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Dirac Bra-Ket Notation

Also known as: Bracket notation · Dirac notation

Dirac split the bracket ⟨φ|ψ⟩ into a 'bra' ⟨φ| and a 'ket' |ψ⟩. A ket is a state vector; a bra is its dual, waiting to be paired with a ket to produce a number — their overlap, which measures how similar two states are. This notation hides the messy integrals and matrices: it works identically whether the state lives in an infinite-dimensional function space or a 2-level qubit. It's the universal shorthand of quantum mechanics.

⟨ϕ∣ψ⟩=∫ϕ∗(x)ψ(x) dx\langle\phi|\psi\rangle=\int\phi^*(x)\psi(x)\,dx
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Two state vectors on the Bloch-like plane; their bra-ket overlap is the projection of one onto the other.

Equivalent forms

A^∣ψ⟩,  ⟨ψ∣A^∣ψ⟩,  ∣ψ⟩=∑ncn∣n⟩\hat{A}|\psi\rangle,\;\langle\psi|\hat{A}|\psi\rangle,\;|\psi\rangle=\sum_n c_n|n\rangle
A single elegant bracket does the job of integrals, sums, and matrix products — write physics once and it works in any Hilbert space.