Section 3.1 Bra-Ket Notation
¶Column matrices play a special role in physics, where they are interpreted as vectors or, in quantum mechanics, states. To remind us of this uniqueness they have their own special notation; introduced by Dirac, called bra-ket notation. In bra-ket notation, a column matrix, called a ket, can be written
\begin{equation}
\left|v\right> := \left(\begin{array}{c}
a\\
b\\
c
\end{array} \right)\text{.}\tag{3.1.1}
\end{equation}
The Hermitian adjoint of this vector is called a bra
\begin{equation}
\left\lt v\right| :=
\left(\left|v\right>\right)^\dagger
=\left(\begin{array}{ccc}
a^*\amp b^*\amp c^*
\end{array} \right)\text{.}\tag{3.1.2}
\end{equation}
If we take \(\left|v\right>\) to be a 3-vector with components \(a\text{,}\) \(b\text{,}\) \(c\) as above, then the inner product of this vector with itself is called a braket
\begin{equation}
\left\lt v|v\right>
= \left(\begin{array}{ccc}
a^*\amp b^*\amp c^*
\end{array} \right)
\left(\begin{array}{c}
a\\
b\\
c
\end{array} \right)
= \left|a\right|^2+\left|b\right|^2+\left|c\right|^2\tag{3.1.3}
\end{equation}
which is the square of the magnitude of the vector as expected.