Illustrated Physics
physics, drawn
Concept

Reduced mass — where it appears

The combination m₁m₂/(m₁+m₂) that a two-body problem behaves as though it had, letting the pair be treated as one body about a fixed centre. It is why positronium is twice the size of hydrogen and muonic hydrogen two hundred times smaller.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A collision with restitution 0.4. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

The point that keeps moving as if nothing had happened

Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.

mechanics · Momentum Why it stops falling in. The energy of an electron confined to a region of radius r around a nucleus, as the sum of two terms with different powers: a confinement energy ħ²/2mr² that rises without limit as the region shrinks, and a Coulomb attraction −Ze²/4πε₀r that falls. At Z = 1 the sum is least at 52.92 pm, where it is -13.61 eV. Both numbers are found by searching the drawn curve and both agree with the Bohr radius over Z and minus Z² Rydbergs to a part in a million. Nothing was quantised to get them. The only quantum input is that confining an electron to a region costs kinetic energy, which is the uncertainty relation and nothing more.

Why an atom is the size it is

A tenth of a nanometre is not a measured constant of nature but the outcome of a competition: confining an electron costs kinetic energy, and the nucleus pays for confinement with attraction. Minimising the sum gives the number, and changing the masses moves it by four orders of magnitude.

quantum · Atomic structure Two lifetimes a thousand times apart. The fraction of positronium atoms surviving in the ground state against time since formation, on a logarithmic time axis, for the spin singlet, para-positronium (red), and the spin triplet, ortho-positronium (blue), in vacuum. Para lives 125.1 ps and ortho 142.1 ns, a ratio of 1135. The lowest-order formulas give 124.5 ps, from α⁵mc²/2ħ, and 138.7 ns, from 2(π² − 9)/9π · α⁶mc²/ħ; the ratio is almost entirely one extra power of the fine-structure constant, the price of a third photon.

The atom with no nucleus

An electron and a positron can bind into an atom whose structure is hydrogen's with every level halved and every size doubled, because neither partner sits still at the centre. The atom then destroys itself, and how fast depends on one symmetry. With the two spins opposed it vanishes in 125 picoseconds as two photons; with them aligned it must make three, and lives 142 nanoseconds — over a thousand times longer — for no reason but that a rule about charge conjugation forbids the shorter route.

quantum · Atomic structure

Named alongside it

The objects these essays reach for when they reach for this one.

AnnihilationAntimatterAtomic structureBinding energyBohr radiusCentre of massCharge conjugationConservation lawsCoulomb forceElastic collisionElectron shellEnergy conservation

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