---
title: "The Path Integral and the Sum Over Histories"
description: "A position-space propagator can be expressed as a path integral with phase factor exp(iS/ℏ), where S is the action and ℏ is the reduced Planck constant. Under suitable semiclassical conditions, statio"
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: 2026-11-12
---

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# The Path Integral and the Sum Over Histories

A position-space propagator can be expressed as a path integral with phase factor exp(iS/ℏ), where S is the action and ℏ is the reduced Planck constant. Under suitable semiclassical conditions, stationary-action paths give leading contributions. The same amplitude framework describes interference and tunnelling.

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

- Identifies the path-integral phase factor and integration domain, and distinguishes amplitudes from probabilities
- Explains stationary-action contributions in a semiclassical approximation and identifies limits of a single-path account
- Explains coherent double-slit interference and tunnelling without assigning damping to individual real-action phase factors

## Lesson: Path amplitudes and the classical limit

For one nonrelativistic particle, K(x_f,t_f;x_i,t_i)=<x_f|U(t_f,t_i)|x_i> is a position-space propagator. Formally, K=∫D[x] exp(iS[x]/ℏ), with endpoint positions and times fixed. S[x]=∫L dt is the action and ℏ=h/(2π) is the reduced Planck constant. Here i²=-1. For real S, exp(iS/ℏ) has modulus one.

D[x] denotes integration over admissible histories, including normalization. A construction such as time slicing defines the formal expression. Include nonclassical histories as well as classical ones. Add complex amplitudes before taking a squared modulus; individual paths do not carry separate probabilities. Applying K to an initial wavefunction gives the final wavefunction, whose squared modulus is position probability density.

*(drawing: The phase factor and integration measure together define the amplitude sum.)*

For neighboring histories, the phase difference is ΔS/ℏ. Rapid phase variation makes nonstationary contributions largely cancel. At a stationary-action path, δS=0 for endpoint-preserving variations, so the first-order phase change vanishes. Nearby contributions can reinforce. Stationarity gives the classical equations of motion; it does not require zero action or the shortest geometric route.

This stationary-phase argument gives an approximation under suitable conditions, not exact deletion of other paths. Large absolute action alone is insufficient; relevant phase variation matters. Several stationary paths can contribute, and their amplitudes must be combined. Turning points and caustics can need different local approximations. These limitations do not make the full path integral undefined.

**Example.** For coherent slit contributions A1 and A2, the screen amplitude is A1+A2 and intensity is proportional to |A1+A2|². Equal contributions with the same phase reinforce. Equal contributions a and -a cancel exactly. Opposite phase alone is insufficient: 2a and -a leave a. Each slit contribution represents a family of histories, not one measured particle route.

The same propagator describes a finite static barrier. Where E<V, no real classical trajectory at energy E crosses the forbidden region, but the quantum transmitted amplitude can be nonzero. For a thick smooth barrier, a semiclassical calculation gives exponential suppression of transmission. That suppression comes from evaluating the amplitude, not damping each individual real-action phase factor. Total energy is conserved.

In a simple alpha-decay model, λ=fT. Here T is barrier transmission probability, λ is decay rate, and f includes attempt frequency and alpha-formation probability. The half-life is t_half=ln(2)/λ. With f fixed, tripling T triples λ and divides the half-life by three. Transmission alone does not determine a half-life. This barrier-penetration model does not require borrowed energy.

**Recap.** Sum amplitudes with phase exp(iS/ℏ). Stationary action guides a semiclassical approximation; interference and tunnelling require the amplitude sum.

*Original lesson, questions, worked solutions and flashcards written for this dataset.*

## 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)

## Optional connections

- [Quantum tunnelling and the transmission probability](https://lightmysky.com/learn/science/quantum-tunnelling-and-the-transmission-probability-mt_tLdPJKx8Id)