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
where {}^\star denotes complex conjugation (sometimes written as z\mapsto\bar{z} instead).
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.
To 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.
If V is a vector space with an inner product \langle-,-\rangle, then this gives us a norm on V by defining \|x\|=\sqrt{\langle x,x\rangle} and thus a metric by defining d(x,y)=\|y-x\|.
We say that a sequence (x_n) in V is Cauchy if its elements “eventually always get closer”, i.e. if for all \varepsilon>0 there exists some N\in\mathbb{N} such that for all m,n>N we have \|x_n-x_m\|<\varepsilon.
We say that a normed space is complete if every Cauchy sequence converges in that space, i.e. if the limits of sequences that should exist actually do exist.
Now one useful fact is the following: on a finite dimensional vector space, all norms are equivalent.
(Note that this does not mean that \|x\|_1=\|x\|_2 for any two norms \|-\|_1 and \|-\|_2, but simply that they “induce the same topology” — feel free to look up the precise definition elsewhere).
This follows from another useful fact: in a finite dimensional vector space, the unit ball is compact.
By a short topological argument, we can use these facts to show that what we claimed, namely that every finite dimensional inner product space is complete (with respect to the norm induced by the inner product, and thus with respect to any norm, since all norms are equivalent).
In the infinite dimensional case these facts are not true, and we have a special name for those inner product spaces which are complete: Hilbert spaces.
So working in the finite dimensional case means that “we do not have to worry about analysis”, in that the completeness property comes for free the moment we have an inner product, and we can freely refer to inner product spaces as Hilbert spaces.
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 (finite) 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}.