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---
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.
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