Editing Operator (physics) (section)

===Wavefunction ===

{{Main|wavefunction}}

The wavefunction must be [[square-integrable]] (see [[Lp space|''L<sup>p</sup>'' spaces]]), meaning:

:<math>\iiint_{\R^3} |\psi(\mathbf{r})|^2 \, d^3\mathbf{r} = \iiint_{\R^3} \psi(\mathbf{r})^*\psi(\mathbf{r}) \, d^3\mathbf{r} < \infty </math>

and normalizable, so that:

:<math>\iiint_{\R^3} |\psi(\mathbf{r})|^2 \, d^3\mathbf{r} = 1 </math>

Two cases of eigenstates (and eigenvalues) are:
* for '''discrete''' eigenstates <math> | \psi_i \rangle </math> forming a discrete basis, so any state is a [[summation|sum]] <math display="block">|\psi\rangle = \sum_i c_i|\phi_i\rangle</math> where ''c<sub>i</sub>'' are complex numbers such that {{!}}''c<sub>i</sub>''{{!}}<sup>2</sup> = ''c<sub>i</sub>''<sup>*</sup>''c<sub>i</sub>'' is the probability of measuring the state <math>|\phi_i\rangle</math>, and the corresponding set of eigenvalues ''a<sub>i</sub>'' is also discrete - either [[Finite set|finite]] or [[countably infinite]]. In this case, the inner product of two eigenstates is given by <math>\langle \phi_i \vert \phi_j\rangle=\delta_{ij}</math>, where <math>\delta_{mn}</math> denotes the [[Kronecker delta|Kronecker Delta]]. However, 
* for a '''continuum''' of eigenstates forming a continuous basis, any state is an [[integral]] <math display="block">|\psi\rangle = \int c(\phi) \, d\phi|\phi\rangle </math> where ''c''(''φ'') is a complex function such that {{!}}''c''(φ){{!}}<sup>2</sup> = ''c''(φ)<sup>*</sup>''c''(φ) is the probability of measuring the state <math>|\phi\rangle</math>, and there is an [[uncountably infinite]] set of eigenvalues ''a''. In this case, the inner product of two eigenstates is defined as <math>\langle \phi' \vert \phi\rangle=\delta(\phi - \phi')</math>, where here <math>\delta(x-y)</math> denotes the [[Dirac delta|Dirac Delta]].