Bra-ket notation

Bra-ket notation is the standard notation used for describing quantum mechanical states. It was invented by Paul Dirac. It is so called because the inner product of two states is denoted by a bracket, ⟨φ|ψ⟩, consisting of a left part, ⟨φ|, called the bra, and a right part, |ψ⟩, called the ket.

In quantum mechanics, the state of a physical system is identified with a vector in a Hilbert space, H. Each vector is called a ket, and written as

<math>|\psi\rangle</math>

where ψ is an arbitrary label for the ket. Each element of the dual space of H (i.e. each continuous linear function from H to the complex numbers C) is known as a bra, and written as

<math>\langle\phi|</math>

where φ is an arbitrary label for the bra. Applying the bra ⟨φ| to the ket |ψ⟩ results in a complex number, called a bra-ket, which we write as

<math>\langle\phi|\psi\rangle.</math>

Every ket |ψ⟩ has a dual bra, written as ⟨ψ|, a continuous linear function on H defined as follows:

<math>\langle\psi|\rho\rangle = ( |\psi\rangle , |\rho\rangle )\;\;

\hbox{for all kets}\ |\rho\rangle </math>

where the right hand side ( , ) denotes the inner product given on the Hilbert space. The notation is justified by the Riesz representation theorem, which states that every bra in the dual space arises from one and only one ket in this fashion.

The bra-ket operation has the following properties:

• Given any bra ⟨φ|, kets |ψ₁⟩ and |ψ₂⟩, and complex numbers c₁ and c₂, then, since bras are linear functionals,

<math>\langle\phi|(c_1|\psi_1\rangle + c_2|\psi_2\rangle) = c_1\langle\phi|\psi_1\rangle + c_2\langle\phi|\psi_2\rangle. </math>

• Given any ket |ψ⟩, bras ⟨φ₁| and ⟨φ₂|, and complex numbers c₁ and c₂, then, by the definition of addition and scalar multiplication of linear functionals,

<math>(c_1 \langle\phi_1| + c_2 \langle\phi_2|)|\psi\rangle = c_1 \langle\phi_1|\psi\rangle + c_2\langle\phi_2|\psi\rangle. </math>

• Given any kets |ψ₁⟩ and |ψ₂⟩, and complex numbers c₁ and c₂, from the properties of the inner product (with "*" denoting the complex conjugate),

<math>

c_1|\psi_1\rangle + c_2|\psi_2\rangle \;\; \hbox{is dual to} \;\; c_1^* \langle\psi_1| + c_2^* \langle\psi_2|. </math>

• Given any bra ⟨φ| and ket |ψ⟩, the inner product axiom gives

<math>\langle\phi|\psi\rangle = \langle\psi|\phi\rangle^*.</math>

If A : H -> H is a linear operator, we can apply A to the ket |ψ⟩ to obtain the ket (A|ψ⟩). The operator also acts on bras: applying the operator A to the bra ⟨φ| results in the bra (⟨φ|A), defined as a linear functional on H by the rule

<math>\langle\phi|A) \; (|\psi\rangle) = \langle\phi| \; (A|\psi\rangle).</math>

This expression is commonly written as

<math>\langle\phi|A|\psi\rangle.</math>

A convenient way to define linear operators on H is given by the outer product: if ⟨φ| is a bra and |ψ⟩ is a ket, the outer product |φ⟩ ⟨ψ| denotes the operator which maps the ket |ρ⟩ to the ket |φ⟩ ⟨ψ|ρ⟩ (here the scalar ⟨ψ|ρ⟩ is written to the right of the vector |φ⟩). One use of the outer product is to construct projection operators[?]. Given a ket |ψ⟩ of norm 1, the orthogonal projection onto the subspace spanned by |ψ⟩ is

<math>|\psi\rangle\langle\psi|</math>

Two Hilbert spaces V and W may form a third space V × W by a tensor product. If |ψ⟩ is a ket in V and |φ⟩ is a ket in W, the tensor product of the two kets is a ket in V × W. This is written variously as

<math>|\psi\rangle|\phi\rangle \; \hbox{or}\; |\psi\rangle \times |\phi\rangle \; \hbox{or}\; |\psi \phi\rangle. </math>