---
title: "Normalisation and Expectation Values"
description: "A finite, nonzero wavefunction norm determines the magnitude of a normalising constant, but leaves its overall phase free. Position averages integrate x against the probability density. Other observab"
canonical: https://lightmysky.com/learn/science/normalisation-and-expectation-values-mt_6Y3DZ8P1qm
source: https://lightmysky.com/learn/science/normalisation-and-expectation-values-mt_6Y3DZ8P1qm.md
retrieved: 2026-11-12
---

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# Normalisation and Expectation Values

A finite, nonzero wavefunction norm determines the magnitude of a normalising constant, but leaves its overall phase free. Position averages integrate x against the probability density. Other observables use their operators.

Subject: Science · Area: Quantum & Modern Physics · Ages 19 to 20
Page: https://lightmysky.com/learn/science/normalisation-and-expectation-values-mt_6Y3DZ8P1qm

## Ready when they can

- Normalises a given wavefunction over its allowed range
- Computes the expectation value of position for a stated state
- Explains why the expectation value need not be a value the measurement can return

## Lesson: One total chance and honest averages

Start with a candidate function u(x). Its squared norm is N = ∫ |u(x)|² dx over the full range. If 0 < N < ∞, set ψ(x) = u(x)/√N. Then ρ(x) = |ψ(x)|² has integral one. Interval probability is ∫ₐᵇ ρ(x) dx. Multiplying ψ by a constant of magnitude one changes its overall phase but leaves these probabilities unchanged.

**Example.** Use dimensionless x. Let u(x) be 1 on [-1, 0], 2 on (0, 1], and zero elsewhere. Its squared norm is 1² × 1 + 2² × 1 = 5, so ψ = u/√5. The two interval probabilities are 1/5 and 4/5. On a constant-density interval, multiply its width by the density. For ψ = Cu, normalisation requires |C|²N = 1; choosing C positive gives C = 1/√N.

*(drawing: The intervals have equal widths but different probabilities. Bar heights show probability, not wavefunction amplitude.)*

For a normalised one-dimensional state, position expectation is E[X] = ∫ xρ(x) dx = ∫ x|ψ(x)|² dx. Require ∫ |x||ψ(x)|² dx < ∞ for a finite mean. Multiply position by its density before integrating. This predicts the average across freshly prepared copies of the same state, not the result of one trial.

Complex conjugation reverses the sign of the imaginary part. The conjugate of ψ = a + ib is a - ib, so |ψ|² = (a - ib)(a + ib) = a² + b². For ψ = 3 + 4i, the density is 9 + 16 = 25. Using ψ² instead can give a complex or negative result. A density may exceed one; its integral over an interval, not its height, is the probability.

For a normalised density with finite ∫ |x|ρ(x) dx, E[X] = ∫ xρ(x) dx. If ρ(-x) = ρ(x), the contributions xρ(x) and -xρ(-x) cancel, so E[X] = 0. The mean need not be a possible location: equal probability in separated symmetric regions can balance in a gap where density is zero.

**Recap.** Normalisation makes ∫ |ψ|² dx = 1. For a finite position mean, integrate x|ψ|². Squared magnitude, not ψ², is the probability density.

*Lesson and questions written for this dataset using standard probability-density and wavefunction-normalisation methods.*

## Practice

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

## Needs first

- [The Wavefunction and the Born Probability Rule](https://lightmysky.com/learn/science/the-wavefunction-and-the-born-probability-rule-mt_elzmbBzmKE)
- [Expectation and Variance by Integration](https://lightmysky.com/learn/mathematics/expectation-and-variance-by-integration-mt_kRSaKgp3bs)

## Optional connections

- [Continuous Random Variables and Density Functions](https://lightmysky.com/learn/mathematics/continuous-random-variables-and-density-functions-mt_jlf4zDTeNm)
- [Expected Value and Variance of a Discrete Random Variable](https://lightmysky.com/learn/mathematics/expected-value-and-variance-of-a-discrete-random-variable-mt_PFdo0Nm3FP)

## Opens up

- [States as Vectors: Dirac Notation and the State Space](https://lightmysky.com/learn/science/states-as-vectors-dirac-notation-and-the-state-space-mt_9g0aNMxQ8M)
- [The Uncertainty Principle from Wave Packets](https://lightmysky.com/learn/science/the-uncertainty-principle-from-wave-packets-mt_DPYnoGksRd)