Outer products
Apart from the inner product \langle u|v\rangle, which is a complex number, we can also form the outer product |u\rangle\langle v|, which is a linear map (operator) on \mathcal{H} (or on \mathcal{H}^\star, depending how you look at it).
This is what physicists like (and what mathematicians dislike!) about Dirac notation: a certain degree of healthy ambiguity.
- The result of |u\rangle\langle v| acting on a ket |x\rangle is |u\rangle\langle v|x\rangle, i.e. the vector |u\rangle multiplied by the complex number \langle v|x\rangle.
- Similarly, the result of |u\rangle\langle v| acting on a bra \langle y| is \langle y|u\rangle\langle v|, i.e. the linear functional \langle v| multiplied by the complex number \langle y|u\rangle.
The product of two maps, A=|a\rangle\langle b| followed by B=|c\rangle\langle d|, is a linear map BA, which can be written in Dirac notation as
BA = |c\rangle\langle d|a\rangle\langle b| = \langle d|a\rangle|c\rangle\langle b|
i.e. the inner product (complex number) \langle d|a\rangle times the outer product (linear map) |c\rangle\langle b|.
Any operator on \mathcal{H} can be expressed as a sum of outer products. Given an orthonormal basis \{|e_i\rangle\}_{i=1,\ldots,n}, any operator which maps the basis vectors |e_i\rangle to vectors |f_i\rangle can be written as \sum_{i=1}^n|f_i\rangle\langle e_i|.
If the vectors \{|f_i\rangle\} also form an orthonormal basis then the operator simply “rotates” one orthonormal basis into another.
These are unitary operators which preserve the inner product.
In particular, if each |e_i\rangle is mapped to |e_i\rangle, then we obtain the identity operator:
\sum_i|e_i\rangle\langle e_i|=\mathbf{1}.
This relation holds for any orthonormal basis, and it is one of the most ubiquitous and useful formulas in quantum theory, known as completeness.
For example, for any vector |v\rangle and for any orthonormal basis \{|e_i\rangle\}, we have
\begin{aligned}
|v\rangle
&= \mathbf{1}|v\rangle
\\&= \sum_i |e_i\rangle\langle e_i|\;|v\rangle
\\&= \sum_i |e_i\rangle\;\langle e_i|v\rangle
\\&= \sum_i v_i|e_i\rangle
\end{aligned}
where v_i=\langle e_i|v\rangle are the components of |v\rangle.
Finally, note that calculating the adjoint of an outer product boils down to just swapping the order:
(|a\rangle\langle b|)^\dagger = |b\rangle\langle a|.