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

**Dirac notation** enables us to do calculations
in a general [Hilbert space](/know/concept/hilbert-space/)
without needing to worry about the space's representation.
It is 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 its dual space $$\mathbb{H}'$$.
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 linear operator
that maps kets $$\ket{V}$$ to other kets $$\ket{V'}$$.
Recall that by definition 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,
which can be thought of as applying an operation to any vector it multiplies:

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

If the kets are instead represented by continuous functions $$f(x)$$ of $$x \in [a, b]$$,
then the bras are *functionals* $$F[u(x)]$$
that take an arbitrary function $$u(x)$$ as an argument and return a scalar:

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

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 is then rather abstract:
a continuous analogue of a matrix:

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

This maybe makes more sense if we surround it
by a bra $$\bra{u}$$ and a ket $$\ket{w}$$ and rearrange:

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



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