Editing Bra–ket notation (section)

===Spinless position–space wave function===

<div class="skin-invert-image">
{{multiple image
 | left
 | footer    = Components of complex vectors plotted against index number; discrete {{math|''k''}} and continuous {{math|''x''}}. Two particular components out of infinitely many are highlighted.
 | width1    = 225
 | image1    = Discrete complex vector components.svg 
 | caption1  = Discrete components {{math|''A''<sub>''k''</sub>}} of a complex vector {{math|1={{ket|''A''}} = Σ<sub>''k''</sub> ''A''<sub>''k''</sub> {{ket|''e<sub>k</sub>''}}}}.
 | width2    = 230
 | image2    = Continuous complex vector components.svg
 | caption2  = Continuous components {{math|''ψ''(''x'')}} of a complex vector {{math|1={{ket|''ψ''}} = ∫ d''x'' ''ψ''(''x''){{ket|''x''}}}}.
}}
</div>

The Hilbert space of a [[Spin (physics)|spin]]-0 point particle  can be represented in terms of a "position [[basis (linear algebra)|basis]]" {{math|{ {{ket|'''r'''}} }<nowiki/>}}, where the label {{math|'''r'''}} extends over the set of all points in [[position space]]. These states satisfy the eigenvalue equation for the [[position operator]]:
<math display="block"> \hat{\mathbf{r}}|\mathbf{r}\rangle = \mathbf{r}|\mathbf{r}\rangle.</math>
The position states are "[[Dirac_delta_function#Quantum_mechanics|generalized eigenvectors]]", not elements of the Hilbert space itself, and do not form a countable orthonormal basis. However, as the Hilbert space is separable, it does admit a countable dense subset within the [[domain of definition]] of its wavefunctions. That is, starting from any ket {{math|{{ket|Ψ}}}} in this Hilbert space, one may ''define'' a complex scalar function of {{math|'''r'''}}, known as a [[wavefunction]],
<math display="block">\Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\Psi\rang \,.</math>

On the left-hand side, {{math|Ψ('''r''')}} is a function mapping any point in space to a complex number; on the right-hand side,
<math display="block">\left|\Psi\right\rangle = \int d^3\mathbf{r} \, \Psi(\mathbf{r}) \left|\mathbf{r}\right\rangle</math> 
is a ket consisting of a superposition of kets with relative coefficients specified by that function.

It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by
<math display="block">\hat A(\mathbf{r}) ~ \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\hat A|\Psi\rang \,.</math>

For instance, the [[momentum]] operator <math>\hat \mathbf {p}</math> has the following coordinate representation,
<math display="block">\hat{\mathbf{p} } (\mathbf{r}) ~ \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r} |\hat \mathbf{p}|\Psi\rang = - i \hbar \nabla \Psi(\mathbf{r}) \,.</math>

One occasionally even encounters an expression such as <math >\nabla |\Psi\rang </math>, though this is something of an [[abuse of notation]]. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis, <math>\nabla \lang\mathbf{r}|\Psi\rang \,,</math>
even though, in the momentum basis, this operator amounts to a mere multiplication operator (by {{math|''iħ'''''p'''}}). That is, to say,
<math display="block"> \langle \mathbf{r} |\hat \mathbf{p} = - i \hbar \nabla \langle \mathbf{r}| ~,</math>
or
<math display="block"> \hat \mathbf{p} = \int d^3 \mathbf{r} ~| \mathbf{r}\rangle ( - i \hbar \nabla) \langle \mathbf{r}| ~.</math>