The *bra-ket notation* or _Dirac notation_ is a way of writing vectors and matrices in [[quantum mechanics]]. The name comes from the English word bracket. The non-matching vertical bar and the triangular symbol are called bra-ket notation $\ket{\cdot}$. It is useful since it allows physicists to quickly spot whether an object is a matrix, a vector or a number. A quantum state in bra-ket notation is usually written as $\ket{\psi}$. It is an alternative way of writing a column vector. Here, we do not use the notation for calculations, but to make it clear when something is a [[Quantum State|quantum state]]. >[!example]- Technical Example > As a concrete example, we can write the two states of a [[Qubit|two-level system]] (or [[qubit]]) as the states $\ket{0}=\begin{pmatrix} 1\\0 \end{pmatrix}$ $\ket{1}=\begin{pmatrix} 1\\0 \end{pmatrix}$ These two states correspond in some sense to the [[Bit|classical bits]] $0$ and $1$ used in computer science. However, we can bring these [[Quantum State|quantum states]] in [[Superposition|superposition]] > $\ket{\psi}=\alpha\ket{0}+\beta\ket{1}=\alpha\begin{pmatrix}1\\0\end{pmatrix}+\beta\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}\alpha\\\beta\end{pmatrix}.$ > Here, we used the two states of a [[Qubit|qubit]], $\ket{0}$ and $\ket{1}$ and the prefactors $\alpha$ and $\beta$ which are just numbers. > This is superposition of the states $\ket{0}$ and $\ket{1}$ is impossible with [[Bit|classical bits]]. >[!read]- Further Reading >- [[Quantum Mechanics]] >- [[Quantum Algorithm]] >- [[Qubit]] >[!ref]- References