Library of human understanding
Equations
Short sentences that hold up whole fields.
Each equation is written in plain characters and followed by what it says in words. The maths is only the shorthand. More equations arrive with each new section of the library.
Quantum Mechanics
The Planck relation · 1900
Light of frequency ν comes in packets of energy E, and the packet size is fixed by one new constant of nature, h. Planck wrote it down to fit the glow of hot objects; it turned out to be the first line of the quantum story.
E energy · ν frequency · h Planck's constant, 6.62607015 × 10⁻³⁴ joule-seconds (exact since 2019)
K = hν − φ
The photoelectric equation · 1905
Shine light on a metal and electrons come out, but only if each packet of light carries enough energy. Brighter light of the wrong colour does nothing. Einstein read this as light really arriving in quanta.
K kinetic energy of the freed electron · φ the energy the metal holds its electrons with
Eₙ = −13.6 eV / n²
Bohr's hydrogen levels · 1913
An electron in hydrogen can only sit at certain energies, numbered n = 1, 2, 3 … Jumping between them releases or absorbs light of exactly matching colours, which is why every element has its own fingerprint of spectral lines.
n whole number · eV electron-volt
The de Broglie wavelength · 1924
Every moving thing has a wavelength: the heavier or faster it is, the shorter it gets. For a tennis ball it is absurdly small. For an electron it is about the size of an atom, so electrons can interfere like waves.
λ wavelength · p momentum (mass × velocity)
The Schrödinger equation · 1926
The rule for how a quantum state changes in time. ψ, the wave function, holds everything that can be known about a system; Ĥ is its energy. Between measurements, ψ evolves smoothly and predictably, and all the strangeness lives in how we read it.
ψ wave function · Ĥ the Hamiltonian (total-energy operator) · ħ = h/2π · i the square root of −1
The Born rule · 1926
The wave function does not say where a particle is. Square its size at a point and you get the probability of finding the particle there. This is where chance enters physics at its foundation.
P probability · |ψ|² the squared magnitude of the wave function
x̂p̂ − p̂x̂ = iħ
Position and momentum do not commute · 1925
In the matrix formulation, the order in which you measure position and momentum matters. This single fact, from Heisenberg, Born and Jordan, is the algebra underneath the uncertainty principle.
The uncertainty principle · 1927
A quantum state cannot have a sharply defined position and a sharply defined momentum at once. This is not clumsy instruments: the two quantities simply are not both sharp in nature. Squeeze one and the other spreads.
Δx spread in position · Δp spread in momentum
The Dirac equation · 1928
Quantum mechanics made compatible with special relativity. It gave the electron its spin automatically and predicted a mirror particle with opposite charge, the positron, found four years later. It was the first prediction of antimatter.
γᵘ Dirac matrices · m electron mass · c speed of light
ψ(a, b) = −ψ(b, a)
The Pauli exclusion principle · 1925
Swap two electrons and the wave function flips sign. A consequence: no two electrons can occupy the same state. It is why atoms have shells, why the periodic table has its shape, and why matter does not collapse into a point.
⟨b|a⟩ = ∫ e^{iS[x]/ħ} 𝒟x
The path integral · 1948
To find how likely a particle is to get from a to b, add up every possible route, each weighted by a phase set by its action S. Most routes cancel; the ones that survive are the paths nature appears to take.
S the action along a path · 𝒟x sum over all paths
An entangled pair (a Bell state) · 1935 / 1964
Two qubits that are each undecided, but whose answers are locked together: measure one and get 0, and the other will give 0; get 1, and it gives 1. No message passes between them. Bell's theorem shows that no scheme of answers fixed in advance, with nothing acting at a distance, can reproduce these results, and experiments confirm it.
|S| ≤ 2 classically, up to 2√2 in quantum theory
The Bell (CHSH) inequality · 1964 / 1969
Combine four correlation measurements into a number S. Any theory in which outcomes are fixed in advance by local hidden facts can never exceed 2. Quantum mechanics can reach about 2.83, and experiments from 1972 onward have seen values above 2.
John Stewart Bell · John Clauser, Alain Aspect and Anton Zeilinger
S(ρ) = −Tr(ρ ln ρ)
Von Neumann entropy · 1927
A measure of how mixed or uncertain a quantum state is. A pure state has zero; the half of an entangled pair, looked at alone, is maximally mixed. It later became the measure of entanglement.
ρ density matrix · Tr trace
no operation U with U|ψ⟩|0⟩ = |ψ⟩|ψ⟩ for every |ψ⟩
The no-cloning theorem · 1982
An unknown quantum state cannot be copied. That is what makes quantum messages impossible to eavesdrop on without a trace, and what makes error correction in a quantum computer so much harder than in a classical one.
Quantum Computing
The qubit · 1995
A quantum bit is not 0 or 1 but a blend of both, with amplitudes α and β. Measuring gives 0 with probability |α|² and 1 with probability |β|², and the blend is gone. The skill of quantum computing is steering the amplitudes before you look.
α, β complex amplitudes
The Hadamard gate · —
The gate that turns a definite 0 into an even blend of 0 and 1. Almost every quantum algorithm starts by applying it to every qubit so that the circuit works on an equal blend of all possible inputs, which interference then sorts out.
CNOT|a, b⟩ = |a, a ⊕ b⟩
The CNOT gate · —
Flip the second qubit if, and only if, the first is 1. Together with single-qubit gates it can build any quantum computation, and applied to a blend it is what creates entanglement.
⊕ addition modulo 2
f(x) = aˣ mod N has period r → gcd(a^{r/2} ± 1, N) shares a factor with N
Shor's period-finding step · 1994
Factoring a large number reduces to finding the period of a repeating sequence. A quantum computer can find that period quickly by interference, where the best known classical methods slow down dramatically as the number grows.
about (π/4)·√N steps to find one item among N
Grover's search · 1996
Searching an unsorted list of N items classically takes about N/2 tries on average. Grover's algorithm needs about √N, by repeatedly nudging amplitude toward the right answer. A real speed-up, but only a quadratic one.
physical error rate p < p_threshold ⇒ logical errors fall as the code grows
The threshold theorem · 1996–1998
If each operation is wrong less often than a critical rate, then spending more qubits on error correction makes the encoded result more reliable, without limit. Whether real hardware sits below that line is the central engineering question of the field.
Physics & Mathematics
Distance in Cartesian coordinates · 1637
Name any point by two numbers: how far across (x) and how far up (y). Its distance from the origin is then Pythagoras on those two numbers. This is Descartes's idea in one line: a picture becomes arithmetic, and arithmetic can be drawn.
x distance across · y distance up · r straight-line distance from the origin (0, 0)
Gibbs free energy · 1876
Will it happen on its own? Take the heat a change releases or absorbs (ΔH), subtract the temperature times how much it spreads things out (TΔS). If what is left, ΔG, is negative, the change goes by itself. Ice melts above 0 °C because there TΔS wins.
ΔG change in free energy · ΔH change in heat content (enthalpy) · T temperature in kelvin · ΔS change in disorder (entropy)