Dirac notation for quantum states

How to read the bra–ket notation?

Published on Nov 15, 2020 by Anton Vasetenkov

Quantum states are usually described using bra–ket notation, also known as Dirac notation.

In Dirac notation, a ket $∣ v ⟩$ represents a state of a quantum system and is a vector in a complex vector space $C^{n}$ . A bra $⟨ f ∣$ denotes a linear map $f : C^{n} \to C$ , and the application of $⟨ f ∣$ to $∣ v ⟩$ is written as $⟨ f ∣ v ⟩$ .

The ket $∣ v ⟩$ can be represented as a column vector:

∣ v ⟩ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ v_{1} v_{2} ⋮ v_{n} ⎦ ⎥ ⎥ ⎥ ⎥ ⎤,

and the bra $⟨ f ∣$ as a row vector:

⟨ f ∣ = [f_{1} f_{2} \dots f_{n}] .

In this representation, a bra next to a ket simply denotes matrix multiplication of a row vector with a column vector:

⟨ f ∣ v ⟩ = [f_{1} f_{2} \dots f_{n}] ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ v_{1} v_{2} ⋮ v_{n} ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ = i = 1 \sum n f_{i} v_{i} .

Dirac notation for quantum states

See also