---
title: "The Path Integral and the Sum Over Histories"
description: "An amplitude can be written as a sum over every path from start to finish, each contributing a phase equal to its action in units of the Planck constant. Paths near the stationary one add in step and "
canonical: https://lightmysky.com/learn/science/the-path-integral-and-the-sum-over-histories-mt_QRLheflB1O
source: https://lightmysky.com/learn/science/the-path-integral-and-the-sum-over-histories-mt_QRLheflB1O.md
retrieved: 2026-09-12
---

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# The Path Integral and the Sum Over Histories

An amplitude can be written as a sum over every path from start to finish, each contributing a phase equal to its action in units of the Planck constant. Paths near the stationary one add in step and the rest cancel, which is why the classical path appears at all.

Subject: Science · Area: Quantum & Modern Physics · Ages 23 to 24
Page: https://lightmysky.com/learn/science/the-path-integral-and-the-sum-over-histories-mt_QRLheflB1O

## Ready when they can

- Writes a propagator as a sum over paths weighted by the phase of the action and says what the weight means
- Recovers the classical trajectory as the stationary-phase limit and states when that limit fails
- Explains double-slit interference and tunnelling in the same language, without switching pictures between them

## Lesson: Every path counts, one path survives

An amplitude can be written as a sum over every path from start to finish. Each path contributes a small arrow, a phase, set by its action in units of a tiny constant.

Big actions spin the arrows fast, so neighboring paths cancel except near the stationary one. When the action is huge next to that constant, only the neighborhood of the stationary path adds up, and that survivor is the classical trajectory. Near barriers or turning points the limit fails.

**Example.** In the double slit, each slit offers a family of paths. Where crest meets crest the arrows line up and the screen is bright; where crest meets trough they cancel and the fringe is dark. The pattern is the sum, not one favored ray.

**Tip.** Tunnelling speaks the same language: paths crossing the barrier count with rapidly shrinking weight. A marble without the energy stays trapped, yet the small surviving weight lets alpha particles leak out, as Gamow showed in 1928.

**Recap.** Add an arrow per path, watch all but the stationary neighborhood cancel, and read interference and tunnelling as the same sum.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Unitary Time Evolution and the Heisenberg Picture](https://lightmysky.com/learn/science/unitary-time-evolution-and-the-heisenberg-picture-mt_3JsTc9VYj8)
- [Hamilton's Principle and the Action](https://lightmysky.com/learn/science/hamiltons-principle-and-the-action-mt_a273REeeGu)
- [Quantum Tunnelling and the Transmission Probability](https://lightmysky.com/learn/science/quantum-tunnelling-and-the-transmission-probability-mt_tLdPJKx8Id)