% Dirac notation


# Dirac notation

*Dirac notation* is a notation to do calculations in a Hilbert space
without needing to worry about the space's representation. It is
basically the *lingua franca* of quantum mechanics.

In Dirac notation there are *kets* $\ket{V}$ from the Hilbert space
$\mathbb{H}$ and *bras* $\bra{V}$ from a dual $\mathbb{H}'$ of the
former. Crucially, the bras and kets are from different Hilbert spaces
and therefore cannot be added, but every bra has a corresponding ket and
vice versa.

Bras and kets can only be combined in two ways: the *inner product*
$\braket{V | W}$, which returns a scalar, and the *outer product*
$\ket{V} \bra{W}$, which returns a mapping $\hat{L}$ from kets $\ket{V}$
to other kets $\ket{V'}$, i.e. a linear operator. Recall that the
Hilbert inner product must satisfy:

$$\begin{aligned}
    \braket{V | W} = \braket{W | V}^*
\end{aligned}$$

So far, nothing has been said about the actual representation of bras or
kets. If we represent kets as $N$-dimensional columns vectors, the
corresponding bras are given by the kets' adjoints, i.e. their transpose
conjugates:

$$\begin{aligned}
    \ket{V} =
    \begin{bmatrix}
        v_1 \\ \vdots \\ v_N
    \end{bmatrix}
    \quad \implies \quad
    \bra{V} =
    \begin{bmatrix}
        v_1^* & \cdots & v_N^*
    \end{bmatrix}
\end{aligned}$$

The inner product $\braket{V | W}$ is then just the familiar dot product $V \cdot W$:

$$\begin{gathered}
    \braket{V | W}
    =
    \begin{bmatrix}
        v_1^* & \cdots & v_N^*
    \end{bmatrix}
    \cdot
    \begin{bmatrix}
        w_1 \\ \vdots \\ w_N
    \end{bmatrix}
    = v_1^* w_1 + ... + v_N^* w_N
\end{gathered}$$

Meanwhile, the outer product $\ket{V} \bra{W}$ creates an $N \cross N$ matrix:

$$\begin{gathered}
    \ket{V} \bra{W}
    =
    \begin{bmatrix}
        v_1 \\ \vdots \\ v_N
    \end{bmatrix}
    \cdot
    \begin{bmatrix}
        w_1^* & \cdots & w_N^*
    \end{bmatrix}
    =
    \begin{bmatrix}
        v_1 w_1^* & \cdots & v_1 w_N^* \\
        \vdots & \ddots & \vdots \\
        v_N w_1^* & \cdots & v_N w_N^*
    \end{bmatrix}
\end{gathered}$$

If the kets are instead represented by functions $f(x)$ of
$x \in [a, b]$, then the bras represent *functionals* $F[u(x)]$ which
take an unknown function $u(x)$ as an argument and turn it into a scalar
using integration:

$$\begin{aligned}
    \ket{f} = f(x)
    \quad \implies \quad
    \bra{f}
    = F[u(x)]
    = \int_a^b f^*(x) \: u(x) \dd{x}
\end{aligned}$$

Consequently, the inner product is simply the following familiar integral:

$$\begin{gathered}
    \braket{f | g}
    = F[g(x)]
    = \int_a^b f^*(x) \: g(x) \dd{x}
\end{gathered}$$

However, the outer product becomes something rather abstract:

$$\begin{gathered}
    \ket{f} \bra{g}
    = f(x) \: G[u(x)]
    = f(x) \int_a^b g^*(\xi) \: u(\xi) \dd{\xi}
\end{gathered}$$

This result makes more sense if we surround it by a bra and a ket:

$$\begin{aligned}
    \bra{u} \!\Big(\!\ket{f} \bra{g}\!\Big)\! \ket{w}
    &= U\big[f(x) \: G[w(x)]\big]
    = U\Big[ f(x) \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big]
    \\
    &= \int_a^b u^*(x) \: f(x) \: \Big(\int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) \dd{x}
    \\
    &= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big)
    \\
    &= \braket{u | f} \braket{g | w}
\end{aligned}$$