On this page
Many predictions in physics start from an experiment that hints at something. Paul Dirac's prediction of antimatter came from mathematics. He wrote an equation for the electron, found it had solutions nobody wanted, and instead of discarding them asked what they meant. This post walks through that reasoning and includes a worked example you can do with JEE-level physics.
Who was Paul Dirac?
The Nobel Prize facts page records that Paul Adrien Maurice Dirac was born on 8 August 1902 in Bristol, England, and died on 20 October 1984 in Tallahassee, Florida. He shared the 1933 Nobel Prize in Physics with Erwin Schrodinger "for the discovery of new productive forms of atomic theory". Our post on Schrodinger's wave equation covers the other half of that prize.
What problem was Dirac trying to solve?
By 1926, quantum mechanics described electrons in atoms well, but only at speeds much lower than light. It ignored special relativity. Dirac wanted an equation that was correct at any speed.
The starting point is the relativistic energy relation, E² = p²c² + m²c⁴. In his Nobel lecture of December 1933, Dirac explains that quantum mechanics requires a wave equation that is linear in energy, not squared, and that relativity then requires it to be linear in momentum too. To make that work he had to introduce new quantities that, he said, "give rise to the spin of the electron". His paper The quantum theory of the electron appeared in the Proceedings of the Royal Society in February 1928.
So spin, which had been added to atomic theory by hand to fit spectra, came out of the equation automatically.
Why did the equation predict a new particle?
Take the square root of E² = p²c² + m²c⁴ and you get two answers, E = +√(p²c² + m²c⁴) and E = −√(p²c² + m²c⁴). In classical physics you simply ignore the negative one. Dirac could not.
In his Nobel lecture he explains why. The equation allows energies greater than mc² or less than −mc², and "if we disturb the electron, we may cause a transition from a positive-energy state of motion to a negative-energy one". The negative states could not be ruled out by a new assumption.
His answer used the Pauli exclusion principle. Suppose nearly all negative-energy states are already filled, one electron per state, and that this uniform sea is unobservable. Then an empty state, a "hole", behaves like a particle with positive energy and positive charge. In the lecture he describes this as "a mirror-image of the electron, having exactly the same mass and opposite charge". He developed the idea in a 1931 paper. As the lecture explains, the theory also predicts that an electron can drop into a hole, so the pair annihilates into radiation, and that radiation can create an electron and a positron.
How was the positron found?
In 1932 Carl Anderson, photographing cosmic-ray tracks in a cloud chamber, found a positively charged particle with a mass apparently equal to the electron's. The Nobel page for Anderson, who shared the 1936 physics prize, calls it the first antiparticle confirmed by experiment. He reported it in The positive electron in Physical Review in March 1933.
Dirac also suggested in his lecture that "negative protons can exist", though they would be harder to make. Emilio Segre and Owen Chamberlain confirmed the antiproton in 1955.
A worked example: the energy needed to make a positron
One way to make antimatter is pair production: a high-energy photon passing near a nucleus turns into an electron and a positron.
Rest energy of one electron. NIST gives mc² = 0.511 MeV. A positron has the same mass.
Minimum photon energy for a pair. You must create two rest masses, so E = 2 × 0.511 = 1.022 MeV. (The nucleus takes a little recoil, which we neglect.)
Wavelength of that photon. Use λ = hc / E with hc ≈ 1240 eV nm. λ = 1240 / (1.022 × 10⁶) nm ≈ 1.21 × 10⁻³ nm, or about 1.21 pm. That is a gamma ray.
The reverse. When an electron and positron at rest annihilate, they usually produce two photons sharing 1.022 MeV, so each carries 0.511 MeV, with wavelength about 2.43 pm.
Common mistakes: using one electron mass instead of two, and forgetting to convert MeV to eV before dividing.
What can a student take from his way of working?
Demand consistency. Dirac's starting point was refusing to accept a theory that broke relativity.
Follow the mathematics, then interpret. The negative solutions were a clue, not an error.
Say what would test you. His theory predicted annihilation and pair creation, both of which experiments could look for.
For more practice with energy and wavelength, try our posts on de Broglie wavelength problems and nuclear binding energy problems. If you get stuck on the first step of a modern physics numerical, the Fermi Problem Library has a step-by-step solver that explains the reasoning, not just the answer.
Work out the minimum photon energy to create a proton-antiproton pair, using 938 MeV for the proton's rest energy.
