Editing Bra–ket notation (section)

==The unit operator==
Consider a complete [[orthonormal]] system (''basis''),
<math display="block">\{ e_i \ | \ i \in \mathbb{N} \} \,,</math>
for a Hilbert space {{math|''H''}}, with respect to the norm from an inner product {{math|{{angbr|·,·}}}}.

From basic [[functional analysis]], it is known that any ket <math>|\psi\rangle </math> can also be written as
<math display="block">|\psi\rangle = \sum_{i \in \mathbb{N}} \langle e_i | \psi \rangle | e_i \rangle,</math>
with {{math|{{bra-ket|·|·}}}} the inner product on the Hilbert space.

From the commutativity of kets with (complex) scalars, it follows that
<math display="block">\sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | = \mathbb{I}</math>
must be the ''identity operator'', which sends each vector to itself.

This, then, can be inserted in any expression without affecting its value; for example
<math display="block">\begin{align}
\langle v | w \rangle &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) | w \rangle \\
                      &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) \left( \sum_{j \in \mathbb{N}} | e_j \rangle \langle e_j |\right)| w \rangle \\
&= \langle v | e_i \rangle \langle e_i | e_j \rangle \langle e_j | w \rangle \,, 
\end{align}</math>
where, in the last line, the [[Einstein summation convention]] has been used to avoid clutter.

In quantum mechanics, it often occurs that little or no information about the inner product {{math|{{bra-ket|''ψ''|''φ''}}}} of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients {{math|1={{bra-ket|''ψ''|''e<sub>i</sub>''}} = {{bra-ket|''e<sub>i</sub>''|''ψ''}}*}} and {{math|{{bra-ket|''e<sub>i</sub>''|''φ''}}}} of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more.

For more information, see [[Resolution of the identity]],<ref>{{harvnb|Sakurai|Napolitano|2021}} Sec 1.2, 1.3</ref>
<math display="block"> {\mathbb I} = \int\!  dx~ | x \rangle \langle x |= \int\! dp ~| p \rangle \langle p |,</math> where <math display="block">|p\rangle = \int dx \frac{e^{ixp / \hbar} |x\rangle}{\sqrt{2\pi\hbar}}.</math>

Since {{math|1={{bra-ket|''x''{{prime}}|''x''}} = ''δ''(''x'' − ''x''{{prime}})}}, plane waves follow, <math display="block"> \langle x | p \rangle = \frac{e^{ixp / \hbar}}{\sqrt{2\pi\hbar}}.</math>

In his book (1958), Ch. III.20, Dirac defines the ''standard ket'' which, up to a normalization, is the translationally invariant momentum eigenstate <math display="inline">|\varpi\rangle=\lim_{p\to 0} |p\rangle</math> in the momentum representation, i.e., <math>\hat{p}|\varpi\rangle=0</math>. Consequently, the corresponding wavefunction is a constant, <math> \langle x|\varpi\rangle \sqrt{2\pi \hbar}=1</math>, and 
<math display="block">|x\rangle= \delta(\hat{x}-x) |\varpi\rangle \sqrt{2\pi \hbar},</math> as well as <math display="block">|p\rangle= \exp (ip\hat{x}/\hbar ) |\varpi\rangle.</math>

Typically, when all matrix elements of an operator such as <math display="block"> \langle x| A |y\rangle </math> are available, this resolution serves to reconstitute the full operator,
<math display="block"> \int dx \, dy \, |x\rangle \langle x| A |y\rangle \langle y | = A \,. </math>