0.8 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.¹⁷ 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 $|v\rangle$.

Not to be confused with “completeness” in the sense of Hilbert spaces.↩︎

Finally, note that calculating the adjoint of an outer product boils down to just swapping the order: $$(|a\rangle\langle b|)^\dagger = |b\rangle\langle a|.$$

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17. Not to be confused with “completeness” in the sense of Hilbert spaces.↩︎