Editing Probability amplitude (section)

===Continuous amplitudes===

A usual presentation of the probability amplitude is that of a [[wave function]] <math>\psi</math> belonging to the {{math|''L''<sup>2</sup>}} space of ([[equivalence class]]es of) [[Square-integrable function|square integrable functions]], i.e., <math>\psi</math> belongs to {{math|''L''<sup>2</sup>(''X'')}} if and only if
:<math>\|\psi\|^{2} = \int_X |\psi(x)|^2\, dx < \infty </math>.
If the [[Normed vector space|norm]] is equal to {{math|1}} and <math>|\psi(x)|^{2}\in\mathbb{R}_{\geq 0}</math> such that
:<math> \int_X |\psi(x)|^2 \,dx \equiv\int_X \,d\mu_{ac}(x) = 1</math>,
then <math>|\psi(x)|^{2}</math> is the [[probability density function]] for a measurement of the particle's position at a given time, defined as the [[Radon–Nikodym derivative]] with respect to the [[Lebesgue measure]] (e.g. on the set {{math| '''R'''}} of all [[real number]]s). As probability is a dimensionless quantity, {{math|{{abs|''&psi;''(''x'')}}<sup>2</sup>}} must have the inverse dimension of the variable of integration {{math|''x''}}. For example, the above amplitude has [[Dimensional analysis|dimension]] [L<sup>−1/2</sup>], where L represents [[length]]. 

Whereas a Hilbert space is separable if and only if it admits a [[countable]] orthonormal basis, the [[range of a function|range]] of a [[Random_variable#Continuous_random_variable|continuous random variable]] <math>x</math> is an [[uncountable set]] (i.e. the probability that the system is "at position <math>x</math>" will always [[almost never|be zero]]). As such, [[eigenstate]]s of an observable need not necessarily be measurable functions belonging to {{math|''L''<sup>2</sup>(''X'')}} (see [[#Normalization|normalization condition]] below). A [[Expectation value (quantum mechanics)#Example in configuration space|typical example]] is the [[position operator]] <math>\hat{\mathrm x}</math> defined as
:<math>\langle x |\hat{\mathrm x}|\Psi\rangle = \hat{\mathrm x}\langle x | \Psi\rangle=x_{0}\psi(x), \quad x \in \mathbb{R},</math>
whose eigenfunctions are [[Dirac delta function#Quantum mechanics|Dirac delta functions]]
:<math>\psi(x)=\delta(x-x_{0})</math>
which clearly do not belong to {{math|''L''<sup>2</sup>(''X'')}}. By replacing the state space by a suitable [[rigged Hilbert space]], however, the rigorous notion of eigenstates from [[self-adjoint operator#Spectral theorem|spectral theorem]] as well as [[Decomposition of spectrum (functional analysis)#Quantum physics|spectral decomposition]] is preserved.{{sfn|de la Madrid Modino|2001|page=97}}