Illustrated Physics
physics, drawn
Series

Exclusion — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The energy ladder of a box. The first 6 energy levels of a box, drawn to scale in E₁, at 1.0, 4.0, 9.0, 16.0, 25.0, 36.0. The levels spread apart as the square of n, so a box's spectrum has no top. The arrow marks a transition: 4 to 3 releases 7.000 E₁.

    No two in the same state, and why matter has volume

    Nothing in the energy levels of an atom says how many electrons may occupy each one. The answer is one per state, it is not derived from any force, and it is the reason a table holds a cup up.

    part 1 · quantum
  2. How much of copper's electron sea a temperature can reach. The probability that a state of a given energy is occupied, in copper, at 4 temperatures, with energy measured in units of the Fermi energy — 7.04 eV here. At absolute zero the curve is a step: every state below the ceiling is full and every state above it is empty. Raising the temperature rounds the step, and rounds it over a range of about kT, which is the whole point — at room temperature kT is 0.0259 eV against a ceiling of 7.04 eV, so the rounding is 1.6 per cent of the way down the sea and everything deeper is untouched. An electron in the deep is not held there by a force; it simply has nowhere to go, because every state it could be promoted to is occupied. at 0 K the step is spread over 0.00 per cent of E_F, at 300 K the step is spread over 1.61 per cent of E_F, at 3000 K the step is spread over 16.13 per cent of E_F, at 20000 K the step is spread over 107.52 per cent of E_F.

    The pressure that is not a temperature

    Copper's conduction electrons are at a temperature of eighty thousand kelvin, in a wire that is at room temperature. That is not a figure of speech, it is what the exclusion principle does to a mole of particles, and it explains the largest unexplained number in the theory of metals.

    part 2 · quantum
  3. The hole that holds exactly one particle. How likely a second particle is to be found a distance away from a first, relative to a gas with no correlation at all, for three cases that differ in nothing but the symmetry of the state under swapping the two labels. There is no interaction anywhere in this calculation: no Coulomb term, no potential, no force. Distinguishable particles give a flat line, which is what no interaction ought to give. Identical fermions dig a hole that reaches exactly zero at zero separation and fills back in over about a wavelength. Identical bosons do the opposite and pile up to twice the density. Integrating the fermion hole gives 0.9992 particles missing from around each one — exactly one, and the sum rule holds at any density, because raising the density narrows the hole in exact proportion. That is what makes the effect worth a name of its own. It is often called an exchange force and it is not a force: nothing carries momentum between the particles, and no term in the energy is proportional to a distance. It is a statement about which states exist. What follows from it is most of chemistry — the reason two atoms with filled shells repel, the reason a metal's electrons cost so much less Coulomb energy than a random arrangement would, and the reason matter takes up room.

    The force with no force in it

    Two identical fermions keep apart and two identical bosons crowd together, and neither is being pushed. The Hamiltonian contains no interaction at all: what produces the hole and the pile is which many-particle states exist, and the hole it digs around each electron holds exactly one particle at any density whatever.

    part 3 · quantum

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