ )
Electron diffraction (Davisson-Germer) Electrons form interference patterns, demonstrating their wave nature (wave-particle duality).
Hund's rule Electrons fill degenerate orbitals singly with parallel spins before pairing, to minimize energy.
Aufbau principle Electrons fill the lowest-energy orbitals first when building up an atom's ground-state configuration. Selection rule for dipole transitions Δ l = ± 1 \Delta l = \pm1 Δ l = ± 1 Δ m l = 0 , ± 1 \Delta m_l = 0, \pm1 Δ m l = 0 , ± 1 X-ray production (characteristic lines) An inner-shell vacancy filled by an outer electron emits a characteristic X-ray; K-alpha is n = 2 → 1 n=2 \to 1 n = 2 → 1
What keeps the electron from falling into the nucleus? The uncertainty principle: confining the electron more tightly raises its momentum (and energy), setting a minimum-energy ground state. Two postulates of special relativity (1) The laws of physics are the same in all inertial frames. (2) The speed of light c c c Lorentz factor γ = 1 1 − v 2 / c 2 \gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}} γ = 1 − v 2 / c 2 1 Time dilation Δ t = γ Δ t 0 \Delta t = \gamma\,\Delta t_0 Δ t = γ Δ t 0 Δ t 0 \Delta t_0 Δ t 0 Length contraction L = L 0 γ L = \dfrac{L_0}{\gamma} L = γ L 0 L 0 L_0 L 0
Proper time The time interval measured by a clock at rest relative to the two events — the shortest possible interval between them. Lorentz transformation x ′ = γ ( x − v t ) , t ′ = γ ( t − v x c 2 ) x' = \gamma(x - vt),\ t' = \gamma\!\left(t - \dfrac{vx}{c^2}\right) x ′ = γ ( x − v t ) , t ′ = γ ( t − c 2 v x ) Relativistic energy E = γ m c 2 E = \gamma mc^2 E = γ m c 2 E 0 = m c 2 E_0 = mc^2 E 0 = m c 2 Relativistic momentum p ⃗ = γ m v ⃗ \vec p = \gamma m\vec v p = γ m v Energy-momentum relation E 2 = ( p c ) 2 + ( m c 2 ) 2 E^2 = (pc)^2 + (mc^2)^2 E 2 = ( p c ) 2 + ( m c 2 ) 2 E = p c E = pc E = p c E = m c 2 E = mc^2 E = m c 2 Relativistic kinetic energy K E = ( γ − 1 ) m c 2 KE = (\gamma - 1)mc^2 K E = ( γ − 1 ) m c 2 1 2 m v 2 \tfrac{1}{2}mv^2 2 1 m v 2 v → c v\to c v → c Relativistic velocity addition u ′ = u + v 1 + u v / c 2 u' = \dfrac{u + v}{1 + uv/c^2} u ′ = 1 + uv / c 2 u + v c c c Invariant spacetime interval Δ s 2 = ( c Δ t ) 2 − Δ x 2 \Delta s^2 = (c\Delta t)^2 - \Delta x^2 Δ s 2 = ( c Δ t ) 2 − Δ x 2
Relativistic Doppler effect Includes time dilation, so it differs from the classical Doppler shift; receding source → redshift, approaching → blueshift.
Twin paradox resolution The traveling twin accelerates and decelerates, breaking the symmetry, so that twin ages less. Why can't a massive object reach c? Its kinetic energy ( γ − 1 ) m c 2 (\gamma-1)mc^2 ( γ − 1 ) m c 2 v → c v\to c v → c
Random vs systematic error Random error scatters measurements (reduced by averaging); systematic error biases them in one direction (a calibration/offset problem). Adding independent uncertainties They add in quadrature: σ = σ 1 2 + σ 2 2 \sigma = \sqrt{\sigma_1^2 + \sigma_2^2} σ = σ 1 2 + σ 2 2 Poisson counting statistics A count of N N N N \sqrt{N} N 1 / N 1/\sqrt{N} 1/ N Standard deviation vs standard error Standard deviation measures spread; the standard error of the mean is σ / N \sigma/\sqrt{N} σ / N
Lock-in amplifier Extracts a small signal at a known reference frequency from heavy noise, by mixing and low-pass filtering, hugely improving signal-to-noise.
Faraday cage A conducting enclosure that shields its contents from external electric fields (charges redistribute to cancel the field inside).
Cryostat with liquid helium Provides thermal insulation to reach and hold temperatures near absolute zero; liquid helium boils at 4.2 K.
Laser (Doppler) cooling Slows and cools atoms using laser light tuned just below an atomic transition, reducing their kinetic energy.
Optical tweezers Trap and move microscopic particles using the radiation pressure (momentum) of a focused laser beam.
Hall effect measurement Measures the carrier density (and sign of charge carriers) in a material from the transverse Hall voltage in a magnetic field.
Time-of-flight mass spectrometry Determines an ion's mass-to-charge ratio from the time it takes to travel a known distance after acceleration.
FTIR spectroscopy advantage Fourier-transform IR offers higher resolution, faster acquisition, and better signal-to-noise (Fellgett's advantage) over dispersive IR.
Oscilloscope Displays voltage versus time, used to view waveforms, measure amplitude, frequency, and phase.
Significant figures rule A computed result is limited by the least-precise input; report uncertainty to one or two significant figures.
Photomultiplier tube Detects single photons by cascading secondary-electron emission, producing a measurable pulse from very weak light.
Geiger-Muller counter Detects ionizing radiation via gas ionization producing electrical pulses; counts follow Poisson statistics.
Nuclear magic numbers 2, 8, 20, 28, 50, 82, 126 — proton or neutron counts that fill nuclear shells and give extra stability. Nuclear binding energy The energy released forming a nucleus from its nucleons; the mass defect times c 2 c^2 c 2 Alpha, beta, gamma decay Alpha emits a X 2 4 X 2 2 2 4 H e \ce{^4_2He} X 2 4 X 2 2 2 4 He Radioactive decay law N ( t ) = N 0 e − λ t N(t) = N_0 e^{-\lambda t} N ( t ) = N 0 e − λ t t 1 / 2 = ln 2 λ t_{1/2} = \dfrac{\ln 2}{\lambda} t 1/2 = λ ln 2
Color confinement Quarks carry color charge and can never be isolated — only color-neutral hadrons (mesons, baryons) are observed.
Quark model Baryons are three quarks (e.g. proton = uud), mesons are a quark-antiquark pair; the six flavors are u, d, c, s, t, b.
The four fundamental forces Strong, electromagnetic, weak, and gravity — mediated by gluons, photons, W/Z bosons, and (hypothetically) gravitons.
Superconductivity Zero electrical resistance below a critical temperature; expels magnetic fields (Meissner effect).
Superfluidity Flow with zero viscosity, seen in liquid helium-4 below 2.17 K — particles flow without losing kinetic energy.
Chandrasekhar limit The maximum white-dwarf mass (≈ 1.4 solar masses) supported by electron degeneracy pressure; above it the core collapses. 
