---
title: "Dirac notation"
sort_title: "Dirac notation"
date: 2021-02-22
categories:
- Quantum mechanics
- Physics
layout: "concept"
---

**Dirac notation** is a notation to do calculations in a [Hilbert space](/know/concept/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 be combined in two ways: the **inner product**
$$\Inprod{V}{W}$$, 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:

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

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 $$\Inprod{V}{W}$$ is then just the familiar dot product $$V \cdot W$$:

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

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:

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

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:

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



## References
1.  R. Shankar,
    *Principles of quantum mechanics*, 2nd edition,
    Springer.