What is Dirac Notation?
Definition
In quantum mechanics, wave functions are vectors and are fundamentally considered column vectors. A column vector is denoted using a right single-angle bracket and is referred to as a ket vector.
$$ \psi = \ket{\psi} = \begin{pmatrix} \psi_{1} \\ \psi_{2} \\ \vdots \\ \psi_{n} \end{pmatrix} $$
The conjugate transpose matrix of $\ket{\psi}$ is denoted using a left single-angle bracket and is referred to as a bra vector.
$$ \psi^{\ast} = \ket{\psi}^{\ast} = \bra{\psi} = \begin{pmatrix} \psi_{1}^{\ast} & \psi_{2}^{\ast} & \cdots & \psi_{n}^{\ast} \end{pmatrix} $$
$^{\ast}$ indicates the complex conjugate transpose.
Explanation
Complex Conjugate Transpose
In physics, typically $^{\ast}$ is explained as a complex conjugate, but in mathematics, $^{\ast}$ encompasses both complex conjugation and matrix transpose. However, from the above definition, it is evident that in physics, this notation indeed carries the meaning of transpose. In other words, in physics, the notation is used in overlapping contexts: when attached to a scalar, it denotes complex conjugation, and when attached to a vector or matrix, it denotes the complex conjugate transpose. If you only think of $^{\ast}$ as meaning complex conjugation, you may not understand why $\psi^{\ast}$ becomes a row vector when $\psi$ is a column vector, so be cautious.
Origin of the Name
The name originates from the English word bracket, which means parenthesis. The word is split in half, taking parts of the word to form their names. Analogously in Korean, it could be likened to 괄-벡터(bra-vector) and 호-벡터(ket-vector).
Relation to Inner Product
In quantum mechanics, it is a notation used to conveniently operate on operators and wave functions (eigenfunctions). The use of single-angle brackets is related to the inner product. In mathematics, the generalized notation for inner product is $\braket{\quad}$. Therefore, when denoting row vectors and column vectors as above, the product of the two vectors itself becomes the inner product, making the notation significant.
$$ \braket{ \psi \vert \phi } = \begin{pmatrix} \psi_{1}^{\ast} & \psi_{2}^{\ast} & \cdots & \psi_{n}^{\ast} \end{pmatrix}\begin{pmatrix} \phi_{1} \\ \phi_{2} \\ \vdots \\ \phi_{n} \end{pmatrix} = \psi_{1}^{\ast}\phi_{1} + \psi_{2}^{\ast}\phi_{2} + \cdots + \psi_{n}^{\ast}\phi_{n} $$
The expectation value of any operator $Q$ is as follows:
$$ \braket{Q}= \int \psi^{\ast} Q \psi dx $$
This can be viewed as the inner product of $\psi$ and $Q\psi$, and thus can be noted as follows:
$$ \braket{\psi \vert Q\psi} $$
Or alternatively, it can be interpreted as the inner product of $Q^{\ast}\psi$ and $\psi$.
$$ \int \psi^{\ast} Q \psi dx = \int (Q^{\ast}\psi)^{\ast} \psi dx = \braket{Q^{\ast}\psi \vert \psi} $$
The following notations all signify the same equation.
$$ \braket{Q} = \braket{\psi \vert Q \vert \psi} = \braket{\psi \vert Q \psi} = \braket{Q^{\ast}\psi \vert \psi} = \int \psi^{\ast} Q \psi dx $$