Editing Probability amplitude (section)

===Discrete amplitudes===

Let <math>\mu_{pp}</math> be [[atom (measure theory)|atomic]] (i.e. the set <math>A\subset X</math> in <math>\mathcal{A}</math> is an ''atom''); specifying the measure of any [[Continuous or discrete variable#Discrete variable|discrete variable]] {{math|''x'' ∈ ''A''}} equal to {{math|1}}. The amplitudes are composed of state vector {{math|{{ket|&Psi;}}}} [[indexed family|indexed]] by {{mvar|A}}; its components are denoted by {{math|''&psi;''(''x'')}} for uniformity with the previous case. If the [[Lp space#General ℓp-space|{{math|''ℓ''<sup>''2''</sup>}}-norm]] of {{math|{{ket|&Psi;}}}} is equal to 1, then {{math|{{abs|''&psi;''(''x'')}}<sup>2</sup>}} is a [[probability mass function]].

A convenient configuration space {{mvar|X}} is such that each point {{mvar|x}} produces some unique value of the observable {{mvar|Q}}. For discrete {{mvar|X}} it means that all elements of the standard basis are [[eigenvector]]s of {{mvar|Q}}. Then <math> \psi (x)</math> is the probability amplitude for the eigenstate {{math|{{ket|''x''}}}}. If it corresponds to a non-[[degenerate energy levels|degenerate]] eigenvalue of {{mvar|Q}}, then <math> |\psi (x)|^2</math> gives the probability of the corresponding value of {{mvar|Q}} for the initial state {{math|{{ket|&Psi;}}}}.

{{math|1={{abs|''&psi;''(''x'')}} = 1}} if and only if {{math|{{ket|''x''}}}} is [[ray (quantum theory)|the same quantum state]] as {{math|{{ket|&Psi;}}}}. {{math|1=''&psi;''(''x'') = 0}} if and only if {{math|{{ket|''x''}}}} and {{math|{{ket|&Psi;}}}} are [[Orthogonality (mathematics)|orthogonal]]. Otherwise the modulus of {{math|''&psi;''(''x'')}} is between 0 and 1.

A discrete probability amplitude may be considered as a [[fundamental frequency]] in the probability frequency domain ([[spherical harmonics]]) for the purposes of simplifying [[M-theory]] transformation calculations.{{citation needed|date=January 2014}} Discrete dynamical variables are used in such problems as a [[Particle in a box|particle in an idealized reflective box]] and [[quantum harmonic oscillator]].{{clarify|reason=Introducing the term "discrete dynamical variable" without context|date=November 2023}}