Bra-ket notation (|ψ⟩)

How to read it

Dirac published it in “A new notation for quantum mechanics”, in the Mathematical Proceedings of the Cambridge Philosophical Society in July 1939 (vol. 35, pp. 416–418). Dirac called the two halves of a bracket the bra and the ket: “As names for the new symbols < and > to be used in speech, I suggest the words bra and ket respectively.” The rule he gave is that a closed bracket ⟨ ⟩ is a number and an unclosed one is a vector. MIT’s Barton Zwiebach describes the notation as widely, though not universally, used; it starts by writing the inner product (u, v) of linear algebra with a vertical bar in place of the comma, ⟨u|v⟩.

Also searched as: bra ket, Dirac notation, ket, |ψ⟩, ⟨ψ|.

The symbols and their code points

Unicode cross-references ⟨ and ⟩ to two look-alike pairs, the older angle brackets 〈 〉 (U+2329, U+232A) and the CJK angle brackets 〈 〉 (U+3008, U+3009). Dirac’s own paper used < and >.

What each part means

A ket |ψ⟩ is a vector, the state itself; Zwiebach calls the ket “just another way to represent the vector”. A bra ⟨φ| belongs to the dual space: if kets are column vectors, bras are row vectors, so a bra followed by a ket, ⟨φ|ψ⟩, multiplies out to a number, the inner product. Written the other way round, |ψ⟩⟨φ| is the outer product, which Zwiebach shows is a linear operator (MIT 22.51 uses the term outer product). The dagger marks the adjoint: the bra that goes with the ket Ω|v⟩ is ⟨v|Ω†. For two systems, |ψ₁⟩ ⊗ |ψ₂⟩ is a product state, and MIT’s 8.06 notes call any state that is not a product state entangled.

Where the Greek letters come in

The labels inside the brackets are often Greek: ψ for a state, as in the wave function, and φ for a second one. A density matrix ρ for a system in the state |ψ⟩ is |ψ⟩⟨ψ| (MIT 8.06).