Dirac notation for quantum states

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 ${\mathbb{C}}^n$. A bra $⟨f|$ denotes a linear map $f:{\mathbb{C}}^n\to \mathbb{C}$, and the application of $⟨f|$ to $|v⟩$ is written as $⟨f|v⟩$.

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

$|v⟩=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right],$

and the bra $⟨f|$ as a row vector:

$⟨f|=\left[\begin{array}{cccc}{f}_1& f_2& \cdots & f_n\end{array}\right].$

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

$⟨f|v⟩=\left[\begin{array}{cccc}{f}_1& f_2& \cdots & f_n\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]={\sum }_{i=1}^n{f}_i{v}_i.$

See also

• On the nature of the wave function - The wave function is an abstract mathematical concept and cannot be "measured" directly. So what is it then?