Dirac Sea — Holes, Antimatter & the Filled Vacuum

Concept: Dirac's vacuum: every negative-energy state E₋ = −√((pc)² + (mc²)²) is already occupied. Lifting one electron across the 2mc² = 1.022 MeV gap costs 2E(p) and leaves a vacancy that behaves as a positron: charge +e, momentum −p_sea, positive energy. Net charge of vacuum + excitations stays exactly zero.
Serves: Relativistic quantum mechanics / intro particle physics; bridges naturally to the semiconductor electron–hole picture used later in solid state.
Inquiry: The sim opens on an undisturbed sea with the field ℰ off. Commit to the drift prediction, then click anywhere on the sea (or press Excite) to pay the exact cost 2E(p) at a momentum you choose, stretch the energy axis to feel the bottomless depth, and switch ℰ on to watch e⁻ and h⁺ separate while net charge holds at zero.
Misconception: Tested rather than asserted: the prediction asks whether a vacancy can move at all, and the real-space strip answers it — with ℰ on, h⁺ drifts opposite e⁻, and a wrong prediction is drawn as a dashed ghost that visibly falls behind the real holes. A hole carries +e and mass m, so it moves and carries current: it is the positron.
Caveats: Dots in the sea are schematic occupancy, not a state density. Speeds use v = pc²/E. Annihilation photons carry off whatever momentum the matter loses.

Principle	Where it appears
Predict before reveal	Opens with an empty sea and ℰ = 0; the three prediction chips sit above the controls, and the field only comes on once the student commits. Nothing is ever disabled — the order does the work.
Misconception targeted structurally	A wrong choice is rendered, not just corrected: dashed ghost h⁺ follow the predicted motion beside the solid real holes until they visibly separate.
Invariants made visible	Net charge is recomputed from the particle list every frame (always 0); net momentum of matter is shown in MeV/c with a note that annihilation photons carry the remainder.
Operational quantities	Every pair prints its exact cost 2E(p) — 2.25 MeV at p = 1 MeV/c; both axes carry units (MeV and MeV/c); internally the ℰ slider applies eℰ = 4.0 MeV/c per second at ℰ = 1.
The Physics sheet on demand	Dispersion relation, pair cost, thermal weight, the hole↔positron map and the drift equations sit behind the ⚛︎ The Physics button, as a chaptered revision sheet with a live worked run.

Dirac's Sea of Electrons

Teach me Hide Text

Speed

Energy vs momentum, lab frame — click the sea to make a pair

e⁻ electron

h⁺ hole (= positron)

empty state

Real space, lab frame — periodic box of length L

Guided Inquiry

1 · The filled sea

Let’s start with Dirac’s big idea. His vacuum isn’t empty space at all: every negative-energy state already holds an electron. For each momentum there’s a positive-energy branch and a negative-energy branch, with a gap between them — you can see both on the plot.

2 · Crossing the gap

Now let’s try to get one out. An electron in the sea sits at energy −E. To occupy the empty positive branch at the same momentum, it has to reach +E. So how much energy do you think must be supplied?

3 · Costlier further out?

Now let’s move along the momentum axis. The cheapest pair is made at rest, p = 0. What do you think a pair will cost when |p| is larger?

4 · Charge of a vacancy

Now look at what gets left behind. Each pair is one electron lifted from the sea plus the vacancy it leaves. What charge do you think that vacancy carries?

5 · Which way does the hole go?

One last experiment. A hole is a missing sea electron. When an electric field is applied, which way do you think the hole will move?

Listen

Controls

Sign alternates each click — or click any point on the sea to pick p yourself.

Field ℰoff

ℰ = applied electric field.

Axis depth±3.0 MeV

Temperature kT0.25 MeV

≈ 2.9×10⁹ K

Readouts

pairs made (electron + hole)	0
cost to make the last pair	—
net charge	0 e
net momentum (matter)	0 MeV/c

Try it: make a pair near p = 0, then click far out on the branch

Two energy branches for every momentum p.

Minimum pair cost = the gap at p = 0.

Thermal odds of a jump of size ΔE (background rate only).

A vacancy in the sea is a positron.

Opposite charges ⇒ opposite drift under ℰ.

The Physics

Dirac's sea of electrons · what this sim shows, the physics behind it, and every control

At a glance

Dirac's equation gives every electron momentum two possible energies, one positive and one negative. This sim lets you disturb the filled negative-energy "sea" and see what the disturbance creates: the reason the idea predicted antimatter.

What you'll learn

Why Dirac's equation has two energy branches, and what it means to fill the lower one.
How the energy needed to lift an electron out of the sea depends on its momentum.
What a vacancy in the sea is, and how it responds to an electric field.
How temperature sets how often pairs appear on their own, and what happens when they meet.

Energy against momentum, with the sea drawn as rows of dots.

What's on screen

Teal curves: the two energy branches. The shaded region under the lower one is the sea.
e⁻ electron (blue): the dots in the sea, and a solid dot on the upper branch once lifted.
h⁺ hole (red ring): a vacancy in the sea, with a dashed twin on the upper branch.
Gold: an arrow while a pair forms, and a γ γ flash when one annihilates.
Strip (below the plot): a periodic box of length L, electrons on the top row, holes on the bottom.
Readouts (collapsed at first): pairs present, cost of the last pair, net charge, net momentum.

See itThe sim lifts one electron out of the undisturbed sea. Watch for: a gold arrow carrying one electron up to the upper branch, and a red ring left behind in the sea.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
⚡ Excite a pair	Lifts one electron out of the sea at p = +1 MeV/c, then −1 MeV/c on the next click. Clicking anywhere on the main plot picks p yourself.	\|p\| up to 3.5 MeV/c
Field ℰ	The applied electric field. Positive and negative push the charges in opposite directions.	−1 to +1 opens at 0 (off)
Axis depth	How far the energy axis reaches above and below zero. It rescales the picture, not the physics.	±2 to ±15 MeV opens at ±3
Temperature kT	How often pairs form by themselves, and how widely they spread in momentum.	0.05–1.5 MeV opens at 0.25
Readouts	Pairs present now, the cost of the last pair made, and the net charge and net momentum of the matter.	panel click to open
⏸ Pause / ▶ Play	Freezes the motion so you can read the picture.	top bar
Speed	Plays the motion faster or slower on screen. The physics stays the same.	0.25×–4× opens at 1×
↻ Reset	Clears the sea back to its undisturbed state and restores the opening settings.	top bar

Fixed: mc² = 0.511 MeV, and the momentum axis runs to ±3.5 MeV/c. Spin is not drawn, and the dots of the sea are schematic.

The filled sea

For a free electron of momentum p, Dirac's equation allows two energies:

Two branches, mirror images of each other. The sim draws exactly this curve.

Nothing in the equation stops an electron from dropping from the upper branch to the lower one, and then lower and lower without limit.
Dirac's answer: every state on the lower branch is already occupied, and the Pauli exclusion principle (one electron per state) blocks the fall.
This filled lower branch is what the theory calls the vacuum. It looks empty because it has no charge and no momentum to report. Only changes to it show up.

Key idea. An electron can sit on the upper branch without falling, because the lower branch is already full.

The sea has no bottom. The sim draws only the top of it, and fades it out at the lower edge of the plot.

See itThe energy axis is stretched to ±15 MeV. Watch for: the shaded sea still running down to the bottom edge of the plot, now at −15 MeV.

What a pair costs

To make a pair, one electron is lifted from −E(p) to +E(p) at the same momentum. The energy supplied is the whole jump:

At rest the cost is 2mc² = 1.022 MeV, the least a pair can cost.

Moving electrons cost more, because the two branches bend away from each other as |p| grows. The extra is kinetic energy, shared by the electron and its partner.

momentum p	cost 2E(p)
0	1.02 MeV
1 MeV/c	2.25 MeV
2 MeV/c	4.13 MeV

Each time a pair is made, the cost appears as a chip on the plot and in the readouts.

See itThe sim makes a pair at p = 0, then one at p = +2 MeV/c. Watch for: the cost readout jumping from 1.02 MeV to 4.13 MeV.

A vacancy is a positron

The lifted electron leaves a vacancy in the sea. Measured against the full sea, the vacancy has the opposite of each property of the missing electron:

Charge. Removing −e from a neutral sea leaves +e.
Energy. Removing an electron of energy −E raises the sea's energy by +E.
Momentum. Removing momentum p leaves −p.

The vacancy behaves as a particle with the electron's mass and charge +e: a positron.

The sim draws one vacancy twice: where the electron left (solid, lower branch) and where the positron sits (dashed, upper branch).

The electron and the positron carry opposite charge and opposite momentum, so a created pair adds 0 e of charge and 0 MeV/c of momentum to the sea.

See itThe sim makes a pair at p = +2 MeV/c. Watch for: a solid red ring where the electron left, a dashed twin at the mirror-image place on the upper branch, and the net charge staying at 0 e.

Opposite charges drift opposite ways

An electric field ℰ pushes the electron one way and the positron the other. The sim adds a drag term −p/τ as a stand-in for collisions, so the momentum levels off instead of growing forever.

The drift law the sim integrates. τ = 0.7 s is the sim's choice, not a property of the sea.

The field shifts every state in the sea by the same momentum, vacancies included. A positron's momentum is minus its vacancy's, so it is shifted the other way.
The levelled-off momentum |p|_∞ = eℰτ is 2.8 MeV/c at ℰ = 1. There v/c = pc/E is about 0.98, so the speed stays below c.
A created pair is neutral, so the field gives it no net force. It only pulls the two apart.

The strip's speed is a display scale: 0.2 box-lengths per second at v = c.

See itThree electron–hole pairs are set up, and the field is switched on. Watch for: the blue e⁻ dots and the red h⁺ dots in the strip moving in opposite directions.

Heat makes pairs, meeting removes them

At temperature kT the sea makes pairs on its own. The chance of a jump that costs ΔE falls as e^−ΔE/kT, so cheap pairs, near p = 0, are the common ones.

The exponential is the physics. The prefactor 1.5 s⁻¹ is the sim's choice, so that events are slow enough to watch.

Halving kT from 0.50 to 0.25 MeV makes pairs about 8 times rarer (0.19 to 0.025 per second). The reverse process also runs: an electron falls into a vacancy, the pair vanishes, and photons carry off the energy and momentum.

At rest each photon carries 0.511 MeV. The sim uses a rate of 0.02 per second for each electron–hole combination.

ℰ kT

These sliders move the sim too.

See itThe sea is heated to kT = 1.0 MeV with no field and no clicks. Watch for: pairs appearing on their own, and soon a gold γ γ flash where one annihilates (the timing is random).

Where it matters

The positron. Dirac's equation led to the prediction of an electron-mass particle with charge +e. Carl Anderson saw it in a cloud chamber in 1932, in cosmic rays.
PET scanners. A positron from a tracer's decay annihilates with an electron. The two 511 keV photons leave in nearly opposite directions, and detecting both locates the annihilation.
Pair production. A photon needs at least 1.022 MeV to make a pair, and a nearby nucleus to take up the recoil.
Semiconductors. A filled band with a gap above it, and a mobile vacancy behaving as a positive charge, is the same picture at electron-volt scale. In silicon the gap is about 1.1 eV.

Go deeper: how real is the sea

Quantum field theory describes the same antiparticles without an infinite filled sea. The hole picture is correct bookkeeping for electrons, but it does not carry over to particles that are not subject to the exclusion principle.
The sim keeps the picture, including its thermal and drift behaviour, but its drag, its rates and the one-electron-per-momentum lattice are teaching devices.

Summary

In one line. Filling the lower branch makes the vacuum, and lifting an electron out of it creates a pair: an electron and a vacancy that acts as a positron, at a cost of 2E(p).

Key points

E = ±√((pc)² + (mc²)²): two branches, the lower one filled.
A pair costs 2E(p), at least 1.022 MeV, and more at larger |p|.
A vacancy has charge +e and opposite momentum: it is a positron.
Opposite charges drift opposite ways; the pair rate rises steeply with kT.

Common mix-ups

"A pair always costs 1.022 MeV." That is the minimum, at p = 0. The cost is 2E(p), and it grows with |p|.
"A vacancy is only an absence, so it has no charge." Taking −e out of a neutral sea leaves +e behind.
"The hole should drift the same way as the electrons." The positron carries the opposite charge, so the field pushes it the opposite way.

Things to try on your own

Click the main plot at p = +3 MeV/c. Compare the cost chip with 2√(3² + 0.511²) MeV.
Set kT to 0.5 MeV, count the pairs that appear in 30 seconds, then repeat at 0.25 MeV. Compare the ratio with section 07.
Set ℰ negative and note which way the electrons move in the strip.