Space-Time Approach to Non-Relativistic Quantum Mechanics
To find where a particle goes, add up every path it could take.
To predict where a quantum particle lands, Feynman said: don't pick its path — add up every path it could possibly take.
The idea, unpacked
In everyday physics a ball follows one path. Feynman's startling claim is that a quantum particle, going from here to there, in effect explores every path at once — the straight one, the looping one, the absurd detour to the Moon and back.
Each path carries a tiny spinning clock hand — a phase. To find the chance of arrival, you add up all those clock hands as little arrows laid tip to tail, and square the length of the result. Most paths point every which way and cancel. The ones that survive are clustered around the single path of 'least action' — which is exactly the path ordinary physics predicts.
Where it came from
As a graduate student at Princeton in the early 1940s, Richard Feynman was hunting for a way to do quantum mechanics from the action — the quantity that, made smallest, gives classical motion. At a beer party, a visiting physicist mentioned that Dirac had written, a decade earlier, that the quantum amplitude was 'analogous to' exp(iL/ħ).
Feynman asked what 'analogous' meant, worked through it at the blackboard that evening, and found it was not merely analogous but proportional — the seed of his whole method. He built it into his 1942 doctoral thesis under John Wheeler, and finally published the polished version in 1948, after the war and the Manhattan Project.
Why it mattered
It gave physics a third way to do quantum mechanics, exactly equivalent to the two textbook versions but often far easier — and, because it is built on the action, the natural language for the quantum theory of fields. It also delivered a picture: the principle of least action, which had long seemed a strange piece of mathematical luck, turned out to be quantum interference in disguise. Out of this approach came Feynman diagrams and much of the modern toolkit of particle physics.
A million clock hands
Imagine every possible route from your door to a friend's house, and on each route a stopwatch whose hand spins as you walk. Lay the final hand positions end to end, as little arrows. Routes that are wildly different have hands pointing all over and largely cancel. But the bundle of routes near the shortest, smoothest one have nearly the same hand position and reinforce. The particle's chance of arriving is the length of the total arrow, squared. Slide ħ in the panel below and watch the arrows curl into a spiral.
Where it sits
This is the third pillar of quantum mechanics, joining Heisenberg's matrices (1925) and Schrödinger's wave equation (1926) — both in this Library — and it reaches back to the principle of least action that runs from Maupertuis and Euler through Lagrange. Looking forward, it becomes the path integral of quantum field theory and underlies almost everything in modern theoretical physics, including the Higgs work also gathered here.
Non-relativistic quantum mechanics is formulated here in a different way. It is, however, mathematically equivalent to the familiar formulation.
In quantum mechanics the probability of an event which can happen in several different ways is the absolute square of a sum of complex contributions, one from each alternative way.