Bra-Ket Notation

linear-algebra

Definition

Bra-Ket Notation

The bra-ket notation is a notation for vectors, inner products, and linear operators in complex inner product spaces.

A vector $x$ is written as a ket $|x⟩$ and its corresponding dual vector is written as a bra $⟨x|$.

The inner product of two vectors $x$ and $y$ is written as

$$⟨x|y⟩=⟨x,y⟩.$$

For a linear operator $T$, one writes

$$⟨x|T|y⟩=⟨x,Ty⟩.$$

Finite Dimensions

In finite-dimensional complex vector spaces, one writes

$$|x⟩=\left(\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right)\text{and}⟨x|=\left(\begin{array}{ccc}\overline{x_1}& \dots & \overline{x_n}\end{array}\right).$$

So the bra is the conjugate transpose of the ket.

Orthonormal Basis

For an orthonormal basis $(|e_i⟩{)}_{i\ge 0}$, every vector can be written as

$$|x⟩=\sum _{i\ge 0}⟨e_i|x⟩|e_i⟩.$$

The coefficient of $|e_i⟩$ is the inner product $⟨e_i|x⟩$.

The operator $|e_i⟩⟨e_i|$ is the orthogonal projection onto the line spanned by $|e_i⟩$. Hence one obtains the completeness relation

$$I=\sum _{i\ge 0}|e_i⟩⟨e_i|.$$

Example

Example

It follows from conjugate linearity in the first argument that

$$⟨\lambda x|y⟩=\overline{\lambda }⟨x|y⟩.$$