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Mechanics

50
dynamics
F=maF = ma

Newton's Second Law

Force equals mass times acceleration: heavier objects need more push.

elasticity
F=−kxF = -kx

Hooke's Law

A spring pushes back proportionally to how far you stretch it.

gravitation
F=GMmr2F = \frac{GMm}{r^2}

Newton's Law of Universal Gravitation

Every mass attracts every other mass; force drops with the square of distance.

energy
KE=12mv2KE = \frac{1}{2}mv^2

Kinetic Energy

Energy of motion: doubles with mass, quadruples with speed.

circular motion
ac=v2ra_c = \frac{v^2}{r}

Centripetal Acceleration

Moving in a circle requires constant inward acceleration; faster or tighter = more.

kinematics
R=v2sin⁡(2θ)gR = \frac{v^2 \sin(2\theta)}{g}

Projectile Range

Launch angle of 45° maximizes range; steeper or flatter angles fall shorter.

energy
Wnet=ΔKE=12mvf2−12mvi2W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2

Work-Energy Theorem

Net work on an object equals the change in its kinetic energy.

oscillations
T=2πmkT = 2\pi\sqrt{\frac{m}{k}}

Simple Harmonic Motion Period

Heavier masses oscillate slower; stiffer springs oscillate faster.

dynamics
p=mvp = mv

Linear Momentum

Momentum measures how hard it is to stop a moving object — it scales with both mass and speed.

dynamics
J=FΔt=ΔpJ = F \Delta t = \Delta p

Impulse-Momentum Theorem

Stopping the same momentum over a longer time means smaller force — the whole point of airbags, crumple zones, and bending your knees.

energy
U=mghU = mgh

Gravitational Potential Energy

Lifting something stores energy in the gravitational field — drop it and the energy reappears as motion.

energy
P=Wt=FvP = \frac{W}{t} = Fv

Mechanical Power

Power is how fast you spend energy. Same work, less time = more power.

rotation
τ=rFsin⁡θ\tau = rF\sin\theta

Torque

Torque is twist. The farther you push from the pivot, and the more perpendicular your push, the more spin you create.

rotation
L=IωL = I\omega

Angular Momentum

Angular momentum is rotational inertia times spin speed. Squeeze inward and you must spin faster to keep it conserved.

rotation
KErot=12Iω2KE_{\text{rot}} = \tfrac{1}{2} I \omega^2

Rotational Kinetic Energy

Spinning things store energy just like moving things — and a small radius increase pays off big because I scales with r².

kinematics
v=2ghv = \sqrt{2gh}

Free Fall Velocity

Drop something from height h in vacuum — by the time it hits, all the gravitational potential mgh has turned into kinetic ½mv².

oscillations
T=2πLgT = 2\pi \sqrt{\frac{L}{g}}

Simple Pendulum Period

Longer pendulums swing slower. The period depends only on length and gravity — not mass and (almost) not on amplitude.

rotational dynamics
I=mr2I = mr^2

Moment of Inertia (Point Mass)

Mass placed far from the rotation axis resists rotation much more than the same mass placed close in.

gravitation
vesc=2GMRv_{\text{esc}} = \sqrt{\frac{2GM}{R}}

Escape Velocity

The speed at which kinetic energy exactly equals the gravitational binding energy — no faster, you fly free.

gravitation
T2=4π2GMa3T^2 = \frac{4\pi^2}{GM} a^3

Kepler's Third Law

Farther orbits take longer — and not linearly: the period grows as the 3/2 power of the orbital radius.

dynamics
Fk=μkNF_k = \mu_k N

Kinetic Friction Force

Once an object is already sliding, friction resists motion with a force proportional to how hard the surfaces are pressed together.

dynamics
Fs,max⁡=μsNF_{s,\max} = \mu_s N

Maximum Static Friction

Static friction adjusts itself to match whatever you push with — up to a hard ceiling set by μ_s N. Cross that line and it gives way.

dynamics
N=mgcos⁡θN = m g \cos\theta

Normal Force on an Incline

The surface only needs to support the part of gravity pointing into it — cos θ of the weight.

dynamics
F∥=mgsin⁡θF_\parallel = m g \sin\theta

Gravity Component Along an Incline

Only the sine-component of gravity accelerates an object down the slope — the cosine-component is canceled by the normal force.

dynamics
T=2m1m2gm1+m2T = \frac{2 m_1 m_2 g}{m_1 + m_2}

Atwood Machine Tension

The tension is twice the harmonic mean of the two weights — equal masses give T = mg (no motion), unequal masses give something in between.

dynamics
v1′=(m1−m2)v1+2m2v2m1+m2v_1' = \frac{(m_1 - m_2) v_1 + 2 m_2 v_2}{m_1 + m_2}

1D Elastic Collision (Final Velocities)

Conserve both momentum and kinetic energy and the velocities of the two bodies swap (when masses are equal) or rearrange linearly in mass ratios.

dynamics
vf=m1v1+m2v2m1+m2v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}

Perfectly Inelastic Collision (1D)

Momentum is conserved, but kinetic energy is not — the bodies stick together and share a single final velocity equal to the system center-of-mass velocity.

dynamics
e=−v1′−v2′v1−v2e = -\frac{v_1' - v_2'}{v_1 - v_2}

Coefficient of Restitution

A single dimensionless number, 0 ≤ e ≤ 1, that says how much of the relative speed survives a collision. e=1 is perfectly elastic; e=0 is perfectly inelastic.

dynamics
xcm=m1x1+m2x2m1+m2x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}

Center of Mass (Two Particles)

A mass-weighted average of positions — the single point that behaves like the whole system for Newton's second law.

rotational dynamics
I=Icm+md2I = I_{cm} + m d^2

Parallel Axis Theorem

Spinning about a shifted axis costs extra rotational inertia equal to m·d² — because every particle now traces a larger circle.

rotational dynamics
v=Rωv = R \omega

Rolling Without Slipping

The contact point of a rolling wheel is instantaneously at rest — so the translation speed of the center equals the tangential speed at the rim.

rotational dynamics
I1ω1=I2ω2I_1 \omega_1 = I_2 \omega_2

Conservation of Angular Momentum

With no external torque, the product Iω is fixed: pull mass closer to the axis and you must spin faster to compensate.

analytical mechanics
S[q]=∫t1t2L(q,q˙,t) dt,δS=0S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t)\, dt, \qquad \delta S = 0

Principle of Stationary Action

Of all imaginable paths between two events, nature takes the one whose action doesn't change under small wiggles.

analytical mechanics
ddt(∂L∂q˙i)−∂L∂qi=0\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0

Euler–Lagrange Equation

Pick any coordinates you like, write energy in, equations of motion come out — no force diagrams needed.

analytical mechanics
pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}

Generalized (Conjugate) Momentum

Each coordinate gets its own momentum — differentiate L by the velocity of that coordinate.

analytical mechanics
H(q,p,t)=∑ipiq˙i−LH(q, p, t) = \sum_i p_i \dot{q}_i - L

The Hamiltonian (Legendre Transform)

Swap each velocity for its momentum; what's left is (usually) the total energy as a function of position and momentum.

analytical mechanics
q˙i=∂H∂pi,p˙i=−∂H∂qi\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}

Hamilton's Canonical Equations

H is a landscape over phase space; states flow along its contour lines, position fed by momentum and momentum drained by force.

analytical mechanics
{f,g}=∑i(∂f∂qi∂g∂pi−∂f∂pi∂g∂qi)\{f, g\} = \sum_i \left( \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i} \right)

Poisson Bracket & Liouville Flow

The bracket measures how two phase-space functions 'stir' each other; pairing anything with H gives its time evolution.

analytical mechanics
∂L∂qi=0  ⇒  ddt∂L∂q˙i=0\frac{\partial L}{\partial q_i} = 0 \;\Rightarrow\; \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} = 0

Noether's Theorem

Every continuous symmetry of the action hands you a conserved quantity — no exceptions, no extra work.

analytical mechanics
L=12(m1+m2)l12θ˙12+12m2l22θ˙22+m2l1l2θ˙1θ˙2cos⁡(θ1−θ2)+(m1+m2)gl1cos⁡θ1+m2gl2cos⁡θ2\mathcal{L} = \tfrac{1}{2}(m_1{+}m_2)l_1^2\dot\theta_1^2 + \tfrac{1}{2}m_2 l_2^2\dot\theta_2^2 + m_2 l_1 l_2 \dot\theta_1\dot\theta_2\cos(\theta_1{-}\theta_2) + (m_1{+}m_2)gl_1\cos\theta_1 + m_2 gl_2\cos\theta_2

Double Pendulum (Lagrangian Chaos)

Two pendulums chained together: the Lagrangian is easy to write, the motion is impossible to predict long-term.

gravitation
Ftidal≈2GMmRd3F_{\text{tidal}} \approx \frac{2GMmR}{d^{3}}

Tidal Force

Gravity weakens with distance, so the Moon pulls Earth's near side harder than its center, and the center harder than the far side. In Earth's frame that difference stretches the planet along the Moon line and squeezes it sideways — two ocean bulges, two high tides a day. Because the effect goes as 1/d³ (not 1/d²), the nearby Moon out-tides the enormous Sun.

collisions
∑imiv⃗i=constant⟺m1u⃗1+m2u⃗2=m1v⃗1+m2v⃗2\sum_i m_i \vec{v}_i = \text{constant} \quad\Longleftrightarrow\quad m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2

Conservation of Linear Momentum

With no outside push, the total momentum of a system can't change — collisions only trade it between parts.

dynamics with drag
vt=2mgρ Cd Av_t = \sqrt{\dfrac{2 m g}{\rho\, C_d\, A}}

Terminal Velocity

You stop accelerating when quadratic air drag grows to exactly match your weight.

rotating frames
a⃗Cor=−2 ω⃗×v⃗\vec{a}_{\text{Cor}} = -2\,\vec{\omega}\times\vec{v}

Coriolis Deflection

In a rotating frame, anything that moves gets pushed sideways — the frame, not a real force, does the deflecting.

rigid body dynamics
τ⃗=dL⃗dt,Ωp=τL=τIω\vec{\tau} = \frac{d\vec{L}}{dt}, \qquad \Omega_p = \frac{\tau}{L} = \frac{\tau}{I\omega}

Gyroscopic Precession

A torque changes angular momentum's direction, not the spin's size, so the axle swings sideways instead of falling.

rigid body dynamics
L⃗=Ifront ω⃗front+Iback ω⃗back=0\vec{L} = I_{\text{front}}\,\vec{\omega}_{\text{front}} + I_{\text{back}}\,\vec{\omega}_{\text{back}} = 0

The Falling-Cat Problem

Change your shape in a cycle and you can reorient with zero angular momentum — geometry rotates you, not spin.

ballistics
R=v2sin⁡2θg,v=2 η MghmR = \frac{v^2 \sin 2\theta}{g}, \qquad v = \sqrt{\frac{2\,\eta\, M g h}{m}}

Physics of the Trebuchet

A heavy counterweight's drop becomes a light stone's speed; range then follows the projectile formula.

oscillations instability
θ¨+2ζnet ωn θ˙+ωn2 θ=0,ζnet=ζstruct−ζaero(U)\ddot{\theta} + 2\zeta_{\text{net}}\,\omega_n\,\dot{\theta} + \omega_n^2\,\theta = 0, \qquad \zeta_{\text{net}} = \zeta_{\text{struct}} - \zeta_{\text{aero}}(U)

Tacoma Narrows: Aeroelastic Flutter

Past a critical wind, the airflow pumps energy in faster than damping drains it, so oscillations blow up.

celestial mechanics
Δϕ=6πGMa (1−e2) c2per orbit\Delta\phi = \frac{6\pi G M}{a\,(1-e^2)\,c^2} \quad \text{per orbit}

Perihelion Precession of Mercury

Relativity bends spacetime just enough that the orbit's closest point creeps forward each lap.

astrophysics
MCh≈ω30 3π2(ℏcG)3/21(μemH)2≈1.44(2μe)2M⊙M_{\text{Ch}} \approx \frac{\omega_3^0\,\sqrt{3\pi}}{2}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^2} \approx 1.44\left(\frac{2}{\mu_e}\right)^2 M_\odot

The Chandrasekhar Limit

Relativistic electron pressure scales like gravity, so above ~1.4 solar masses gravity always wins.

Electromagnetism

52
electrostatics
F=ke∣q1q2∣r2F = k_e \frac{|q_1 q_2|}{r^2}

Coulomb's Law

Electric force between two charges falls off as the square of the distance between them.

electrostatics
E⃗=14πε0qr2r^\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r}

Electric Field of a Point Charge

Each charge creates a field that tells other charges how much force they would feel.

electrostatics
∮E⃗⋅dA⃗=Qencε0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

Gauss's Law

The total electric flux through a closed surface equals the enclosed charge divided by ε₀.

electrostatics
V=14πε0qrV = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}

Electric Potential (Point Charge)

Potential is the energy per unit charge — it falls off as 1/r, not 1/r².

electrostatics
C=ε0AdC = \varepsilon_0 \frac{A}{d}

Parallel Plate Capacitor

Bigger plates and smaller gaps store more charge per volt.

circuits
V=IRV = IR

Ohm's Law

Voltage is the push, resistance is the friction, current is how much flows.

magnetostatics
dB⃗=μ04πI dl⃗×r^r2d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2}

Biot–Savart Law

Each bit of current creates a magnetic field perpendicular to both the current and distance.

magnetostatics
∮B⃗⋅dl⃗=μ0(Ienc+ε0dΦEdt)\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)

Ampère's Law (with Maxwell's correction)

Magnetic field loops around currents; total circulation equals enclosed current times μ₀.

electromagnetic induction
E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}

Faraday's Law of Induction

A changing magnetic flux through a loop induces a voltage that opposes the change.

electromagnetic dynamics
F⃗=q(E⃗+v⃗×B⃗)\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

Lorentz Force Law

Electric fields push charges; magnetic fields deflect moving charges sideways.

magnetostatics
∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0

Gauss's Law for Magnetism

Magnetic field lines always close on themselves — no isolated north or south poles.

electrostatics
U=12CV2U = \tfrac{1}{2} C V^2

Energy Stored in a Capacitor

To charge a capacitor you must push charge against a growing voltage — the work done is stored as electric field energy between the plates.

circuits
V(t)=V0(1−e−t/τ),τ=RCV(t) = V_0 \left(1 - e^{-t/\tau}\right), \quad \tau = RC

RC Circuit Charging

Current is largest at t = 0 when the capacitor is empty; as voltage builds up, current decays, asymptotically filling the capacitor to V₀.

magnetostatics
F⃗=IL⃗×B⃗\vec{F} = I \vec{L} \times \vec{B}

Magnetic Force on a Current-Carrying Wire

Each moving charge in the wire feels a Lorentz force; summed over all of them, the wire experiences a net push perpendicular to both the current and the field.

magnetostatics
B=μ0nIB = \mu_0 n I

Magnetic Field Inside a Solenoid

Each turn contributes a circular field; many tightly packed turns add up inside the tube and cancel outside — leaving a near-uniform interior field.

magnetostatics
L=μ0n2V=μ0N2AℓL = \mu_0 n^2 V = \mu_0 \frac{N^2 A}{\ell}

Self-Inductance of a Solenoid

When current changes, flux through every loop changes, inducing an EMF that opposes the change. The geometric constant tying flux to current is the inductance.

magnetostatics
U=12LI2U = \tfrac{1}{2} L I^2

Energy Stored in an Inductor

To establish a current in an inductor you must do work against the back-EMF. That work survives as magnetic field energy stored in the coil's interior.

circuits
f=12πLCf = \frac{1}{2\pi \sqrt{LC}}

LC Oscillation Frequency

Energy sloshes between the capacitor's electric field and the inductor's magnetic field — a pure electromagnetic version of a mass-on-a-spring.

electromagnetic waves
S⃗=1μ0E⃗×B⃗\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}

Poynting Vector

Wherever both E and B exist, energy flows perpendicular to both. The Poynting vector tells you how much energy crosses a unit area per second, and in what direction.

electromagnetic induction
ε=BLv\varepsilon = B L v

Motional EMF

Free charges inside the moving rod feel a magnetic force qv×B that pushes them along the rod — positive charge piles up at one end until the resulting electric field balances the push. That charge separation is a battery made of motion.

electromagnetic dynamics
fc=qB2πmf_c = \frac{q B}{2\pi m}

Cyclotron Frequency

The magnetic force qvB always points to the center, bending the path into a circle. Faster particles ride bigger circles, but the extra path length exactly cancels the extra speed — every lap takes the same time.

circuits
I(t)=V0R(1−e−t/τ),τ=LRI(t) = \frac{V_0}{R}\left(1 - e^{-t/\tau}\right), \quad \tau = \frac{L}{R}

RL Circuit Current Growth

An inductor hates changes in current: its back-EMF L·dI/dt initially cancels the battery entirely, so current starts at zero and creeps up. As the current settles, the back-EMF dies away and the resistor takes over.

electromagnetic induction
VsVp=NsNp\frac{V_s}{V_p} = \frac{N_s}{N_p}

Ideal Transformer Equation

Both coils share the same changing flux through the iron core, so each turn of wire sees the same induced voltage. Voltage per coil is just volts-per-turn × turns — the ratio of turns is the ratio of voltages.

electromagnetic waves
c=1μ0ε0c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}

Speed of Electromagnetic Waves

A changing electric field makes a magnetic field (Ampère–Maxwell); a changing magnetic field makes an electric field (Faraday). The two regenerate each other in a self-sustaining ripple whose speed is fixed by how strongly each effect couples — ε₀ and μ₀.

electromagnetic dynamics
∮B⃗⋅dl⃗=μ0(I+ε0dΦEdt)\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I + \varepsilon_0 \frac{d\Phi_E}{dt} \right)

Ampère–Maxwell Law

Maxwell noticed Ampère's law contradicts itself at a charging capacitor: a loop around the wire sees current, but the same loop with its surface bulged through the gap sees none. His fix: a changing electric field acts exactly like a current — the displacement current — and creates magnetic field just the same.

electromagnetic waves
P=q2a26πε0c3P = \frac{q^2 a^2}{6 \pi \varepsilon_0 c^3}

Larmor Formula

A charge at rest has radial field lines. Accelerate it, and the news of its new motion spreads outward at c — the field lines develop a traveling kink. That kink is a transverse field pulse carrying energy: radiation. No acceleration, no kink, no radiation.

circuits
vd=InAqv_d = \frac{I}{n A q}

Drift Velocity

Electrons in a wire already zip around at ~10⁶ m/s thermally, but in random directions that cancel. An applied field adds a tiny net drift on top — a slow river current under a churning sea. The signal is fast (a field change at near light speed) but the carriers themselves barely move.

circuits
ρ=ρ0 [1+α(T−T0)]\rho = \rho_0\,[1 + \alpha (T - T_0)]

Temperature Dependence of Resistivity

Resistance comes from electrons scattering off the lattice. Heat makes the ions vibrate harder, so electrons collide more often and drift less freely — resistivity climbs nearly linearly over ordinary temperature ranges.

magnetostatics
τ=μBsin⁡θ,μ=NIA\tau = \mu B \sin\theta,\quad \mu = N I A

Magnetic Dipole Moment & Torque

A current loop is a tiny bar magnet: its moment μ points along the loop's axis. A uniform field can't push it (forces cancel) but it twists it, trying to align μ with B — exactly how a compass needle swings north.

electromagnetic waves
∇ ⁣⋅ ⁣E⃗=ρε0, ∇ ⁣⋅ ⁣B⃗=0, ∇ ⁣× ⁣E⃗=−∂tB⃗, ∇ ⁣× ⁣B⃗=μ0J⃗+μ0ε0∂tE⃗\nabla\!\cdot\!\vec{E}=\tfrac{\rho}{\varepsilon_0},\ \nabla\!\cdot\!\vec{B}=0,\ \nabla\!\times\!\vec{E}=-\partial_t\vec{B},\ \nabla\!\times\!\vec{B}=\mu_0\vec{J}+\mu_0\varepsilon_0\partial_t\vec{E}

Maxwell's Equations (Unified)

Two of the four equations say charges make diverging E-fields and that there are no magnetic charges; the other two say a changing B makes a curling E and a changing E (or a current) makes a curling B. That mutual feedback lets the fields regenerate each other and sail off as light — no charges required.

circuits
ΔV=I Rseg=I ρ LA\Delta V = I\,R_{seg} = I\,\rho\,\frac{L}{A}

Why Birds on Power Lines Don't Get Shocked

Current follows the path of least resistance. The bird's body (tens of kΩ) sits in parallel with a few centimeters of thick wire (micro-ohms). The wire is millions of times more conductive, so essentially all the current stays in the metal and the bird carries a vanishing trickle. Shock needs a voltage difference across you — and across two nearby points on one wire that difference is microscopic.

magnetostatics
VH=IBnqtV_H = \frac{I B}{n q t}

Hall Effect

Moving carriers in a perpendicular field feel a sideways Lorentz push and pile up on one edge. The charge buildup creates a transverse electric field until it exactly cancels the magnetic deflection. The leftover voltage across the strip reveals both the sign and the density of the carriers — a direct count of charge carriers.

electrostatics
E⃗inside=0\vec{E}_{inside} = 0

Why a Faraday Cage Shields You

A conductor's free charges move until they kill any field inside the metal. Put a cavity inside and those charges arrange on the outer surface to cancel the external field throughout the hollow — leaving the interior field-free. For oscillating waves a mesh works too, as long as the holes are far smaller than the wavelength, because the induced surface currents re-radiate a canceling field.

applications
f=γB02πf = \frac{\gamma B_0}{2\pi}

How MRI Imaging Works (Larmor)

Every proton is a tiny spinning magnet. In a strong field B₀ it doesn't just align — it precesses like a wobbling top, at a frequency set only by the field. Hit it with a radio pulse at exactly that frequency and it absorbs energy, tips over, then relaxes back while broadcasting a faint radio signal. A field gradient makes the frequency a position label — turning the echo into an image.

ac circuits
δ=2ωμσ=1πfμσ\delta = \sqrt{\frac{2}{\omega\mu\sigma}} = \frac{1}{\sqrt{\pi f \mu \sigma}}

Skin Effect in Conductors

An alternating current sets up a changing magnetic field inside the conductor, which by Faraday's law drives eddy currents that oppose the flow in the core and reinforce it near the surface. The faster the oscillation, the more the current is squeezed into a thin surface layer of thickness δ — the skin depth.

plasma physics
sin⁡2α1B1=sin⁡2α2B2\frac{\sin^2\alpha_1}{B_1} = \frac{\sin^2\alpha_2}{B_2}

Physics of the Aurora

A charged particle spirals around a magnetic field line. As it moves toward a pole the field gets stronger and tighter, and conservation of the magnetic moment (the first adiabatic invariant) forces its spiral to widen in pitch until it stops and reflects — a magnetic mirror. Particles with small enough pitch angle slip through the mirror and slam into the upper atmosphere, exciting oxygen and nitrogen that glow green and red.

electromagnetic waves
fc=c2a(TE10)f_c = \frac{c}{2a}\quad(\text{TE}_{10})

Waveguide Cutoff Frequency

A guided wave can be pictured as a plane wave zig-zagging between the conducting walls. To satisfy the boundary conditions, exactly a half-wavelength (for TE₁₀) must fit across the width a. If the free-space wavelength is too long — frequency too low — it can't fit at any bounce angle, so the wave can't propagate and instead decays exponentially: it's evanescent.

theoretical
e g=2πnℏ=nh  ⇒  g=nhee\,g = 2\pi n \hbar = n h \;\Rightarrow\; g = \frac{n h}{e}

Dirac Magnetic Monopole Quantization

Wrap an electron's quantum wavefunction around a monopole and its phase must come back to itself — single-valuedness. The phase picked up is set by the magnetic flux from the monopole; demanding it be a multiple of 2π forces the product of electric and magnetic charge to be quantized. Turn it around: even one monopole makes every electric charge a multiple of a basic unit.

circuits
∑kVk=0\sum_{k} V_k = 0

Kirchhoff's Voltage Law

Voltage is electrical height. Go around a closed loop and every rise (battery) must be exactly cancelled by the drops (resistors) — you can't gain energy by walking in a circle. It is energy conservation wearing a circuit hat.

induction
E=−LdIdt\mathcal{E} = -L\frac{dI}{dt}

Self-Inductance

A coil stores energy in its magnetic field and hates sudden change. Push current in and it pushes back; cut current and it strains to keep it flowing — electrical inertia, the magnetic analogue of mass.

ac circuits
ω0=1LC\omega_0 = \frac{1}{\sqrt{LC}}

RLC Series Resonance

An inductor and capacitor swap energy back and forth like a pendulum swapping speed and height. At one special frequency their opposing reactances cancel, the circuit looks purely resistive, and the current rings loudest.

ac circuits
XL=ωL=2πfLX_L = \omega L = 2\pi f L

Inductive Reactance

The faster AC tries to change, the harder a coil's back-EMF pushes against it. Reactance is that opposition — zero for DC, growing straight-line with frequency, like air resistance that gets worse the faster you go.

capacitance
C=QVC = \frac{Q}{V}

Capacitance

Capacitance is how much charge a conductor pair will swallow for each volt you push — a bucket size for electric charge. Bigger plates, closer together, with the right filling, hold more per volt.

circuits
τ=RC\tau = RC

RC Time Constant

Resistance throttles the current that fills (or drains) the capacitor, so their product sets one natural tick. In one τ the gap to the final value shrinks by 63%; after ~5τ it's essentially done.

ac circuits
f0=12πLCf_0 = \frac{1}{2\pi\sqrt{LC}}

LC Resonant Frequency

The capacitor stores energy in its electric field, the inductor in its magnetic field, and they trade it endlessly. Their sizes set the period exactly as mass and spring set a pendulum's swing.

circuits
IL=VthRth+RLI_L = \frac{V_{th}}{R_{th} + R_L}

Thévenin's Theorem

Any linear two-terminal network, no matter how complex, behaves at its terminals exactly like a single voltage source V_th in series with a single resistor R_th. Replace the whole box with two numbers.

circuits
R1R2=R3R4\frac{R_1}{R_2} = \frac{R_3}{R_4}

Wheatstone Bridge

Two voltage dividers fed from the same source; when their midpoints sit at the same voltage, no current crosses the bridge. Balance turns an unknown resistor into a ratio of three known ones.

induction
L=μ0N2AℓL = \frac{\mu_0 N^2 A}{\ell}

Solenoid Inductance

Each turn both makes flux and catches it. Double the turns and you double the field AND double the linkage — hence the square. Pack them on a long, fat coil for the most inductance.

magnetostatics
τ=NBIAsin⁡θ\tau = N B I A \sin\theta

How Electric Motors Spin

A current loop is a magnet; a field twists it to align, just like a compass needle. Keep flipping the current (commutator) so it never settles, and the loop spins forever — that's a motor.

induction
ε=NBAωsin⁡(ωt)\varepsilon = N B A \omega \sin(\omega t)

How Generators Make Electricity

As a coil rotates, the flux through it rises and falls sinusoidally; Faraday's law says the EMF is the rate of change of that flux — so the output is a sine wave whose peak grows with how fast you spin.

induction
fhum=2flinef_{hum} = 2 f_{line}

Why Transformers Hum

The core's iron physically stretches in a magnetic field (magnetostriction), and the strain depends on B². Since B² peaks twice per AC cycle — once each polarity — the mechanical vibration, and the sound, comes out at double the line frequency.

ac circuits
Z0=LCZ_0 = \sqrt{\frac{L}{C}}

Transmission-Line Characteristic Impedance

A long line is a ladder of tiny inductors and capacitors. A wave entering it sees a constant ratio of voltage to current — √(L/C) — set by the per-metre L and C. Match a load to it and there's no reflection.

Thermodynamics

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gas laws
PV=nRTPV = nRT

Ideal Gas Law

Pressure times volume is proportional to temperature for a fixed amount of gas.

energy conservation
ΔU=Q−W\Delta U = Q - W

First Law of Thermodynamics

Energy in (heat) minus energy out (work) equals the change stored inside.

heat transfer
q=−kdTdxq = -k \frac{dT}{dx}

Fourier's Law of Heat Conduction

Heat flows from hot to cold, faster through better conductors and steeper gradients.

thermal properties
ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T

Linear Thermal Expansion

Materials grow longer when heated — by an amount proportional to their length and temperature rise.

heat engines
η=1−TCTH\eta = 1 - \frac{T_C}{T_H}

Carnot Efficiency

No engine can beat the efficiency set by the ratio of its cold and hot reservoir temperatures.

radiation
P=σAT4P = \sigma A T^4

Stefan-Boltzmann Law

Hot objects radiate energy as light — and the power skyrockets with temperature (fourth power!).

entropy
ΔS=∫δQrevT\Delta S = \int \frac{\delta Q_{\text{rev}}}{T}

Entropy Change

Entropy measures how much energy has spread out — it always increases in the universe overall.

statistical mechanics
f(v)=4πn(m2πkBT)3/2v2e−mv22kBTf(v) = 4\pi n \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}

Maxwell-Boltzmann Speed Distribution

Gas molecules have a spread of speeds — most cluster near a peak, with a long tail of fast outliers.

statistical mechanics
S=kBln⁡WS = k_B \ln W

Boltzmann Entropy

Entropy counts the number of microscopic ways a macroscopic state can be realized — more ways means higher entropy.

phase transitions
dPdT=LTΔv\frac{dP}{dT} = \frac{L}{T \Delta v}

Clausius-Clapeyron Equation

Steeper vapor-pressure curve where latent heat is high — phase boundary tilt is governed by entropy of vaporization.

calorimetry
Cv=(∂U∂T)VC_v = \left(\frac{\partial U}{\partial T}\right)_V

Heat Capacity at Constant Volume

Heat capacity at constant volume measures the energy needed to raise temperature when no work is done — pure internal energy storage.

calorimetry
Cp=(∂H∂T)PC_p = \left(\frac{\partial H}{\partial T}\right)_P

Heat Capacity at Constant Pressure

At constant P, some heat goes into work done expanding the gas — so C_p is always larger than C_v by exactly R (for ideal gases).

thermodynamic potentials
G=H−TSG = H - TS

Gibbs Free Energy

Gibbs free energy is the maximum non-PV work extractable from a system at constant T and P — and it must decrease for spontaneous processes.

thermodynamic potentials
F=U−TSF = U - TS

Helmholtz Free Energy

Helmholtz energy is the maximum work extractable at constant T and V. Like Gibbs but without the PV term — appropriate for sealed rigid containers.

blackbody radiation
λmax⁡T=b\lambda_{\max} T = b

Wien's Displacement Law

Hotter objects emit shorter-wavelength radiation; the peak wavelength is inversely proportional to temperature.

blackbody radiation
Bλ(T)=2hc2λ51ehc/λkBT−1B_\lambda(T) = \frac{2 h c^2}{\lambda^5} \frac{1}{e^{hc/\lambda k_B T} - 1}

Planck Radiation Law

Quantized photon energies cap the high-frequency emission of a blackbody, solving the ultraviolet catastrophe.

Statistical Mechanics
Z=∑ie−βEi,β=1kBTZ = \sum_i e^{-\beta E_i}, \quad \beta = \frac{1}{k_B T}

Canonical Partition Function

Z counts microstates weighted by their Boltzmann suppression; hotter systems explore more states.

Quantum Statistical Mechanics
f(E)=1e(E−μ)/(kBT)+1f(E) = \frac{1}{e^{(E-\mu)/(k_B T)} + 1}

Fermi-Dirac Distribution

Each quantum state can hold at most one fermion; the chemical potential μ acts as a 'cut-off' — states below are occupied, above are empty.

Quantum Statistical Mechanics
⟨ni⟩=1eβ(εi−μ)−1\langle n_i \rangle = \frac{1}{e^{\beta(\varepsilon_i - \mu)} - 1}

Bose-Einstein Distribution

Bosons actively prefer to share states — the more particles already in a state, the more likely the next one joins (stimulated emission).

Statistical Mechanics
S=NkB[ln⁡(VN(4πmE3Nh2)3/2)+52]S = Nk_B\left[\ln\left(\frac{V}{N}\left(\frac{4\pi m E}{3Nh^2}\right)^{3/2}\right) + \frac{5}{2}\right]

Sackur-Tetrode Equation

Entropy counts accessible microstates in units of h³ per degree of freedom; the 5/2 is the 3/2 kinetic + 1 from volume/particle indistinguishability.

Statistical Mechanics
Ω=−kBTln⁡Ξ,Ξ=∑N,ie−β(EN,i−μN)\Omega = -k_B T \ln \Xi, \quad \Xi = \sum_{N,i} e^{-\beta(E_{N,i} - \mu N)}

Grand Canonical Potential (Landau Free Energy)

Ω is the free energy cost of a system that can bleed particles; minimise it to find the equilibrium particle number at given T and μ.

Statistical Mechanics
⟨qi∂H∂qi⟩=kBT\left\langle q_i \frac{\partial H}{\partial q_i} \right\rangle = k_B T

Equipartition Theorem

Classical thermal fluctuations distribute energy democratically: every quadratic term in H gets the same share k_BT/2.

Non-equilibrium Statistical Mechanics
SFF(ω)=2kBT Re[χ(ω)]S_{FF}(\omega) = 2k_B T\, \text{Re}[\chi(\omega)]

Fluctuation-Dissipation Theorem

A system that dissipates energy (resistance) must also fluctuate spontaneously (noise) at the same rate — you can't have one without the other at finite temperature.

Statistical Mechanics
⟨Ekin⟩=−12⟨Epot⟩\langle E_{\text{kin}} \rangle = -\frac{1}{2}\langle E_{\text{pot}} \rangle

Virial Theorem (Statistical Mechanics)

In a bound system, twice the kinetic energy always equals the negative of the potential energy — energy is partitioned by the power law of the force.

Kinetic Theory
f(v)=4πn(m2πkBT)3/2v2exp⁡ ⁣(−mv22kBT)f(v) = 4\pi n\left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 \exp\!\left(-\frac{mv^2}{2k_B T}\right)

Maxwell Speed Distribution (3D)

The v² factor (surface area of velocity sphere) competes with the Boltzmann suppression e^{−mv²/2k_BT} to create a peaked distribution; the peak shifts right as T rises.

Laws of Thermodynamics
ΔSuniverse=ΔSsys+ΔSsurr≥0\Delta S_{\text{universe}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \geq 0

Second Law of Thermodynamics

Isolated systems drift toward disorder; entropy only ever goes up.

Laws of Thermodynamics
(A∼C)∧(B∼C)  ⟹  (A∼B)(A \sim C) \wedge (B \sim C) \implies (A \sim B)

Zeroth Law of Thermodynamics

If two things each match a thermometer, they match each other.

Laws of Thermodynamics
lim⁡T→0S(T)=S0=kBln⁡g0\lim_{T \to 0} S(T) = S_0 = k_B \ln g_0

Third Law of Thermodynamics

Cool toward 0 K and entropy freezes to a constant you can never fully remove.

Thermodynamic Processes
PVγ=const,γ=CpCvP V^{\gamma} = \text{const}, \quad \gamma = \frac{C_p}{C_v}

Adiabatic Process

No heat in or out, so work done on the gas becomes its internal energy.

Equations of State
(P+an2V2)(V−nb)=nRT\left(P + \frac{a n^2}{V^2}\right)(V - n b) = n R T

Van der Waals Equation

Real molecules take up space (b) and attract each other (a), bending the ideal gas law.

Heat Engines & Cycles
COPref=QcW=QcQh−Qc≤TcTh−Tc\text{COP}_{\text{ref}} = \frac{Q_c}{W} = \frac{Q_c}{Q_h - Q_c} \leq \frac{T_c}{T_h - T_c}

How a Refrigerator Moves Heat Uphill

Spend work to pump heat from cold to hot; the colder the inside, the more work each joule costs.

Real Gases & Phase Changes
μJT=(∂T∂P)H=1Cp[T(∂V∂T)P−V]\mu_{\text{JT}} = \left(\frac{\partial T}{\partial P}\right)_H = \frac{1}{C_p}\left[T\left(\frac{\partial V}{\partial T}\right)_P - V\right]

Joule-Thomson Effect

Throttle a real gas at constant enthalpy and it cools — below the inversion temperature.

Phase Transitions
dTmdP=T ΔVmeltLf≈−7.4×10−3 K/atm\frac{dT_m}{dP} = \frac{T\,\Delta V_{\text{melt}}}{L_f} \approx -7.4\times10^{-3}\ \text{K/atm}

Why Ice Skates Glide

A thin film of liquid water, not pressure alone, lets the blade slide.

Statistical Mechanics
Werase≥kBTln⁡2 per bitW_{\text{erase}} \geq k_B T \ln 2 \ \text{per bit}

Maxwell's Demon and Information

Sorting molecules looks free, but erasing the demon's memory pays the full entropy bill.

Statistical Mechanics
ni+1neni=2gi+1gi(2πmekBTh2)3/2e−χ/kBT\frac{n_{i+1} n_e}{n_i} = \frac{2 g_{i+1}}{g_i}\left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} e^{-\chi/k_B T}

Saha Ionization Equation

Hotter, thinner gas is more ionized; the Boltzmann factor e^(−χ/kT) sets the balance.

Heat Engines & Cycles
ηOtto=1−1rγ−1,r=V1V2\eta_{\text{Otto}} = 1 - \frac{1}{r^{\gamma - 1}}, \quad r = \frac{V_1}{V_2}

Otto Cycle Efficiency

Efficiency depends only on how much you compress the fuel-air mix before ignition.

Statistical Mechanics
ΔSmix=−NkB∑ixiln⁡xi  →  0 for identical species\Delta S_{\text{mix}} = -N k_B \sum_i x_i \ln x_i \;\to\; 0 \text{ for identical species}

Gibbs Paradox

Mixing distinct gases adds entropy; mixing identical ones adds none.

Waves & Optics

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refraction
n1sin⁡θ1=n2sin⁡θ2n_1 \sin\theta_1 = n_2 \sin\theta_2

Snell's Law

Light bends toward the normal when entering a denser medium.

reflection
θi=θr\theta_i = \theta_r

Law of Reflection

Light bounces off a mirror at the same angle it arrives.

wave mechanics
v=fλv = f\lambda

Wave Speed Equation

Wave speed equals how many wavelengths pass a point each second.

geometric optics
1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Thin Lens Equation

Reciprocals of object and image distances always add up to the lens power.

interference
dsin⁡θ=mλd \sin\theta = m\lambda

Young's Double Slit Interference

Two overlapping wave sources create bright and dark bands.

diffraction
asin⁡θ=mλa \sin\theta = m\lambda

Single Slit Diffraction

A narrow slit spreads light into a pattern of bright and dark bands.

polarization
tan⁡θB=n2n1\tan\theta_B = \frac{n_2}{n_1}

Brewster's Angle

At one special angle, reflected light becomes perfectly polarized.

wave mechanics
f′=fv+vov−vsf' = f \frac{v + v_o}{v - v_s}

Doppler Effect

Moving toward a wave source compresses waves; moving away stretches them.

polarization
I=I0cos⁡2θI = I_0 \cos^2\theta

Malus's Law

Only the component of the electric field aligned with the polarizer axis gets through; the rest is absorbed.

diffraction
θmin=1.22λD\theta_{min} = 1.22 \frac{\lambda}{D}

Rayleigh Criterion

Two point sources are just resolvable when the central maximum of one falls on the first dark ring of the other.

diffraction
dsin⁡θ=mλd \sin\theta = m \lambda

Diffraction Grating Equation

Light constructively interferes whenever the path difference between adjacent slits equals an integer number of wavelengths.

geometric optics
1f=(n−1)(1R1−1R2)\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)

Lensmaker's Equation

Focal length depends on how strongly light bends at each curved surface, governed by the index step and surface curvature.

geometrical optics
1f=1v+1u\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

Mirror Equation

Object distance, image distance, and focal length lock together — change one and the others must rearrange.

electromagnetic waves
n=cvn = \frac{c}{v}

Refractive Index Definition

The refractive index tells you the factor by which light slows down inside a material.

electromagnetic waves
I=12cϵ0E02I = \frac{1}{2} c \epsilon_0 E_0^2

Intensity of an Electromagnetic Wave

Intensity scales with the square of the electric field amplitude — doubling the field quadruples the power flow.

quantum optics
E=hf=hcλE = h f = \frac{h c}{\lambda}

Photon Energy (Planck-Einstein Relation)

Higher frequency means each light particle carries more punch — color literally equals energy.

wave mechanics
∂2y∂t2=v2∂2y∂x2\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}

Wave Equation

The curvature of the wave in space drives its acceleration in time — sharper bends snap back faster.

wave mechanics
fn=nv2Lf_n = \frac{n v}{2 L}

Standing Wave Frequencies

Only wavelengths that fit a whole number of half-wave humps between the two fixed ends survive — those are the notes.

refraction
θc=arcsin⁡(n2n1)\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)

Critical Angle (Total Internal Reflection)

Past a certain angle, light heading from dense to rare can't refract out — it gets reflected back perfectly.

superposition
fbeat=∣f1−f2∣f_{beat} = |f_1 - f_2|

Beat Frequency

Two waves of nearly equal frequency drift in and out of phase. When they line up they add (loud); when opposed they cancel (silent). The loudness pulses at the difference frequency.

geometric optics
m=−didom = -\frac{d_i}{d_o}

Lateral Magnification

Magnification compares image height to object height. A negative sign means the image is inverted; its magnitude tells you how many times larger or smaller the image is.

interference
2nt=(m+12)λ2 n t = \left(m + \tfrac{1}{2}\right)\lambda

Thin-Film Interference

Light reflecting off the top and bottom surfaces of a film travels different path lengths. A half-wave phase flip at the top surface makes the half-integer condition give bright reflection — and different colors satisfy it at different thicknesses.

diffraction
nλ=2dsin⁡θn\lambda = 2 d \sin\theta

Bragg's Law

Atomic planes act like a stack of partial mirrors. Reflections from successive planes add up only when their extra path length is a whole number of wavelengths — that condition pins the angle for each wavelength.

geometric optics
NA=nsin⁡θmaxNA = n \sin\theta_{max}

Numerical Aperture

Numerical aperture is the sine of the widest cone of light an optical system can gather or emit, scaled by the medium's index. Bigger NA means more light collected and finer resolving power.

wave propagation
vg=dωdkv_g = \frac{d\omega}{dk}

Group Velocity

A localized wave packet is a sum of many frequencies. The crests move at the phase velocity, but the packet's envelope — where the energy and information ride — moves at the group velocity, the slope of the dispersion curve.

interferometry
ΔN=2 Δdλ\Delta N = \frac{2\,\Delta d}{\lambda}

Michelson Interferometer Fringe Shift

Light in one arm makes a round trip, so moving the mirror by Delta d changes the path by 2*Delta d. Each whole wavelength of extra path slides the fringe pattern by exactly one fringe — turning displacement into a count.

wave propagation
v=rt⇒r=v tv = \frac{r}{t} \quad\Rightarrow\quad r = v\,t

Huygens' Principle

Every point a wave reaches becomes the launch point of a new little wave. Add all those wavelets up and their leading edge is where the wave goes next. Where speed changes, the wavelets shrink and the front tilts — that bend is refraction.

wave kinematics
vp=ωk=fλv_p = \frac{\omega}{k} = f\lambda

Phase Velocity

Pick one crest and follow it. In one period T it advances exactly one wavelength lambda, so its speed is lambda/T = f*lambda. Equivalently, the point of constant phase moves at omega/k.

geometric optics
OPL=n d\text{OPL} = n\,d

Optical Path Length

Inside glass light crawls slower, so a 1 mm slab packs in more wave crests than 1 mm of vacuum. Multiplying the real thickness d by the refractive index n converts physical distance into 'how much phase the wave accumulated' — that is what interference cares about.

acoustics
L=10 log⁡10 ⁣(II0)L = 10\,\log_{10}\!\left(\frac{I}{I_0}\right)

Sound Intensity Level (Decibels)

Ears respond to ratios, not differences. The decibel measures how many factors of ten your sound sits above the faintest audible intensity I0 = 1e-12 W/m^2. Ten times the power is +10 dB; a hundred times is +20 dB.

acoustics
Z=ρ cZ = \rho\,c

Acoustic Impedance

Impedance measures how hard a medium pushes back when sound tries to move it: dense, stiff media (high rho, high c) resist more. When a wave meets a different impedance, part of it reflects — the bigger the mismatch, the more bounces back.

dispersion
n(λ)=A+Bλ2n(\lambda) = A + \frac{B}{\lambda^2}

Optical Dispersion

A material's electrons resonate at ultraviolet frequencies. Visible light pushes those electrons, and the closer its frequency is to resonance (toward blue), the more strongly the medium responds — raising the index. Shorter wavelength means larger n, so blue bends more.

interference
rm=m λ Rr_m = \sqrt{m\,\lambda\,R}

Newton's Rings

The air gap under a curved lens thickens with distance from the contact point. Wherever the round-trip through the gap equals a whole number of wavelengths you get a dark ring (the half-wave reflection flip makes the center dark). Because the gap grows quadratically with radius, the rings crowd together outward.

scattering
I∝1λ4I \propto \frac{1}{\lambda^4}

Why the Sky Is Blue (Rayleigh)

A passing light wave jiggles the electrons in a tiny air molecule, which then re-radiate in all directions. Higher-frequency (blue) light jiggles them harder, and the re-radiated power scales as frequency to the fourth — so blue scatters about 5-10 times more than red, painting the whole sky blue.

geometric optics
θ=4 arcsin⁡ ⁣(sin⁡θin)−2θi\theta = 4\,\arcsin\!\left(\frac{\sin\theta_i}{n}\right) - 2\theta_i

How Rainbows Form

Each ray that enters a spherical drop bends in, bounces off the far wall, and bends out. Rays piling up at the angle of maximum deviation make a bright cone. Because the drop refracts red and blue by slightly different amounts, the cone's edge splits into colors — and you see an arc.

geometric optics
n(y) sin⁡θ(y)=constn(y)\,\sin\theta(y) = \text{const}

Why Mirages Appear on Hot Roads

Refractive index drops as the air gets hotter near the ground. A nearly horizontal ray heading down keeps hitting ever-less-dense air, bending a little more each time until it turns and heads back up — total internal reflection spread out smoothly. Your brain projects it straight back, placing sky on the ground.

reflection and refraction
Rs=∣n1cos⁡θi−n2cos⁡θtn1cos⁡θi+n2cos⁡θt∣2R_s = \left|\frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}\right|^2

Fresnel Equations

At a boundary the wave's electric field must stay continuous. Solving that boundary condition for the two polarizations (s = perpendicular, p = parallel) gives how much reflects. The p-component vanishes entirely at Brewster's angle; near grazing both shoot up to total reflection.

variational optics
δ∫ABn(s) ds=0\delta \int_A^B n(s)\,ds = 0

Fermat's Principle of Least Time

Among all conceivable routes from A to B, light takes the one where small wiggles do not change the travel time to first order. Neighboring paths then arrive nearly in phase and reinforce; all others interfere away. Snell's and the reflection law are just this condition written out.

Quantum

47
wave particle duality
λ=hp\lambda = \frac{h}{p}

de Broglie Wavelength

Momentum and wavelength are inversely related through Planck's constant — big things have unmeasurably tiny wavelengths.

quantization
E=hfE = hf

Photon Energy (Planck Relation)

Light energy is quantized: each photon carries a fixed packet of energy set by its frequency.

quantization
Kmax=hf−ϕK_{max} = hf - \phi

Photoelectric Effect

A photon gives all its energy to one electron; the work function is the minimum escape cost.

foundations
Δx Δp≥ℏ2\Delta x \, \Delta p \geq \frac{\hbar}{2}

Heisenberg Uncertainty Principle

Position and momentum are conjugate — pinning one down spreads the other.

atomic physics
En=−13.6 eVn2E_n = -\frac{13.6 \text{ eV}}{n^2}

Bohr Hydrogen Energy Levels

Bound electrons can only sit on a quantized energy ladder; jumping down emits a photon.

foundations
iℏ∂Ψ∂t=H^Ψi\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi

Time-Dependent Schrödinger Equation

The Hamiltonian generates time evolution of the wavefunction in complex Hilbert space.

bound states
En=n2π2ℏ22mL2E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}

Particle in a 1D Box

Standing waves must fit inside the box; only integer half-wavelengths are allowed.

relativistic qm
(iγμ∂μ−mc/ℏ)ψ=0(i\gamma^\mu \partial_\mu - mc/\hbar)\psi = 0

Dirac Equation

A first-order relativistic wave equation whose solutions naturally carry spin and antiparticles.

wave mechanics
H^ψ=Eψ\hat{H}\psi = E\psi

Time-Independent Schrödinger Equation

A wavefunction is allowed only if applying the Hamiltonian gives back the same wavefunction scaled by a number — that number is the energy.

wave mechanics
En=ℏω(n+12)E_n = \hbar\omega\left(n + \tfrac{1}{2}\right)

Quantum Harmonic Oscillator

Energy comes in equal steps of hbar*omega, with a built-in floor of hbar*omega/2 — the zero-point motion required by Heisenberg.

atomic spectra
1λ=RH(1n12−1n22)\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Light emerges when an electron drops between two energy rungs — the wavelength is set by the difference of two inverse squares.

many body
Ψ(r1,r2)=−Ψ(r2,r1)\Psi(\mathbf{r}_1,\mathbf{r}_2) = -\Psi(\mathbf{r}_2,\mathbf{r}_1)

Pauli Exclusion Principle

Swap two identical fermions and the wavefunction flips sign — so the wavefunction vanishes if they share the same state.

spin
S^i=ℏ2σi\hat{S}_i = \tfrac{\hbar}{2}\sigma_i

Spin-1/2 Operators

Three 2×2 matrices encode every spin-1/2 measurement — they don't commute, which is why spin in x and y can't be known simultaneously.

wave mechanics
T≈exp⁡ ⁣(−2ℏ∫ab2m(V(x)−E) dx)T \approx \exp\!\left(-\frac{2}{\hbar}\int_a^b\sqrt{2m\bigl(V(x)-E\bigr)}\,dx\right)

Quantum Tunneling Probability

The wavefunction decays exponentially inside a barrier — make the barrier thinner or shorter and a measurable tail emerges on the far side.

spin
Fz=μz ∂Bz∂zF_z = \mu_z\,\frac{\partial B_z}{\partial z}

Stern–Gerlach Deflection

A magnetic dipole in a field gradient feels a force along the gradient; quantum spin offers only two values of mu_z, so the beam splits in two.

open systems
ρ^=∑ipi ∣ψi⟩⟨ψi∣\hat{\rho} = \sum_i p_i\,|\psi_i\rangle\langle\psi_i|

Density Matrix

Replace one wavefunction with a weighted bookkeeper of many — diagonal entries are probabilities, off-diagonal entries are coherences that decoherence destroys.

foundations
P(x) dx=∣ψ(x)∣2 dxP(x)\,dx = |\psi(x)|^2\,dx

Born Rule

The squared magnitude of the wavefunction is the probability map for measurement outcomes.

operator algebra
[x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar

Canonical Commutation Relation

Order matters: measuring x then p differs from p then x by exactly iℏ — the seed of uncertainty.

angular momentum
∣L∣=l(l+1) ℏ,Lz=mℏ|\mathbf{L}| = \sqrt{l(l+1)}\,\hbar, \qquad L_z = m\hbar

Quantization of Angular Momentum

Angular momentum comes in rungs of ℏ: its length is √(l(l+1))ℏ and its z-shadow is mℏ.

spin dynamics
ωL=γB\omega_L = \gamma B

Larmor Precession

A spin in a magnetic field precesses like a gyroscope, at a rate exactly proportional to the field.

dynamics
d⟨x^⟩dt=⟨p^⟩m,d⟨p^⟩dt=−⟨∂V∂x⟩\frac{d\langle \hat{x}\rangle}{dt} = \frac{\langle \hat{p}\rangle}{m}, \qquad \frac{d\langle \hat{p}\rangle}{dt} = -\left\langle \frac{\partial V}{\partial x}\right\rangle

Ehrenfest Theorem

Quantum averages obey Newton-like equations — the wavepacket's center moves classically.

two level systems
Pe(t)=Ω2Ω2+Δ2 sin⁡2 ⁣(12Ω2+Δ2  t)P_e(t) = \frac{\Omega^2}{\Omega^2 + \Delta^2}\,\sin^2\!\left(\tfrac{1}{2}\sqrt{\Omega^2+\Delta^2}\;t\right)

Rabi Oscillations

A driven two-level system cycles between ground and excited state; detuning caps the swing.

entanglement
S=E(a,b)−E(a,b′)+E(a′,b)+E(a′,b′),∣S∣≤2S = E(a,b) - E(a,b') + E(a',b) + E(a',b'), \qquad |S| \le 2

Bell–CHSH Inequality

Local hidden variables cap a four-correlation sum at 2; entangled particles push it to 2√2.

particle physics
Lint=−e ψˉγμψAμ\mathcal{L}_{\text{int}} = -e\,\bar{\psi}\gamma^{\mu}\psi A_{\mu}

Standard Model Interactions

Every particle reaction in nature is built from a handful of vertices — an electron emitting a photon, a quark emitting a W boson, a gluon splitting. Glue these elementary moves together and you get beta decay, annihilation, Compton scattering, pair production. The bookkeeping rules are absolute: electric charge, lepton number, and baryon number in must equal out. If a reaction conserves them all, somewhere in the universe it happens.

foundations
iℏ ∂∂t Ψ(r,t)=H^ Ψ(r,t)i\hbar\,\frac{\partial}{\partial t}\,\Psi(\mathbf{r},t) = \hat{H}\,\Psi(\mathbf{r},t)

The Schrödinger Equation

Newton's law tells a particle where to go next given a force; the Schrödinger equation tells a *wavefunction* how to evolve given an energy operator. The whole state of a quantum system lives in Ψ — a complex amplitude over every possible configuration — and the Hamiltonian Ĥ acts like a clock, rotating that amplitude forward in time. Everything you can ever measure is hidden inside |Ψ|². It is the F = ma of the quantum world: a single deterministic rule for an object that is anything but.

quantum optics
g(ν)=σ(ν) (N2−N1)g(\nu) = \sigma(\nu)\,(N_2 - N_1)

How a Laser Achieves Coherence

Normally a photon hitting an atom is more likely to be absorbed than to trigger emission, because the ground state is more populated — light dies out. Einstein showed that an excited atom struck by a photon can be stimulated to emit a *clone* photon: same frequency, phase and direction. To make light grow instead of fade you need more atoms up than down — a 'population inversion' — which can't happen at thermal equilibrium, so you pump energy in. Put that gain medium between two mirrors and the cloned photons bounce, stimulate more clones, and a coherent beam explodes out of the noise.

condensed matter
ψnk(r)=eik⋅r unk(r),unk(r+R)=unk(r)\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}\,u_{n\mathbf{k}}(\mathbf{r}),\qquad u_{n\mathbf{k}}(\mathbf{r}+\mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})

Why Metals Conduct (Band Theory)

Bring two atoms together and the Pauli principle splits each shared level into two. Bring 10²³ atoms together and each level splits into 10²³ sub-levels so closely spaced they form a continuous 'band'. Electrons fill these bands from the bottom up. If the topmost occupied band is only partly full, electrons sit right next to empty states and an electric field nudges them freely — a metal. If a band is completely full and the next is far above across a wide gap, no nearby empty states exist, electrons can't move, and you have an insulator. A small gap gives a semiconductor.

foundations
Δφ=qℏ∮A⋅dℓ=q ΦBℏ\Delta\varphi = \frac{q}{\hbar}\oint \mathbf{A}\cdot d\boldsymbol{\ell} = \frac{q\,\Phi_B}{\hbar}

Aharonov-Bohm Effect

Classically a charged particle feels only forces, and where the magnetic field B is zero there is no force — so nothing should happen. The Aharonov–Bohm effect says otherwise. Send electrons around both sides of a thin, perfectly shielded solenoid. Outside the solenoid B = 0 everywhere, yet the two paths enclose magnetic flux, and the vector potential A is not zero there. Each path picks up a quantum phase from A, and when the beams recombine their interference fringes *shift* — controlled entirely by flux the electrons never touched. It proved the potentials A and φ, long thought to be mere mathematical bookkeeping, are physically real in quantum mechanics.

foundations
∣Ψ−⟩=12(∣↑⟩A∣↓⟩B−∣↓⟩A∣↑⟩B)|\Psi^-\rangle = \frac{1}{\sqrt{2}}\big(|\uparrow\rangle_A|\downarrow\rangle_B - |\downarrow\rangle_A|\uparrow\rangle_B\big)

The EPR Paradox

Prepare two particles in a single entangled state — say total spin zero — and send them far apart. The pair has no definite individual spins; only the *correlation* is fixed. Measure particle A along any axis and you instantly know B will give the opposite, no matter the distance. EPR said: either the answer was secretly predetermined (a 'hidden variable' quantum theory left out — so it's incomplete), or measuring A really does reach across space to set B (which violates locality). Einstein bet on hidden variables. Bell later showed the two options make *different* numerical predictions — and experiments side with quantum mechanics, not Einstein.

foundations
K(b,a)=∫D[x(t)]  e iℏS[x(t)]K(b,a) = \int \mathcal{D}[x(t)]\; e^{\,\frac{i}{\hbar} S[x(t)]}

Path-Integral Formulation

Ask how a particle gets from A to B and quantum mechanics answers: it takes *every* path at once. Each conceivable trajectory contributes an arrow (a complex phase) of equal length but angle set by that path's classical action S divided by ℏ. Add up all the arrows. Wild, jagged paths have wildly varying phases that cancel by destructive interference; only paths near the one that makes S stationary add up in step. That surviving bundle *is* the classical trajectory — so Newton's principle of least action emerges as the place where quantum phases stop cancelling. It reformulates all of quantum mechanics as a single 'sum over histories.'

Foundations
[x^,p^]=x^p^−p^x^=iℏ[\hat{x},\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar

Canonical Commutation [x, p]

In classical physics, measuring position then momentum gives the same answer as momentum then position — order doesn't matter. In quantum mechanics it does: swap the order of x̂ and p̂ and you don't get zero, you get iℏ. That tiny leftover is the mathematical seed of the entire uncertainty principle. Because position and momentum operators refuse to commute, no state can have both sharply defined. The whole strangeness of quantum mechanics is encoded in this one non-zero bracket.

Foundations
⟨A^⟩=∫ψ∗ A^ ψ dx=⟨ψ∣A^∣ψ⟩\langle \hat{A}\rangle=\int \psi^*\,\hat{A}\,\psi\,dx=\langle\psi|\hat{A}|\psi\rangle

Expectation Value of an Observable

A quantum measurement gives random results, but repeat it on many identically prepared systems and the results have a definite average. The expectation value is that average — sandwich the operator between the wavefunction and its conjugate and integrate. For position it's literally the center of mass of |ψ|². It is not the 'expected' single outcome (you may never measure it) but the mean of a huge ensemble of measurements.

Foundations
∫−∞∞∣ψ(x)∣2 dx=1\int_{-\infty}^{\infty}|\psi(x)|^2\,dx=1

Wavefunction Normalization

The particle is somewhere, so the total probability of finding it anywhere must be exactly 1. Since |ψ(x)|² is the probability density, adding it up over all space has to equal one. This condition fixes the otherwise-free overall scale of the wavefunction. If your ψ integrates to some number N, just divide by √N and it's normalized. Any physically meaningful state must be square-integrable — plane waves that don't die off need special (delta-function) treatment.

Foundations
A^=A^†  ⇒  a∈R,  ⟨ϕm∣ϕn⟩=δmn\hat{A}=\hat{A}^{\dagger}\;\Rightarrow\; a\in\mathbb{R},\;\langle\phi_m|\phi_n\rangle=\delta_{mn}

Hermitian Operators & Real Eigenvalues

Measured quantities are always real numbers — you never read '3+2i volts' off a meter. Quantum mechanics guarantees this by demanding every observable be a Hermitian (self-adjoint) operator. Hermiticity forces the eigenvalues (the possible measurement outcomes) to be real and the eigenstates to be orthogonal, so different outcomes are perfectly distinguishable. It's the algebraic condition that makes operators safe to call 'observables.'

Foundations
⟨ϕ∣ψ⟩=∫ϕ∗(x)ψ(x) dx\langle\phi|\psi\rangle=\int\phi^*(x)\psi(x)\,dx

Dirac Bra-Ket Notation

Dirac split the bracket ⟨φ|ψ⟩ into a 'bra' ⟨φ| and a 'ket' |ψ⟩. A ket is a state vector; a bra is its dual, waiting to be paired with a ket to produce a number — their overlap, which measures how similar two states are. This notation hides the messy integrals and matrices: it works identically whether the state lives in an infinite-dimensional function space or a 2-level qubit. It's the universal shorthand of quantum mechanics.

Exactly Solvable Systems
a^=mω2ℏ(x^+imωp^),    En=ℏω(n+12)\hat{a}=\sqrt{\tfrac{m\omega}{2\hbar}}\Big(\hat{x}+\tfrac{i}{m\omega}\hat{p}\Big),\;\;E_n=\hbar\omega\left(n+\tfrac12\right)

Ladder (Creation/Annihilation) Operators

Instead of solving the harmonic oscillator's differential equation, Dirac found a shortcut: two operators that step you up and down the energy ladder. â lowers the energy by one quantum ℏω (destroys a quantum), ↠raises it (creates one). Apply â to the lowest state and you get zero — that floor is why there's a ½ℏω zero-point energy. The whole spectrum falls out from algebra alone, and the same trick builds all of quantum field theory's particles.

Angular Momentum & Spin
L^2∣l,m⟩=ℏ2l(l+1)∣l,m⟩,    L^z∣l,m⟩=ℏm∣l,m⟩\hat{L}^2|l,m\rangle=\hbar^2 l(l+1)|l,m\rangle,\;\;\hat{L}_z|l,m\rangle=\hbar m|l,m\rangle

Angular Momentum Eigenvalues

Angular momentum in quantum mechanics can't point just anywhere or take any size — it's quantized. The total magnitude squared comes in lumps ℏ²l(l+1), and its projection on any axis is a whole (or half) multiple of ℏ. Curiously, the magnitude √(l(l+1)) always exceeds the maximum projection l, so the vector can never fully align with an axis — it must always tilt, a direct fingerprint of the uncertainty principle for the angular-momentum components.

Uncertainty & Wave-Particle
ΔE Δt≳ℏ2\Delta E\,\Delta t\gtrsim\frac{\hbar}{2}

Energy-Time Uncertainty

A state that lasts only a short time cannot have a sharply defined energy. The shorter its lifetime Δt, the fuzzier its energy ΔE. This is why short-lived particles and excited atomic states have a 'width' — their energy is spread out. It also lets the vacuum briefly borrow energy to create virtual particles, as long as they vanish fast enough. Unlike position-momentum, time isn't an operator here; Δt is a characteristic timescale of change.

Foundations
lim⁡n→∞(quantum)=(classical)\lim_{n\to\infty}\text{(quantum)}=\text{(classical)}

Correspondence Principle

Quantum mechanics can't just contradict the classical physics we see every day — it must reproduce it in the right limit. Bohr's principle says: for large quantum numbers (highly excited states) or when actions are huge compared to ℏ, quantum predictions blur into classical ones. A hydrogen electron in orbit n=1000 radiates almost exactly like a classical charge going around a loop. It's the guardrail that kept early quantum theory tethered to known physics.

Exactly Solvable Systems
ktan⁡(ka)=κ  (even),k=2mE/ℏ,  κ=2m(V0−E)/ℏk\tan(ka)=\kappa\;(\text{even}),\quad k=\sqrt{2mE}/\hbar,\;\kappa=\sqrt{2m(V_0-E)}/\hbar

Finite Square Well

Trap a particle in a box with walls of finite height V₀ and two things change versus the infinite box: there are only finitely many bound states, and the wavefunction leaks past the walls, decaying exponentially into the 'forbidden' region. Matching the wiggly inside solution smoothly to the leaking outside solution only works at special energies — the transcendental condition k tan(ka)=κ. That leakage is the seed of quantum tunneling.

foundations
A^ ∣an⟩=an ∣an⟩\hat{A}\,|a_n\rangle = a_n\,|a_n\rangle

Operators and Observables

Every measurable quantity is a Hermitian operator; its eigenvalues are the only possible readings.

foundations
∣ψ⟩=c1∣ψ1⟩+c2∣ψ2⟩,∣c1∣2+∣c2∣2=1|\psi\rangle = c_1|\psi_1\rangle + c_2|\psi_2\rangle,\quad |c_1|^2 + |c_2|^2 = 1

Quantum Superposition

Add valid states with complex weights and you get another valid state that can interfere with itself.

foundations
j=ℏm Im ⁣(ψ∗∂ψ∂x)j = \frac{\hbar}{m}\,\mathrm{Im}\!\left(\psi^{*}\frac{\partial \psi}{\partial x}\right)

Probability Current Density

The imaginary part of psi-star times its gradient is the flow of probability, obeying a continuity law.

angular momentum
[L^x,L^y]=iℏ L^z[\hat{L}_x, \hat{L}_y] = i\hbar\,\hat{L}_z

Angular Momentum Operators

The three angular-momentum components don't commute, so only its magnitude and one axis are sharp together.

angular momentum
S^i=ℏ2 σi\hat{S}_i = \frac{\hbar}{2}\,\sigma_i

Pauli Spin Matrices

Three 2x2 matrices generate every spin-1/2 observable and every single-qubit rotation.

angular momentum
Ylm(θ,ϕ)=Nlm Plm(cos⁡θ) eimϕY_l^m(\theta,\phi) = N_l^m\,P_l^m(\cos\theta)\,e^{im\phi}

Spherical Harmonics

The eigenfunctions of angular momentum on a sphere set the lobed shapes of atomic orbitals.

hydrogen atoms
n=1,2,3,… ;  l=0,…,n−1;  ml=−l,…,+l;  ms=±12n=1,2,3,\dots;\ \ l=0,\dots,n-1;\ \ m_l=-l,\dots,+l;\ \ m_s=\pm\tfrac12

Atomic Quantum Numbers

Four integers (n, l, m_l, m_s) label every electron and, with Pauli's rule, build the periodic table.

Relativity

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special relativity
γ=11−v2/c2\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Lorentz Factor

Quantifies how much time, length, and mass distort as velocity approaches c.

special relativity
Δt=γΔt0\Delta t = \gamma \Delta t_0

Time Dilation

A moving clock ticks slower than a stationary one, as seen by an outside observer.

special relativity
L=L0/γL = L_0 / \gamma

Length Contraction

Objects in motion appear shorter along their direction of travel.

special relativity
p=γmvp = \gamma m v

Relativistic Momentum

Momentum diverges as velocity approaches c, preventing massive objects from reaching light speed.

special relativity
E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

Energy–Momentum Relation

The full relativistic relation linking total energy, momentum, and rest mass.

special relativity
u=u′+v1+u′v/c2u = \frac{u' + v}{1 + u'v/c^2}

Relativistic Velocity Addition

Velocities don't simply add at high speeds; the result never exceeds c.

special relativity
f=f01−β1+βf = f_0 \sqrt{\frac{1 - \beta}{1 + \beta}}

Relativistic Doppler Effect

Receding sources redshift, approaching sources blueshift — with a relativistic gamma correction.

general relativity
rs=2GMc2r_s = \frac{2GM}{c^2}

Schwarzschild Radius

The radius at which escape velocity equals c — the event horizon of a non-rotating black hole.

special relativity
E=mc2E = mc^2

Mass-Energy Equivalence

Mass and energy are interchangeable currencies; any object at rest carries an enormous reservoir of energy because c^2 is huge.

special relativity
K=(γ−1)mc2K = (\gamma - 1) m c^2

Relativistic Kinetic Energy

Kinetic energy is the extra energy a body gains by moving — it diverges as v approaches c, not 1/2 mv^2.

special relativity
x′=γ(x−vt),t′=γ ⁣(t−vxc2)x' = \gamma(x - vt), \quad t' = \gamma\!\left(t - \frac{vx}{c^2}\right)

Lorentz Transformation

Space and time mix between observers in relative motion — the geometry of spacetime is a hyperbolic rotation, not a Galilean shear.

general relativity
dτdt=1−2GMrc2\frac{d\tau}{dt} = \sqrt{1 - \frac{2GM}{rc^2}}

Gravitational Time Dilation

Gravity slows time. The deeper you are in a gravitational potential, the slower your clock runs compared to a distant observer.

special relativity
s2=−c2Δt2+Δx2+Δy2+Δz2s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2

Spacetime Interval

Spacetime has its own Pythagorean theorem — but with one minus sign. The interval is the only true 'distance' all observers agree on.

special relativity
dτ=dt1−v2/c2=dtγd\tau = dt\sqrt{1 - v^2/c^2} = \frac{dt}{\gamma}

Proper Time

Proper time is what you'd read on a wristwatch you carry with you — the invariant 'age' of any worldline.

general relativity
Δλλ=GMrc2\frac{\Delta\lambda}{\lambda} = \frac{GM}{rc^2}

Gravitational Redshift

Photons climbing out of a gravity well lose energy — their wavelength stretches and clocks at the bottom appear to tick slower.

black hole thermodynamics
TH=ℏc38πGMkBT_H = \frac{\hbar c^3}{8\pi G M k_B}

Hawking Temperature

Quantum effects at a black hole's event horizon make it radiate like a blackbody — the temperature is inversely proportional to its mass.

special relativity
τtraveler=τEarth1−v2c2\tau_{\text{traveler}} = \tau_{\text{Earth}}\sqrt{1 - \frac{v^2}{c^2}}

The Twin Paradox

Send one twin on a fast round trip and bring her home: she returns younger than the twin who stayed. There's no paradox — the situations aren't symmetric. The traveling twin must turn around, and that acceleration breaks the symmetry, so it is unambiguously she who logs less proper time along her bent path through spacetime.

special relativity
cos⁡θ′=cos⁡θ+β1+βcos⁡θ\cos\theta' = \frac{\cos\theta + \beta}{1 + \beta\cos\theta}

Relativistic Aberration of Light

Run fast enough and the sky rearranges itself: light that arrived from your sides crowds into the direction you're heading, like rain on a windshield slanting forward as you accelerate. At near-light speed almost the entire sky compresses into a bright spot dead ahead — the relativistic headlight effect.

special relativity
Iobs=δ 3+α Iemit,δ=1γ(1−βcos⁡θ)I_{\text{obs}} = \delta^{\,3+\alpha}\,I_{\text{emit}},\qquad \delta = \frac{1}{\gamma(1 - \beta\cos\theta)}

Relativistic Beaming (Doppler Boosting)

A source moving toward you doesn't just blue-shift — it gets dramatically brighter, because aberration funnels its photons forward, time dilation packs more of them per second, and the Doppler shift lifts each photon's energy. These three effects multiply into a steep δ⁴ dependence, so a jet pointed at you can outshine an identical one pointed away by factors of thousands.

special relativity
E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

Energy–Momentum Four-Vector

Energy and momentum aren't separate bookkeeping — they're the time and space components of one four-vector, just as duration and length are facets of spacetime. Different observers disagree on E and on p, but they all agree on the length of that four-vector, and that invariant length is the particle's rest mass. The mass is the part of energy-momentum nobody can boost away.

general relativity
h=2Gc4d Q¨h = \frac{2G}{c^4 d}\,\ddot{Q}

Gravitational-Wave Strain

Accelerating masses ripple spacetime itself, and those ripples stretch and squeeze every length they pass through by a fractional amount h — the strain. The catch is the factor c⁴ in the denominator: it makes h staggeringly tiny. Two merging black holes a billion light-years away wobble LIGO's 4 km arms by less than a thousandth the width of a proton, which is exactly what it detected.

general relativity
ΩLT=2GJc2r3\Omega_{\text{LT}} = \frac{2GJ}{c^2 r^3}

Frame Dragging (Lense–Thirring Effect)

A spinning mass doesn't just curve spacetime — it drags it around, winding the very fabric of space into a slow vortex like a spoon stirring honey. A gyroscope held nearby is gently twisted by the rotation even though no force touches it; near a spinning black hole the dragging becomes so violent that, inside the ergosphere, standing still is physically impossible.

general relativity
θ=4GMc2b\theta = \frac{4GM}{c^{2}b}

General Relativity: Light Bending & Curved Spacetime

Mass curves spacetime, and light follows the straightest possible path through that curved geometry — so starlight grazing the Sun bends by 1.75 arcseconds, exactly twice what Newton's gravity-on-light would give. The same curvature, turned up, gives gravitational lensing, Einstein rings, ripples in spacetime itself (gravitational waves), and at the extreme, black holes from which no path leads out.

general relativity
mi=mg  ⇒  g⃗local≡−a⃗m_i = m_g \;\Rightarrow\; \vec{g}_{\text{local}} \equiv -\vec{a}

Einstein Equivalence Principle

Inside a sealed elevator you cannot tell gravity from acceleration: a falling lab and a coasting rocket give identical physics. Because gravitational and inertial mass are the same number, everything falls at the same rate — so 'gravity' is locally just the geometry of an accelerating frame.

special relativity
ϕ=tanh⁡−1 ⁣(vc),ϕtot=ϕ1+ϕ2\phi = \tanh^{-1}\!\left(\frac{v}{c}\right),\qquad \phi_{\text{tot}} = \phi_1 + \phi_2

Rapidity (Additive Velocity Parameter)

Velocities don't add in relativity — but rapidity does. Define phi = arctanh(v/c) and successive boosts just sum like ordinary angles. A Lorentz boost is literally a hyperbolic rotation of spacetime, and rapidity is its rotation angle.

special relativity
Uμ=dxμdτ=γ (c, v⃗)U^{\mu} = \frac{dx^{\mu}}{d\tau} = \gamma\,(c,\ \vec{v})

Four-Velocity

Everything moves through spacetime at the speed of light. At rest, all your motion points along time; speed up and you trade time-motion for space-motion. The four-velocity is that fixed-length arrow tangent to your worldline, with magnitude always c.

special relativity
Δt′=γ ⁣(Δt−v Δxc2)\Delta t' = \gamma\!\left(\Delta t - \frac{v\,\Delta x}{c^2}\right)

Relativity of Simultaneity

Two events that happen 'at the same time' for you happen at different times for someone moving past you. Simultaneity is not absolute — it tilts with velocity. In a moving frame, the leading clock reads earlier (leading clocks lag), which is the real root of the twin and ladder paradoxes.

special relativity
(mc2)2=E2−(pc)2(mc^2)^2 = E^2 - (pc)^2

Invariant Mass (Energy-Momentum Relation)

Energy and momentum each change between frames, but the combination E^2 - (pc)^2 does not — it is a fixed invariant equal to (mc^2)^2. Mass is the 'length' of the energy-momentum four-vector, the same for every observer, and for a system it can exceed the sum of its parts' masses.

applications
Δtt=ΔUc2⏟GR, faster−v22c2⏟SR, slower≈+38 μs/day\frac{\Delta t}{t} = \underbrace{\frac{\Delta U}{c^2}}_{\text{GR, faster}} - \underbrace{\frac{v^2}{2c^2}}_{\text{SR, slower}} \approx +38\ \mu\text{s/day}

Why GPS Needs Both Relativities

A GPS satellite clock runs fast by 45 microseconds/day because it sits higher in Earth's gravity well (GR), and slow by 7 microseconds/day because it orbits fast (SR). The net +38 microseconds/day, uncorrected, would shoot your position off by ~11 km per day. GPS only works because engineers pre-tune the clocks for relativity.

general relativity
θE=4GMc2 DLSDLDS\theta_E = \sqrt{\frac{4GM}{c^2}\,\frac{D_{LS}}{D_L D_S}}

Gravitational Lensing & the Einstein Radius

Mass bends light, so a galaxy acts like a lens: a background source directly behind it smears into a glowing ring, and slightly off-axis into multiple arcs and images. The ring's radius — the Einstein radius — measures the lens's total mass, including dark matter you cannot otherwise see.

general relativity
Δt=2GMc3 ln⁡ ⁣(4r1r2b2)\Delta t = \frac{2GM}{c^3}\,\ln\!\left(\frac{4 r_1 r_2}{b^2}\right)

Shapiro Time Delay (Fourth Test of GR)

Light slows down (in coordinate time) as it passes deep in a gravity well — not because its local speed changes, but because spacetime is curved. A radar pulse grazing the Sun on its way to Venus and back returns up to ~240 microseconds late. Irwin Shapiro proposed this as the 'fourth classical test' of general relativity.

general relativity
Ωgeo=32 GMc2r3 (r⃗×v⃗)\Omega_{\text{geo}} = \frac{3}{2}\,\frac{GM}{c^2 r^3}\,(\vec{r}\times\vec{v})

Geodetic Precession (Gravity Probe B)

Carry a spinning gyroscope around a massive body and its axis slowly tips — not from any torque, but because the space it moves through is curved. After one orbit the gyroscope points in a slightly new direction. Gravity Probe B measured Earth's geodetic precession at 6.6 arcseconds per year, exactly as general relativity predicts.

general relativity
θ=4GM⊙c2R⊙=1.75′′\theta = \frac{4GM_\odot}{c^2 R_\odot} = 1.75''

Physics of the 1919 Eclipse

During the total solar eclipse of 29 May 1919, Eddington photographed stars near the darkened Sun and found their apparent positions shifted outward by ~1.75 arcseconds — twice the Newtonian value. Einstein's general relativity predicted exactly that doubling. Overnight, the result made Einstein world-famous.

general relativity
h=ΔLL∼10−21,fGW∝M˙chirph = \frac{\Delta L}{L} \sim 10^{-21},\qquad f_{GW} \propto \dot{M}_{\text{chirp}}

How LIGO Detected Gravitational Waves

Two black holes spiralling together stretch and squeeze spacetime, sending out gravitational waves. LIGO's 4 km laser arms change length by less than one-thousandth of a proton's width (strain ~10^-21). On 14 Sept 2015 LIGO caught the 'chirp' of two ~30-solar-mass black holes merging a billion light-years away.

general relativity
r±=GMc2(1±1−a∗2),a∗=JcGM2r_\pm = \frac{GM}{c^2}\left(1 \pm \sqrt{1-a_*^2}\right),\quad a_* = \frac{Jc}{GM^2}

Kerr Metric for Rotating Black Holes

A spinning black hole is not just a deeper well — it drags spacetime into a vortex. The Kerr solution has two horizons and an outer 'ergosphere' where nothing can stand still. Its spin sets a maximum (a_* = 1); push past it and the horizon vanishes, exposing a forbidden naked singularity.

special relativity
ds2=−c2 dt2+dx2+dy2+dz2ds^2 = -c^2\,dt^2 + dx^2 + dy^2 + dz^2

Minkowski Metric

The single rule that replaces Pythagoras in spacetime: time enters with a minus sign, so the 'distance' between two events can be negative, zero, or positive. That one sign is the whole of special relativity.

special relativity
A⋅B=ημνAμBν=−A0B0+A⃗⋅B⃗A\cdot B = \eta_{\mu\nu}A^\mu B^\nu = -A^0B^0 + \vec{A}\cdot\vec{B}

Four-Vectors and Invariants

Package the four numbers of an event, a velocity or a momentum into one object whose 'length' every observer agrees on. The bookkeeping of relativity collapses into a single dot product with one minus sign.

special relativity
c2t2=x2+y2+z2c^2 t^2 = x^2 + y^2 + z^2

Light Cone & Causal Structure

Every event sits at the tip of a double cone of light. Only what lies inside the past cone can have caused it, and only what lies inside the future cone can it affect. Everything else is 'elsewhere' — forever causally disconnected.

relativistic dynamics
E=γmc2=mc21−v2/c2E = \gamma m c^2 = \frac{mc^2}{\sqrt{1 - v^2/c^2}}

Relativistic Total Energy

A moving object's total energy is its rest energy scaled up by the Lorentz factor. As v approaches c, gamma runs to infinity, so no finite energy can ever push a massive body to light speed.

general relativity
ds2=−(1−rsr)c2dt2+dr21−rsr+r2dΩ2ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dt^2 + \frac{dr^2}{1-\frac{r_s}{r}} + r^2 d\Omega^2

Schwarzschild Metric

The exact geometry of empty space around any non-spinning mass. The (1 - r_s/r) factor squeezes time and stretches radial distance more and more as you approach the horizon, where it hits zero.

general relativity
rH=2GMc2wheregtt(rH)=0r_H = \frac{2GM}{c^2}\quad\text{where}\quad g_{tt}(r_H)=0

Event Horizon

A surface of no return, not a wall. At the horizon the outward escape speed equals light speed, so even light aimed straight out just hovers — everything inside is causally sealed off from the rest of the universe.

cosmology
v=H0 dv = H_0\, d

Hubble-Lemaître Law

Every galaxy is receding, and the farther one is, the faster it flees — not because we are special, but because space itself is stretching uniformly. The proportionality constant is the current expansion rate.

cosmology
1+z=a(tobs)a(temit)=λobsλemit1 + z = \frac{a(t_{\text{obs}})}{a(t_{\text{emit}})} = \frac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}}

Cosmological Redshift

Light does not lose energy climbing out of anything — the space it travels through simply stretches, and the wave stretches with it. The factor by which wavelengths grow equals the factor by which the universe grew during the light's journey.

general relativity
d2xμdτ2+Γαβμdxαdτdxβdτ=0\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0

Geodesic Equation

Gravity is not a force — it is the shape of spacetime. A freely falling body just coasts along the straightest possible path, and the Christoffel symbols encode how 'straight' bends when the geometry is curved.

general relativity
ds2=gμν dxμdxνds^2 = g_{\mu\nu}\,dx^\mu dx^\nu

The Metric Tensor

The metric is the ruler-and-clock field of the universe: at every point it tells you how to convert coordinate steps into real measured distances and times. In flat space it is Minkowski; where mass is present it warps, and that warping IS gravity.

special relativity
L=L0γ=L01−v2c2L = \frac{L_0}{\gamma} = L_0\sqrt{1-\frac{v^2}{c^2}}

Ladder (Pole-Barn) Paradox

The 'paradox' is really a lesson about simultaneity. Length contraction lets a long ladder fit inside a short barn in one frame; in the ladder's frame it never fits, because the two doors do not close at the same time. Nothing is contradictory — the frames just slice spacetime differently.

special relativity
fobs=fsrc1−β1+β(receding)f_{\text{obs}} = f_{\text{src}}\sqrt{\frac{1-\beta}{1+\beta}}\quad(\text{receding})

Longitudinal Doppler Shift

When a source moves straight toward or away from you, two effects stack: the classical bunching/stretching of wavefronts AND time dilation of the source's own clock. The square-root form is what you get when both are included exactly.

special relativity
NN0=exp⁡ ⁣(−dγ β c τ0)\frac{N}{N_0} = \exp\!\left(-\frac{d}{\gamma\,\beta\,c\,\tau_0}\right)

Why Cosmic Muons Reach the Ground

Muons are the most tangible proof of time dilation. Without relativity almost none would survive the fall; because their internal clock runs slow (or, in their frame, the atmosphere is contracted), a large fraction make it to sea level. The exponential decay law does the counting.

special relativity
fobs=fsrcγ=fsrc1−β2f_{\text{obs}} = \frac{f_{\text{src}}}{\gamma} = f_{\text{src}}\sqrt{1-\beta^2}

Transverse Doppler Effect

Even when a source flies straight ACROSS your line of sight — with zero radial velocity — you still see it redshifted, purely because its clock is time-dilated. There is no classical counterpart; this shift is relativity, naked.

Modern Physics

27
special relativity
E=mc2E = mc^2

Mass-Energy Equivalence

Mass is a highly concentrated form of energy; the speed of light squared is the conversion factor.

special relativity
γ=11−v2/c2\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Lorentz Factor

As speed approaches c, time stretches and lengths contract by the factor γ.

quantum mechanics
Kmax=hν−ϕK_{max} = h\nu - \phi

Photoelectric Effect

Light comes in quanta of energy hν; only photons above the work-function threshold can free electrons.

quantum mechanics
λ=hp\lambda = \frac{h}{p}

De Broglie Wavelength

Every particle has a wavelength inversely proportional to its momentum — heavy/fast things have wavelengths so small they're unobservable.

quantum mechanics
Δx⋅Δp≥ℏ2\Delta x \cdot \Delta p \geq \frac{\hbar}{2}

Heisenberg Uncertainty Principle

Position and momentum cannot both be sharply defined; nature enforces a fundamental fuzziness at small scales.

quantum mechanics
−ℏ22m∇2ψ+Vψ=Eψ-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi

Schrödinger Equation (Time-Independent)

The total energy operator acting on the wavefunction returns the energy times the same wavefunction — an eigenvalue problem for reality.

atomic physics
En=−13.6 eVn2E_n = -\frac{13.6\,\text{eV}}{n^2}

Bohr Energy Levels (Hydrogen)

Electrons in hydrogen are stuck on a ladder of negative energies; the gaps determine the colors of light atoms emit.

nuclear physics
N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}

Radioactive Decay Law

Each nucleus has a fixed probability per unit time of decaying, producing a smooth exponential decline in the population.

quantum mechanics
Δλ=hmec(1−cos⁡θ)\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Compton Scattering

Photons carry momentum p = h/λ. When one collides with a free electron, conservation of energy + momentum forces the photon to give up energy — its wavelength grows by an amount that depends only on the scattering angle.

atomic physics
a0=4πε0ℏ2mee2a_0 = \frac{4\pi\varepsilon_0 \hbar^2}{m_e e^2}

Bohr Radius

A balance between two demands: Coulomb attraction wants the electron as close to the proton as possible, but the uncertainty principle penalizes localization (smaller box → bigger momentum → bigger kinetic energy). The minimum-energy compromise sits at a₀.

atomic physics
1λ=RH(1n12−1n22)\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Each integer n labels an allowed electron energy level (the staircase). A photon's wavelength encodes the energy difference between two steps. Two integers → all hydrogen spectral lines.

quantum mechanics
B(λ,T)=2hc2λ5⋅1ehc/(λkBT)−1B(\lambda, T) = \frac{2hc^2}{\lambda^5}\cdot\frac{1}{e^{hc/(\lambda k_B T)} - 1}

Planck Radiation Law

Light energy is quantized in packets of size hν. At low frequency, packets are cheap → many emitted (Rayleigh-Jeans). At high frequency, each packet costs more than kT → exponentially suppressed. The peak balance gives the body's color.

quantum mechanics
λmax⁡T=b\lambda_{\max} T = b

Wien's Displacement Law

Hotter → more energetic photons → shorter peak wavelength. The exact peak comes from differentiating Planck's law and solving a transcendental equation; the answer is a universal product b.

quantum mechanics
j=σT4j = \sigma T^4

Stefan-Boltzmann Law

Integrating the Planck spectrum over all wavelengths gives the total emitted power. Two factors of T from the peak shift (Wien) and two more from the bandwidth growth combine to a T⁴ scaling.

particle physics
Eγmin⁡=2mec2E_{\gamma}^{\min} = 2 m_e c^2

Pair Production Threshold

Mass-energy equivalence says creating two particles of mass m_e demands at least 2m_e·c² of energy. But a lone photon can't do it — momentum conservation forbids it. A nearby nucleus (or another photon) absorbs the recoil and unlocks the process.

special relativity
Δt=γΔt0=Δt01−v2/c2\Delta t = \gamma \Delta t_0 = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}}

Time Dilation

The speed of light is the same for all observers. To keep that fixed when one observer moves relative to another, time itself must stretch — moving clocks run slow.

special relativity
L=L0γ=L01−v2/c2L = \frac{L_0}{\gamma} = L_0 \sqrt{1 - v^2/c^2}

Length Contraction

Just as moving clocks dilate, moving rulers contract. Both follow from c being invariant: lengths along motion direction shrink by 1/γ. Lengths perpendicular to motion are unchanged.

Birth of Quantum Theory
Bλ(λ,T)=2hc2λ5 1ehc/λkBT−1B_\lambda(\lambda,T)=\frac{2hc^2}{\lambda^5}\,\frac{1}{e^{hc/\lambda k_B T}-1}

Planck's Blackbody Radiation Law

Heat any object and it glows — dull red, then orange, then white-hot. Classical physics predicted the glow should carry infinite energy at short wavelengths (the 'ultraviolet catastrophe'). Planck fixed it by a desperate guess: light energy comes in discrete lumps hf, not a continuum. That single assumption bends the spectrum back down at short wavelengths and matches every glowing object perfectly. It was the first crack in classical physics and the birth of the quantum.

Birth of Quantum Theory
Kmax=hf−ϕ=eV0K_{max}=hf-\phi=eV_0

Work Function & Stopping Potential

Shine light on a metal and electrons pop out — but only if the light's color is bluer than a threshold, no matter how bright. Einstein explained it: light arrives as packets of energy hf. One packet kicks out one electron, but the electron must first pay an 'exit toll' φ to escape the metal. Whatever energy is left over becomes the electron's kinetic energy. Brightness adds more packets (more electrons) but never more energy per electron — that only comes from higher frequency.

Birth of Quantum Theory
λ=hp=hmv\lambda=\frac{h}{p}=\frac{h}{mv}

Wave-Particle Duality (de Broglie)

If light — long thought a wave — can act like particles (photons), de Broglie asked the reverse: can matter act like waves? He assigned every particle a wavelength λ = h/p. For a baseball this wavelength is absurdly tiny and invisible; for an electron it's about the size of an atom, so electrons diffract and interfere just like light. Everything is both, but Planck's constant is so small that the wave side only shows up for the very light and very slow.

Atomic Models & Spectra
dσdΩ=(Z1Z2e216πε0E)21sin⁡4(θ/2)\frac{d\sigma}{d\Omega}=\left(\frac{Z_1Z_2e^2}{16\pi\varepsilon_0 E}\right)^2\frac{1}{\sin^4(\theta/2)}

Rutherford Scattering

Rutherford fired alpha particles at gold foil expecting them to pass almost straight through the 'plum-pudding' atom. Instead a few bounced almost straight back — 'as if you fired a 15-inch shell at tissue paper and it came back'. The only way to deflect a heavy fast alpha so sharply is a tiny, dense, positive core: the nucleus. The 1/sin⁴(θ/2) law is the fingerprint of a point Coulomb charge, and its success pinned down the nuclear atom.

Condensed Matter Foundations
Eg=Ec−Ev,n∝e−Eg/2kBTE_g = E_c - E_v,\qquad n\propto e^{-E_g/2k_BT}

Band Theory of Solids

Bring one atom's sharp energy level close to another's and they split in two; bring 10²³ atoms together in a crystal and the levels smear into continuous bands separated by forbidden gaps. Whether a material conducts depends on where the electrons stop filling: land in the middle of a band and it's a metal; fill a band exactly to a gap and it's an insulator or, if the gap is small, a semiconductor. The band gap E_g is the single number that decides.

Condensed Matter Foundations
σ=ne2τm,vd=−eτmE\sigma=\frac{n e^2 \tau}{m},\qquad \mathbf{v}_d=-\frac{e\tau}{m}\mathbf{E}

Drude Free-Electron Model

Drude pictured a metal as a gas of free electrons rattling around fixed positive ions, like pinballs. Apply a voltage and the electrons accelerate, but keep crashing into ions every time τ; between crashes they pick up a small average drift velocity. Balancing the electric push against the collision drag gives Ohm's law and a formula for conductivity. It's crude — it ignores quantum statistics — yet it nails the form of σ and even predicts the Hall effect.

Foundational Constants & Relations
q=Ne,N∈Zq = N e,\qquad N\in\mathbb{Z}

Quantization of Electric Charge

Charge doesn't come in arbitrary amounts — it comes in whole-number multiples of one indivisible unit e. Millikan proved it by suspending tiny charged oil droplets between charged plates: turn up the voltage until an electron's electric pull exactly balances gravity. Every drop's charge came out as an integer times the same tiny number, 1.6×10⁻¹⁹ C. You never find half an electron's worth. Charge is granular, like money in indivisible pennies.

Named Cases
α=e24πε0ℏc≈1137.036\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\approx\frac{1}{137.036}

Fine-Structure Constant

α is the single dimensionless number that measures how strongly light and electrons couple — the strength of electromagnetism itself. Because it's a pure number (~1/137) with no units, every civilization in the universe would measure the same value. It sets the fine splitting of spectral lines, the speed of an electron in the first Bohr orbit (αc), and why atoms are the size they are. Feynman called it 'one of the greatest damn mysteries of physics.'

Atomic Models & Spectra
1λ=RH(122−1n2),  n=3,4,5,…\frac{1}{\lambda}=R_H\left(\frac{1}{2^2}-\frac{1}{n^2}\right),\; n=3,4,5,\dots

Balmer Series

Hydrogen's visible glow is four sharp lines — red, cyan, blue, violet. In 1885 a Swiss schoolteacher, Balmer, found a simple formula fitting their wavelengths using integers, with no physical theory at all. Every line is an electron falling to the n=2 level from a higher orbit; the bigger the drop, the bluer the light. Bohr later explained why the integers appear. The Balmer series is the training-wheels case that cracked open atomic structure.

Named Cases
μB=eℏ2me≈9.274×10−24 J/T\mu_B=\frac{e\hbar}{2m_e}\approx 9.274\times10^{-24}\,\text{J/T}

Bohr Magneton

An electron orbiting or spinning is a tiny current loop, and every current loop is a magnet. The Bohr magneton is the natural unit of that magnetism — the magnetic moment of an electron with one quantum of angular momentum. Atomic magnets come in multiples of this unit, so μ_B is to magnetism what e is to charge: the fundamental brick. It sets the scale of the Zeeman effect, electron spin resonance, and the strength of ferromagnets.

Nuclear & Particle

29
relativistic energy
E=mc2E = mc^2

Mass–Energy Equivalence

Mass is a highly concentrated form of energy; converting even a tiny mass releases enormous energy.

radioactivity
N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}

Radioactive Decay Law

Undecayed nuclei fall exponentially; each has a fixed decay probability per unit time.

radioactivity
T1/2=ln⁡2λT_{1/2} = \frac{\ln 2}{\lambda}

Half-Life Relation

Half-life is a fixed isotope property—independent of the sample size.

nuclear structure
B=Δm c2B = \Delta m \, c^2

Nuclear Binding Energy

The missing mass when nucleons bind into a nucleus appears as the energy holding them together.

nuclear reactions
Q=(mi−mf)c2Q = (m_i - m_f) c^2

Q-Value of Nuclear Reaction

Lighter products mean the missing mass exits as kinetic energy and radiation.

relativistic energy
E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

Relativistic Energy–Momentum Relation

Energy and momentum combine to form a Lorentz invariant — the rest mass of the particle.

particle physics
λ=hp\lambda = \frac{h}{p}

de Broglie Wavelength (Relativistic)

A particle’s wavelength shrinks with momentum—more momentum probes smaller scales.

radioactivity
log⁡10T1/2=aZQ+b\log_{10} T_{1/2} = a \frac{Z}{\sqrt{Q}} + b

Geiger–Nuttall Law

Alpha half-lives vary exponentially with decay energy via quantum tunneling.

nuclear structure
B(A,Z)=aVA−aSA2/3−aCZ2A1/3−aA(A−2Z)2A+δ(A,Z)B(A,Z) = a_V A - a_S A^{2/3} - a_C \frac{Z^2}{A^{1/3}} - a_A \frac{(A-2Z)^2}{A} + \delta(A,Z)

Semi-Empirical Mass Formula

Five competing terms—volume, surface, Coulomb, asymmetry, pairing—model the nuclear binding energy.

particle physics
dσdΩ=12α2rc2(ω′ω)2[ωω′+ω′ω−sin⁡2θ]\frac{d\sigma}{d\Omega} = \frac{1}{2} \alpha^2 r_c^2 \left(\frac{\omega'}{\omega}\right)^2 \left[\frac{\omega}{\omega'} + \frac{\omega'}{\omega} - \sin^2\theta\right]

Klein–Nishina Cross Section

QED correction to Thomson scattering: photon cross-section shrinks at high energy.

photon interactions
Δλ=hmec(1−cos⁡θ)\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Compton Scattering Wavelength Shift

A photon hitting an electron transfers momentum, so it leaves with a longer wavelength — light behaves like a particle.

atomic structure
a0=4πϵ0ℏ2mee2a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m_e e^2}

Bohr Radius

The natural size of an atom emerges from balancing electron kinetic energy with Coulomb attraction to the nucleus.

atomic structure
1λ=RH(1n12−1n22)\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Hydrogen emits sharp, discrete colors because its electron only inhabits quantized energy levels — the spectrum is the atom's barcode.

photon interactions
Kmax=hf−ϕK_{max} = h f - \phi

Photoelectric Effect Equation

Light comes in packets of energy hf; only packets above the work-function threshold can free an electron — no matter how intense the beam.

radioactivity
A=λNA = \lambda N

Radioactive Activity-Decay Constant Relation

Activity is just the number of unstable atoms times how likely each is to decay per second.

scattering
dσdΩ=(Z1Z2e216πϵ0E)21sin⁡4(θ/2)\frac{d\sigma}{d\Omega} = \left(\frac{Z_1 Z_2 e^2}{16\pi\epsilon_0 E}\right)^2 \frac{1}{\sin^4(\theta/2)}

Rutherford Scattering Cross Section

Rare large-angle scattering proves the atom has a tiny, dense, positively charged core — the nucleus.

nuclear force
V(r)=−g2e−mrc/ℏrV(r) = -g^2 \frac{e^{-m r c / \hbar}}{r}

Yukawa Potential

A short-ranged attractive force whose range is set by the mass of the exchanged particle — heavier mediators = shorter range.

transition rates
Γi→f=2πℏ∣Mfi∣2ρ(Ef)\Gamma_{i \to f} = \frac{2\pi}{\hbar} |M_{fi}|^2 \rho(E_f)

Fermi's Golden Rule

The transition rate between quantum states is proportional to the squared coupling times the number of final states available.

photon interactions
Eγmin⁡=2mec2E_{\gamma}^{\min} = 2 m_e c^2

Pair Production Threshold Energy

A photon must carry at least the combined rest energy of the electron and positron it creates — 1.022 MeV.

radioactive decay
Qα=(mP−mD−mα)c2Q_\alpha = (m_P - m_D - m_\alpha)c^2

Alpha Decay Q-Value

The kinetic energy carried away by the alpha particle (and the recoiling daughter) equals the rest-mass difference between parent and decay products, expressed via E = mc².

weak interaction
dNdE=GF2∣M∣22π3ℏ7c5F(Z,E)pE(Q−E)2\frac{dN}{dE} = \frac{G_F^2 |M|^2}{2\pi^3 \hbar^7 c^5} F(Z,E) p E (Q - E)^2

Fermi Beta Decay Spectrum

Three-body decay (nucleus + electron + neutrino) shares the released energy Q, so the electron's spectrum is continuous; the (Q-E)² factor is the neutrino phase space.

scattering
λ=1nσ\lambda = \frac{1}{n\sigma}

Mean Free Path

If the medium has n targets per unit volume each presenting cross-sectional area σ, the average distance between interactions is 1/(nσ).

nuclear structure
Bˉ=B(A,Z)A\bar{B} = \frac{B(A,Z)}{A}

Binding Energy per Nucleon

Dividing total binding energy by mass number gives the per-nucleon glue strength; the curve peaks near A=56 (Fe/Ni), explaining why fusion liberates energy below A=56 and fission above.

radioactive decay
τ=1λ\tau = \frac{1}{\lambda}

Mean Lifetime of Radioactive Nuclei

Mean lifetime is the expectation value of the exponential decay distribution; equivalently, the time after which N drops to N₀/e.

nuclear structure
R=R0A1/3R = R_0 A^{1/3}

Nuclear Radius Formula

Nuclei have nearly constant density, so volume scales linearly with A and radius scales as A^(1/3) — like balls of incompressible nuclear fluid.

scattering
σT=8π3(e24πε0mec2)2\sigma_T = \frac{8\pi}{3}\left(\frac{e^2}{4\pi\varepsilon_0 m_e c^2}\right)^2

Thomson Scattering Cross Section

An EM wave shakes a free electron, which re-radiates a dipole pattern; the total scattering rate corresponds to an effective area σ_T set by the classical electron radius r_e.

radiation
v>cn  ⇔  cos⁡θC=1nβv > \frac{c}{n} \;\Leftrightarrow\; \cos\theta_C = \frac{1}{n\beta}

Cherenkov Radiation Threshold

When a charged particle moves faster than the phase velocity of light in the medium (c/n), it sets off a coherent optical shockwave at angle θ_C — the blue glow in reactor pools.

particle physics
p=uud,n=udd,ZAX:  A=Z+Np = uud, \quad n = udd, \quad {}^{A}_{Z}X: \; A = Z + N

Building Atoms from the Standard Model

Everything you've ever touched is three particles: up quarks, down quarks, and electrons. Two ups and a down make a proton (+1); one up and two downs make a neutron (0). The proton count Z picks the element, the neutron count N picks the isotope, and the electron count sets the charge. Stray too far from the stable Z–N balance and the weak force flips a quark — beta decay — to restore it.

fission
Nn=N0 kn,k=ν pN_{n} = N_{0}\,k^{n}, \qquad k = \nu \, p

Nuclear Chain Reaction

One neutron splits a uranium-235 nucleus, releasing ~200 MeV and ν ≈ 2.4 fresh neutrons. If, on average, more than one of those goes on to cause another fission (k > 1), the population explodes geometrically — 2, 4, 8, … 2⁸⁰ in microseconds. A reactor is the art of pinning k at exactly 1.000; a bomb is k ≈ 2 with nothing in the way.

Fluid Mechanics

34
kinematics
A1v1=A2v2A_1 v_1 = A_2 v_2

Continuity Equation

What flows in must flow out — narrow pipes force faster flow.

dynamics
P+12ρv2+ρgh=constP + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}

Bernoulli's Equation

Pressure, kinetic, and potential energy per unit volume sum to a constant along a streamline.

statics
P=P0+ρghP = P_0 + \rho g h

Hydrostatic Pressure

Pressure grows linearly with depth because of the weight of fluid above.

statics
Fb=ρVgF_b = \rho V g

Archimedes' Principle

Buoyant force equals the weight of fluid displaced.

dynamics
v=2ghv = \sqrt{2 g h}

Torricelli's Law

Falling fluid trades height for speed, just like a dropped ball.

viscous flow
Re=ρvLμRe = \frac{\rho v L}{\mu}

Reynolds Number

Ratio of inertial to viscous forces — high Re means inertia wins, turbulence reigns.

viscous flow
Fd=6πηrvF_d = 6\pi \eta r v

Stokes' Law

Slow, syrupy flow around a tiny sphere produces drag linear in speed.

viscous flow
Q=πr4ΔP8ηLQ = \frac{\pi r^4 \Delta P}{8 \eta L}

Poiseuille's Law

Pipe flow scales with the FOURTH power of radius — narrowing matters massively.

pipe flow
ΔP=fLDρv22\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}

Darcy-Weisbach Equation

Pressure loss in a pipe scales with length, kinetic energy, and a fudge factor for roughness.

statics
F1A1=F2A2\frac{F_1}{A_1} = \frac{F_2}{A_2}

Pascal's Principle

Pressure applied to a confined fluid is transmitted everywhere — undiminished.

rheology
τ=ηdvdy\tau = \eta \frac{dv}{dy}

Newton's Law of Viscosity

Friction inside a fluid is proportional to how fast layers slide past each other.

interfacial
ΔP=2γr\Delta P = \frac{2 \gamma}{r}

Young-Laplace Equation

Curved interfaces compress what's inside — smaller curvature, bigger squeeze.

interfacial
h=2γcos⁡θρgrh = \frac{2 \gamma \cos\theta}{\rho g r}

Jurin's Law (Capillary Rise)

Surface tension pulls liquid up a thin tube until gravity catches up — narrower tube, taller climb.

high re flow
Fd=12CdρAv2F_d = \frac{1}{2} C_d \rho A v^2

Drag Force (Quadratic)

Push air aside fast enough and it pushes back — quadratically.

compressible flow
M=vaM = \frac{v}{a}

Mach Number

Mach number is your speed measured in 'speeds of sound' — at M=1 you outrun your own pressure waves.

governing equations
ρ(∂v∂t+v⋅∇v)=−∇p+μ∇2v+f\rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v}\right) = -\nabla p + \mu\nabla^2\mathbf{v} + \mathbf{f}

Navier-Stokes Equation

Newton's second law written for a fluid parcel: mass times acceleration equals pressure forces plus viscous friction plus body forces.

governing equations
ρDvDt=−∇p+ρg\rho\frac{D\mathbf{v}}{Dt} = -\nabla p + \rho\mathbf{g}

Euler Equation (Inviscid Flow)

With no viscosity, a fluid parcel accelerates purely from pressure differences and gravity.

dynamics
p1−p2=12ρ(v22−v12)p_1 - p_2 = \frac{1}{2}\rho\left(v_2^2 - v_1^2\right)

Venturi Effect

Where a flow speeds up through a constriction, its pressure must drop — Bernoulli in a pipe.

dimensionless numbers
Fr=vgLFr = \frac{v}{\sqrt{gL}}

Froude Number

The ratio of how fast the fluid moves to how fast a gravity wave can travel — it sets whether disturbances can run upstream.

dimensionless numbers
We=ρv2LσWe = \frac{\rho v^2 L}{\sigma}

Weber Number

Whether a moving blob of fluid holds together by surface tension or is torn apart by inertia.

aerodynamics
L′=ρvΓL' = \rho v \Gamma

Kutta-Joukowski Lift Theorem

Lift equals density times speed times circulation — net swirl around the wing forces the air down and the wing up.

open channel
v=1nR2/3S1/2v = \frac{1}{n}R^{2/3}S^{1/2}

Manning's Equation

Open-channel flow speed grows with depth (hydraulic radius) and slope, and falls with roughness.

aerodynamics
FL=ρ v Γ,Γ=2πr2ωF_L = \rho\, v\, \Gamma,\qquad \Gamma = 2\pi r^2 \omega

Magnus Effect

Spin drags a thin layer of air around the ball, speeding the flow on one side and slowing it on the other. By Bernoulli the fast side is low-pressure, so the ball is pushed sideways — the same circulation-times-speed law that gives a wing its lift.

flow measurement
v=2 (p0−p)ρv = \sqrt{\frac{2\,(p_0 - p)}{\rho}}

Pitot Tube Airspeed

Bring the moving air to a dead stop at the tube's nose and all its kinetic energy turns into extra pressure. The size of that extra (dynamic) pressure tells you how fast the air was going.

open channel flow
y2y1=12(1+8 Fr12−1)\frac{y_2}{y_1} = \tfrac{1}{2}\left(\sqrt{1 + 8\,Fr_1^2} - 1\right)

Hydraulic Jump

When shallow water moves faster than its own surface waves (Fr > 1) it can't 'feel' the slower deep water ahead, so it piles up in a sudden standing wall — the liquid analogue of a sonic shock — dumping the excess energy as turbulence.

transient flow
Δp=ρ c Δv\Delta p = \rho\, c\, \Delta v

Water Hammer (Joukowsky)

The moving column of water has momentum; stop it suddenly and that momentum has nowhere to go but into pressure. A compression wave rockets back up the pipe at the speed of sound in water, hammering every fitting it passes.

hydrostatics
GM=IV−BG‾GM = \frac{I}{V} - \overline{BG}

Metacentric Height (Ship Stability)

When a ship heels, the underwater shape shifts the buoyancy force sideways to act through a point called the metacentre. If that point sits above the centre of gravity, buoyancy twists the ship back upright; if below, it capsizes.

everyday aerodynamics
Δp=12ρ (vout2−vin2)\Delta p = \tfrac{1}{2}\rho\,(v_{out}^2 - v_{in}^2)

Why Shower Curtains Billow Inward

The falling spray drags air downward, so air moves fast inside the shower and slow outside. By Bernoulli, fast-moving air has lower pressure, so the higher outside pressure simply pushes the lightweight curtain inward.

biofluid dynamics
Δp=12ρ (v22−v12)\Delta p = \tfrac{1}{2}\rho\,(v_2^2 - v_1^2)

How Prairie-Dog Burrows Self-Ventilate

Wind speeds up with height, so it blows faster over the taller mound. Faster air means lower pressure there (Bernoulli), and the pressure gap between the two openings continuously sucks fresh air through the tunnel — a passive pump.

aerodynamics
FD=12 CD ρ v2A,Re=v DνF_D = \tfrac{1}{2}\,C_D\,\rho\,v^2 A,\qquad Re = \frac{v\,D}{\nu}

Why Golf Balls Have Dimples (Drag Crisis)

A smooth ball's boundary layer separates early, leaving a fat low-pressure wake that drags it back. Dimples deliberately trip the layer turbulent so it clings farther around the ball, shrinking the wake and roughly halving the drag — so the ball flies about twice as far.

vortex dynamics
v r=const  ⇒  v=Γ2πrv\,r = \text{const}\;\Rightarrow\; v = \frac{\Gamma}{2\pi r}

Why Draining Water Swirls (Free Vortex)

As a fluid ring spirals inward toward the drain its radius shrinks, and like a skater pulling in their arms it must spin faster to conserve angular momentum. The rising speed near the hole carves the familiar funnel-shaped surface dip.

wake dynamics
f=St vDf = \frac{St\,v}{D}

Kármán Vortex Street

As fluid sweeps past a blunt body it can't cling to the back, so it rolls up into vortices that peel off alternately from each side. That regular left-right shedding pushes the body sideways at frequency f — the hum of a wire and the sway of a chimney.

pressure vessels
σmax=σ(1+2ab)\sigma_{max} = \sigma\left(1 + \frac{2a}{b}\right)

Why Airplane Windows Are Round

Stress flows through a skin like water through a channel; force it around a sharp corner and it bunches up just as water speeds at a constriction. A circle spreads the flow gently (peak 3×), but a square corner spikes the stress until a crack starts.

turbulence
E(k)=C ε2/3 k−5/3E(k) = C\,\varepsilon^{2/3}\,k^{-5/3}

Kolmogorov Turbulent Energy Cascade

Energy fed in at large scales is handed down, eddy by eddy, to ever-smaller whirls without loss, until the smallest eddies are so fine that viscosity finally smears their motion into heat. Across that 'inertial range' only ε matters, fixing the universal −5/3 slope.

Mathematical Physics

37
vector calculus
∇f=∂f∂x x^+∂f∂y y^+∂f∂z z^\nabla f = \frac{\partial f}{\partial x}\,\hat{\mathbf{x}} + \frac{\partial f}{\partial y}\,\hat{\mathbf{y}} + \frac{\partial f}{\partial z}\,\hat{\mathbf{z}}

Gradient

The gradient packages all the partial derivatives into one vector that points in the direction of steepest ascent, with length equal to that maximum slope.

vector calculus
∇⋅F=∂Fx∂x+∂Fy∂y+∂Fz∂z\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}

Divergence

Divergence measures the net outward flux per unit volume from an infinitesimal box — positive at a source, negative at a sink, zero where flow just passes through.

vector calculus
∇×F=(∂Fz∂y−∂Fy∂z, ∂Fx∂z−∂Fz∂x, ∂Fy∂x−∂Fx∂y)\nabla \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\ \frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x},\ \frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)

Curl

Curl measures the local spin of a field — the circulation per unit area around an infinitesimal loop, with direction along the axis of rotation by the right-hand rule.

vector calculus
∇2f=∇⋅∇f=∂2f∂x2+∂2f∂y2+∂2f∂z2\nabla^2 f = \nabla\cdot\nabla f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}

Laplacian

The Laplacian compares the value at a point to the average of its neighbors: positive where the point sits in a valley, negative where it caps a hill, zero where it equals the surrounding average (harmonic).

vector calculus
∮∂VF⋅dA=∫V(∇⋅F) dV\oint_{\partial V} \mathbf{F}\cdot d\mathbf{A} = \int_V (\nabla\cdot\mathbf{F})\, dV

Divergence (Gauss) Theorem

The total outward flux through a closed surface equals the sum of all the sources and sinks enclosed — surface bookkeeping equals volume bookkeeping.

vector calculus
∮∂SF⋅dr=∫S(∇×F)⋅dA\oint_{\partial S} \mathbf{F}\cdot d\mathbf{r} = \int_S (\nabla\times\mathbf{F})\cdot d\mathbf{A}

Stokes' Theorem

The circulation of a field around a closed loop equals the total curl 'flux' threading any surface that the loop bounds — boundary swirl equals interior swirl.

complex analysis
eiθ=cos⁡θ+i sin⁡θe^{i\theta} = \cos\theta + i\,\sin\theta

Euler's Formula

Multiplying by e^{iθ} rotates the complex plane by angle θ. The exponential of an imaginary number is a point on the unit circle, decomposing naturally into cosine (real) and sine (imaginary) parts.

complex analysis
∂u∂x=∂v∂y,∂u∂y=−∂v∂x\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

Cauchy–Riemann Equations

For a complex function f = u + iv to be differentiable (analytic), the rate of change must be the same in every direction. That single requirement collapses into two coupled equations relating the partials of the real and imaginary parts.

fourier analysis
f(x)=a02+∑n=1∞[ancos⁡nπxL+bnsin⁡nπxL]f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi x}{L} + b_n\sin\frac{n\pi x}{L}\right]

Fourier Series

Any reasonable periodic function is an infinite weighted sum of sines and cosines (or complex exponentials). The coefficients are the amounts of each pure frequency present — the function's recipe in tones.

fourier analysis
f^(k)=∫−∞∞f(x) e−ikx dx\hat{f}(k) = \int_{-\infty}^{\infty} f(x)\, e^{-i k x}\, dx

Fourier Transform

The Fourier transform measures how much of each frequency e^{ikx} a signal contains, by correlating the signal against every pure wave. It turns convolution into multiplication and derivatives into multiplication by ik.

fourier analysis
F{f∗g}=f^(k) g^(k)\mathcal{F}\{f * g\} = \hat{f}(k)\,\hat{g}(k)

Convolution Theorem

Convolution in one domain is plain multiplication in the other. A smearing operation that costs an integral over all shifts becomes a single product once you switch to the frequency domain.

partial differential equations
∂2u∂t2=c2 ∇2u\frac{\partial^2 u}{\partial t^2} = c^2\,\nabla^2 u

Wave Equation

Wherever the field curves in space (nonzero ∇²u), it accelerates in time — restoring toward its neighbors' average. That feedback launches disturbances that travel at speed c without changing shape.

partial differential equations
∂u∂t=D ∇2u\frac{\partial u}{\partial t} = D\,\nabla^2 u

Heat Equation

The temperature at a point rises in proportion to how much colder it is than its neighbors' average (the Laplacian). Heat flows down gradients, smoothing bumps and erasing fine detail irreversibly.

special functions
∫−∞∞f(x) δ(x−a) dx=f(a)\int_{-\infty}^{\infty} f(x)\,\delta(x-a)\,dx = f(a)

Dirac Delta Function

The delta function is an idealized spike with zero width and unit area. Its defining job is the sifting property: integrated against any function, it samples that function at a single point.

Probability & Stochastic Processes
P(A∣B)=P(B∣A) P(A)P(B)P(A\mid B)=\frac{P(B\mid A)\,P(A)}{P(B)}

Bayes' Theorem

Bayes' theorem is how you update a belief when new evidence arrives. Start with a prior — how likely a hypothesis was before — multiply by how well the hypothesis predicts the new data (the likelihood), and renormalize. The counterintuitive punchline: even a very accurate test for a rare condition mostly returns false positives, because the tiny prior overwhelms the likelihood. It flips 'probability of data given hypothesis' into the thing you actually want: 'probability of hypothesis given data.'

Theorems & Identities
(x+y)n=∑k=0n(nk)xn−kyk(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}

Binomial Theorem

Expanding (x+y)ⁿ, each term picks either x or y from each of the n factors; the coefficient counts how many ways to pick k y's — that's 'n choose k', Pascal's triangle. Newton's leap was letting the exponent be any real number, turning the finite sum into an infinite series that gives (1+x)^α. Physicists live on the first two terms: (1+x)^α ≈ 1+αx for small x, the workhorse approximation behind nearly every 'to first order' calculation.

Theorems & Identities
∣⟨u,v⟩∣≤∥u∥ ∥v∥|\langle u,v\rangle|\le \lVert u\rVert\,\lVert v\rVert

Cauchy-Schwarz Inequality

The overlap of two vectors can never exceed the product of their lengths — geometrically because ⟨u,v⟩ = ‖u‖‖v‖cosθ and cosθ ≤ 1. Equality happens only when the vectors are parallel. This humble bound is secretly everywhere: it guarantees correlation coefficients stay within ±1, it underlies the triangle inequality, and in quantum mechanics it's the step that turns operator commutators into the Heisenberg uncertainty principle.

Probability & Stochastic Processes
1n∑i=1nXi−μσ  →d  N(0,1)\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{X_i-\mu}{\sigma}\;\xrightarrow{d}\;\mathcal{N}(0,1)

Central Limit Theorem

Add up many small independent random effects — no matter what weird shape each one has — and their sum looks like a bell curve. This is why the normal distribution is everywhere: measurement errors, heights, thermal noise, diffusion. The individual quirks wash out; only the mean and variance survive. The averaged quantity's spread shrinks as 1/√n, which is why more measurements give proportionally better precision.

Probability & Stochastic Processes
P(k)=(nk)pk(1−p)n−kP(k)=\binom{n}{k}p^{k}(1-p)^{n-k}

Binomial Distribution

Flip a biased coin n times; the binomial distribution gives the probability of exactly k heads. Each specific sequence with k heads has probability p^k(1−p)^{n−k}, and there are C(n,k) such sequences. The distribution peaks near np, its mean, and spreads as √(np(1−p)). It's the discrete ancestor of the bell curve: for large n it becomes Gaussian (de Moivre–Laplace), and for rare events (small p, large n) it becomes Poisson.

Complex Analysis
f(z0)=12πi∮Cf(z)z−z0 dzf(z_0)=\frac{1}{2\pi i}\oint_{C}\frac{f(z)}{z-z_0}\,dz

Cauchy's Integral Formula

This is complex analysis's near-magical result: if you know a holomorphic function only on a closed loop, you know it everywhere inside. The value at any interior point is a weighted average around the boundary. It means analytic functions are absurdly rigid — pin down a curve and the interior is forced. It also hands you all the derivatives from the same boundary data, proving holomorphic functions are infinitely differentiable automatically.

Complex Analysis
∮Cf(z) dz=0\oint_{C}f(z)\,dz=0

Cauchy's Integral Theorem

If a complex function is holomorphic (smooth in the complex sense) throughout a region, then integrating it around any closed loop gives exactly zero. Equivalently, the integral between two points doesn't care which path you take. This is the complex cousin of a conservative force field having zero circulation. It's the foundation stone: from it flow Cauchy's integral formula, the residue theorem, and the whole contour-integration toolkit physicists use to crack hard integrals.

Complex Analysis
∂u∂x=∂v∂y,∂u∂y=−∂v∂x\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}

Analytic (Holomorphic) Functions

A complex function is analytic if it's differentiable in the complex sense — and that single requirement is astonishingly strong. Complex differentiability forces the real and imaginary parts to satisfy the Cauchy–Riemann equations, which make each part harmonic (a solution of Laplace's equation). Analytic functions are automatically infinitely differentiable and equal to their own Taylor series. It's a world where 'differentiable once' secretly means 'perfect forever' — nothing like real calculus.

Linear Algebra
p(λ)=det⁡(A−λI)=0p(\lambda)=\det(A-\lambda I)=0

Characteristic Polynomial

To find a matrix's eigenvalues — the special scaling factors along which it merely stretches vectors — you ask when A−λI collapses a direction to zero, i.e. when its determinant vanishes. That determinant, as a function of λ, is the characteristic polynomial. Its roots are the eigenvalues. Physicists call it the 'secular equation' because it first arose computing slow (secular) perturbations of planetary orbits. Its coefficients encode invariants: the trace and determinant of A.

Special Functions
x2y′′+xy′+(x2−ν2)y=0,y=Jν(x)x^2 y'' + x y' + (x^2-\nu^2)y=0,\quad y=J_\nu(x)

Bessel Functions

Bessel functions are what sine and cosine become in cylindrical worlds. Whenever a problem has circular symmetry — a vibrating drumhead, waves in a round pipe, light through a circular aperture — the radial part obeys Bessel's equation and the solutions J_ν(x) appear. They wiggle like damped sinusoids whose amplitude fades as 1/√x and whose zeros aren't evenly spaced. Those zeros set the drum's overtones and the diffraction rings you see around a point of light.

asymptotics perturbation
f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Taylor Expansion

Near a point a, a smooth function looks like its tangent line; add curvature and it looks like a parabola; add the next derivative and it hugs the curve further out. Each term corrects the previous approximation using one more derivative.

theorems identities
∫abf(x) dx=F(b)−F(a),F′(x)=f(x)\int_a^b f(x)\,dx = F(b) - F(a), \quad F'(x)=f(x)

Fundamental Theorem of Calculus

The area accumulated under f up to x is a function A(x). Nudging x by dx adds a sliver of area f(x)dx, so dA/dx = f(x). Accumulation and rate-of-change are inverse operations.

theorems identities
f′(c)=f(b)−f(a)b−a,c∈(a,b)f'(c) = \frac{f(b)-f(a)}{b-a}, \quad c \in (a,b)

Mean Value Theorem

The average rate of change over an interval is the slope of the secant. If the curve is smooth, its instantaneous slope must equal that average at least once — the tangent runs parallel to the secant somewhere inside.

probability stochastic
p(x)=1σ2π e−(x−μ)22σ2p(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Gaussian (Normal) Distribution

The exponential of a negative square gives a symmetric hump centered at μ. σ sets the spread, and the prefactor is exactly what makes the total probability one. Squaring the deviation punishes outliers smoothly, producing the ubiquitous bell.

linear algebra
Av=λvA\mathbf{v} = \lambda\mathbf{v}

Eigenvalue Equation

An eigenvector of A is a direction the matrix only stretches (by λ), never rotates. Those directions are the coordinate system in which A acts as simple scaling — the skeleton of the transformation.

linear algebra
det⁡ ⁣(abcd)=ad−bc\det\!\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Determinant

The determinant is the signed volume-scaling factor of the linear map. |det| tells you how much areas/volumes grow; its sign tells you whether orientation is preserved or flipped; zero means the map crushes space into a lower dimension.

symmetry algebra
R(θ)=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Rotation Matrices

A rotation keeps lengths and angles fixed and the origin put — so its columns must be perpendicular unit vectors. In 2D those columns are (cosθ, sinθ) and its 90°-turned partner, giving the familiar cosine–sine block.

special functions
Γ(z)=∫0∞tz−1e−t dt\Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t}\,dt

Gamma Function

The integral weighs powers of t against an exponential cutoff. Integration by parts reproduces the factorial recursion Γ(z+1)=zΓ(z), so the smooth curve inherits the factorial's stair-step growth while filling in every value between.

special functions
Pn(x)=12nn!dndxn(x2−1)nP_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n

Legendre Polynomials

Legendre polynomials are what you get by orthogonalizing 1, x, x², … on the interval [-1,1]. They are the angular building blocks of any problem with spherical symmetry — each degree n captures one more level of angular detail.

calculus of variations
ddt ⁣(∂L∂q˙)−∂L∂q=0\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot q}\right) - \frac{\partial L}{\partial q} = 0

Euler–Lagrange Equation

A path that minimizes (or extremizes) an integral can't be improved by any tiny wiggle — the first-order change must vanish. Demanding that for every possible wiggle forces the integrand's derivatives into this exact balance, the equation of motion.

vector calculus
∮∂D(P dx+Q dy)=∬D ⁣(∂Q∂x−∂P∂y)dA\oint_{\partial D}(P\,dx + Q\,dy) = \iint_D\!\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA

Green's Theorem

The circulation of a vector field around a closed curve counts the net swirl it encloses. Interior swirls of adjacent tiny loops cancel on shared edges, leaving only the outer boundary — so the boundary integral equals the area integral of the curl.

numerical methods
xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Newton–Raphson Method

Near a root the curve looks like its tangent. So follow the tangent to where it crosses zero and use that as the next guess. Each step squares the error, giving quadratic convergence once you're close.

ordinary differential equations
dydx=f(x) g(y)  ⇒  ∫dyg(y)=∫f(x) dx\frac{dy}{dx} = f(x)\,g(y) \;\Rightarrow\; \int\frac{dy}{g(y)} = \int f(x)\,dx

Separable ODE

If the rate factorizes into an x-only piece times a y-only piece, you can herd all the y's to one side and all the x's to the other, then integrate each side independently. One integration on each side solves it.