---
title: "States as Vectors: Dirac Notation and the State Space"
description: "A pure quantum state is represented by a vector in a complex Hilbert space. Its wavefunction gives components in a chosen representation. Bra-ket notation expresses states, overlaps and changes of rep"
canonical: https://lightmysky.com/learn/science/states-as-vectors-dirac-notation-and-the-state-space-mt_9g0aNMxQ8M
source: https://lightmysky.com/learn/science/states-as-vectors-dirac-notation-and-the-state-space-mt_9g0aNMxQ8M.md
retrieved: 2026-11-12
---

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# States as Vectors: Dirac Notation and the State Space

A pure quantum state is represented by a vector in a complex Hilbert space. Its wavefunction gives components in a chosen representation. Bra-ket notation expresses states, overlaps and changes of representation without changing physical predictions.

Subject: Science · Area: Quantum & Modern Physics · Ages 22 to 23
Page: https://lightmysky.com/learn/science/states-as-vectors-dirac-notation-and-the-state-space-mt_9g0aNMxQ8M

## Ready when they can

- Writes a state as a superposition of basis kets and reads off the amplitudes
- Converts between a ket and the wavefunction that represents it in a chosen basis
- Uses an inner product to compute an overlap and says what that number predicts

## Lesson: One quantum state, many languages

A pure state is represented by a ket |ψ> in a complex Hilbert space. Its bra <ψ| is the conjugate transpose. The position wavefunction is ψ(x) = <x|ψ>; the ket is representation-independent. Full-line position and momentum eigenkets are generalised, not unit vectors. Position kets use <x|x'> = δ(x - x'). The delta selects a value under an integral: ∫ δ(x - x')g(x') dx' = g(x) for a smooth test function g. Momentum kets use the same convention.

**Example.** In a discrete orthonormal basis, |ψ> = a|0> + b|1>. The normalisation condition is |a|² + |b|² = 1. Measuring in this basis gives probabilities |a|² and |b|². For |ψ> = (|0> + i|1>)/√2, both probabilities are 1/2. A coefficient's phase can matter for other measurements even when its squared magnitude stays the same.

*(drawing: These bars show probabilities in the stated discrete basis. They do not show the complex amplitudes.)*

In an orthonormal basis, conjugate the bra coefficients, multiply corresponding coefficients and add: <φ|ψ> = conj(c)a + conj(d)b. Here ψ has coefficients a,b; φ has c,d; conj means complex conjugate. The bracket is one complex number. <ψ|ψ> = |a|² + |b|² is squared norm, equal to 1 when normalised. For normalised states, |<φ|ψ>|² gives projection probability; zero overlap means orthogonality.

A finite-dimensional basis is a linearly independent set spanning the space. Spanning lets every vector be expressed using the set; independence means no vector is redundant. An orthonormal basis additionally has unit lengths and zero pairwise overlaps. Ordinary bases need not be orthonormal.

A passive change of representation rewrites the same ket using different components. It does not rotate or otherwise change the prepared state. Rewriting both the state and a fixed observable consistently leaves every prediction unchanged. This applies when changing between energy and position representations.

Let |0>, |1> be orthonormal and |±> = (|0> ± |1>)/√2. A normalised state a|0> + b|1> has ± coefficients (a + b)/√2 and (a - b)/√2. Squared coefficient magnitudes give probabilities for the basis being measured. Measuring in the original basis still gives |a|² and |b|². New coefficients describe a different measurement basis, not changed probabilities for a fixed measurement.

**Recap.** A ket represents a pure state. In a discrete orthonormal measurement basis, squared amplitude magnitudes give probabilities; |ψ(x)|² is a position density.

*Lesson and questions written for this dataset using standard bra-ket notation and inner-product methods.*

## Practice

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

## Needs first

- [Normalisation and Expectation Values](https://lightmysky.com/learn/science/normalisation-and-expectation-values-mt_6Y3DZ8P1qm)
- [Operators, Eigenvalues and Measurement](https://lightmysky.com/learn/science/operators-eigenvalues-and-measurement-mt_E9C_9DlwD2)

## Optional connections

- [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk)
- [Hilbert Spaces and Orthogonal Projection in Infinite Dimensions](https://lightmysky.com/learn/mathematics/hilbert-spaces-and-orthogonal-projection-in-infinite-dimensions-mt_xmKq8SAgvy)

## Opens up

- [Observables as Hermitian Operators and Their Spectra](https://lightmysky.com/learn/science/observables-as-hermitian-operators-and-their-spectra-mt_TXCN8uiWqs)