---
title: "The Wavefunction and the Born Probability Rule"
description: "For a single particle in a pure state, a position wavefunction gives complex amplitudes. Their squared magnitude gives position probability density. Integrating this density over a region predicts the"
canonical: https://lightmysky.com/learn/science/the-wavefunction-and-the-born-probability-rule-mt_elzmbBzmKE
source: https://lightmysky.com/learn/science/the-wavefunction-and-the-born-probability-rule-mt_elzmbBzmKE.md
retrieved: 2026-11-12
---

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# The Wavefunction and the Born Probability Rule

For a single particle in a pure state, a position wavefunction gives complex amplitudes. Their squared magnitude gives position probability density. Integrating this density over a region predicts the probability of detecting the particle there.

Subject: Science · Area: Quantum & Modern Physics · Ages 18 to 20
Page: https://lightmysky.com/learn/science/the-wavefunction-and-the-born-probability-rule-mt_elzmbBzmKE

## Ready when they can

- States that the squared magnitude of the wavefunction gives position probability density and explains how interval probability is obtained
- Interprets an interference pattern built one particle at a time
- Distinguishes the wavefunction itself from the probability density it produces

## Lesson: Where the particle might be

Quantum mechanics uses a wave function, written psi, for a single particle in a pure state. Here we consider position along a line and leave spin out. Psi is a complex-valued mathematical description used to predict measurements, including where the particle may be detected. It is not something you can see or touch. Quantum mechanics also explains phone chips and lasers.

The Born rule gives position density |psi(x)|², the squared magnitude of psi. Probability in an interval is the area under this density, found by integrating over that interval. For constant density, this area is a rectangle: density times width. Multiplying density by width is an approximation when the density varies little.

A normalized wave function has a total position probability of one. Integrate |psi(x)|² over all space to find that total. Normalization expresses certainty that a detection position is somewhere in space, not a specific position in advance.

**Example.** Send identically prepared electrons through two open slits without which-slit information. Keep only one electron in the apparatus at a time. Each detection gives one dot. Psi does not predict its exact position. Many dots build an interference pattern described by the squared magnitude of the combined amplitude. The pattern does not require several electrons to be present together.

*(drawing: Use squared magnitude for density. It is real and nonnegative; an ordinary complex square need not be.)*

With both slits open and no which-slit information, add the amplitudes from the two slits before taking the squared magnitude. Amplitudes can add or cancel. Adding the separate single-slit probabilities misses this interference. The Born rule predicts detection probabilities; it alone does not settle whether a particle had a definite unmeasured path.

Compare interval probabilities using their areas, not just density at one point. For constant density, probability equals density times width. For two equal-width intervals, the one with higher constant density has greater probability and is expected to collect more detections over many identical preparations. Finite totals can fluctuate.

**Recap.** Take the squared magnitude of psi for position density. Integrate over an interval to find its probability; normalized density integrates to one over all space.

*Lesson corrections and replacement questions authored for this dataset.*

## Practice

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

## Needs first

- [The Imaginary Unit and Complex Arithmetic](https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550)
- [The Compton Effect and Photon Momentum](https://lightmysky.com/learn/science/the-compton-effect-and-photon-momentum-mt_wt7kdaof_P)
- [de Broglie Waves and Electron Diffraction](https://lightmysky.com/learn/science/de-broglie-waves-and-electron-diffraction-mt_Y8NthsmPcv)

## Opens up

- [Normalisation and Expectation Values](https://lightmysky.com/learn/science/normalisation-and-expectation-values-mt_6Y3DZ8P1qm)