Bras and kets
An inner product on a vector space V (over the complex numbers) is a function that assigns to each pair of vectors |u\rangle,|v\rangle\in V a complex number \langle u|v\rangle, and satisfies the following conditions:
- \langle u|v\rangle=\langle v|u\rangle^\star;
- \langle v|v\rangle\geqslant 0 for all |v\rangle;
- \langle v|v\rangle= 0 if and only if |v\rangle=0.
The inner product must also be linear in the second argument but antilinear in the first argument:
\langle c_1u_1+c_2u_2|v\rangle = c_1^\star\langle u_1|v\rangle+c_2^\star\langle u_2|v\rangle
for any complex constants c_1 and c_2.
With any physical system we associate a complex vector space with an inner product, known as a Hilbert space \mathcal{H}.
The inner product between vectors |u\rangle and |v\rangle in {\mathcal{H}} is written as \langle u|v\rangle.
For example, for column vectors |u\rangle and |v\rangle in \mathbb{C}^n written as
|u\rangle
= \begin{bmatrix}u_1\\u_2\\\vdots\\u_n\end{bmatrix}
\qquad
|v\rangle
= \begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}
their inner product is defined by
\langle u|v\rangle
= u_1^\star v_1 + u_2^\star v_2+\ldots + u_n^\star v_n.
Following Dirac, we may split the inner product into two ingredients:
\langle u|v\rangle
\longrightarrow \langle u|\,|v\rangle.
Here |v\rangle is a ket vector, and \langle u| is called a bra vector, or a bra, and can be represented by a row vector:
\langle u|
= \begin{bmatrix}u_1^\star,&u_2^\star,&\ldots,&u_n^\star\end{bmatrix}.
The inner product can now be viewed as the result of the matrix multiplication:
\begin{aligned}
\langle u|v\rangle
&= \begin{bmatrix}u_1^\star,&u_2^\star,&\ldots,&u_n^\star\end{bmatrix}
\cdot \begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}
\\&= u_1^\star v_1 + u_2^\star v_2 + \ldots + u_n^\star v_n.
\end{aligned}
Bras are vectors: you can add them, and multiply them by scalars (which, here, are complex numbers), but they are vectors in the space {\mathcal{H}}^\star which is dual to \mathcal{H}.
Elements of {\mathcal{H}}^\star are linear functionals, that is, linear maps from \mathcal{H} to \mathbb{C}.
A linear functional \langle u| acting on a vector |v\rangle in \mathcal{H} gives a complex number \langle u|v\rangle.
All Hilbert spaces of the same dimension are isomorphic, so the differences between quantum systems cannot be really understood without additional structure. This structure is provided by a specific algebra of operators acting on \mathcal{H}.