Editing Probability amplitude (section)

==Mathematical formulation==
{{See also | Bound state #Definition}}
In a formal setup, the state of an isolated physical system in [[Mathematical formulation of quantum mechanics#Description of the state of a system|quantum mechanics]] is represented, at a fixed time <math>t</math>, by a [[Quantum state|state vector]] {{math|{{ket|&Psi;}}}} belonging to a [[Hilbert space#Separable spaces|separable]] complex [[Hilbert space#Quantum mechanics|Hilbert space]]. Using [[bra–ket notation#Usage in quantum mechanics|bra–ket notation]] the relation between state vector and "position [[Quantum state#Basis states of one-particle systems|basis]]" <math>\{|x\rangle\}</math> of the Hilbert space can be written as<ref>The spanning set of a Hilbert space does not suffice for defining coordinates as wave functions form rays in a projective Hilbert space (rather than an ordinary Hilbert space). See: [[Projective space#Frame|Projective frame]]</ref> 
:<math> \psi (x) = \langle x|\Psi \rangle</math>.
Its relation with an [[observable]] can be elucidated by generalizing the quantum state <math>\psi</math> to a [[measurable function]] and its [[Partial function|domain of definition]] to a given [[Measure space#Important classes of measure spaces|{{math|''&sigma;''}}-finite measure space]] <math>(X, \mathcal A, \mu)</math>. This allows for a refinement of [[Lebesgue's decomposition theorem]], decomposing ''μ'' into three mutually singular parts
:<math> \mu = \mu_{\mathrm{ac}} + \mu_{\mathrm{sc}} + \mu_{\mathrm{pp}}</math>
where ''μ''<sub>ac</sub> is absolutely continuous with respect to the Lebesgue measure, ''μ''<sub>sc</sub> is singular with respect to the Lebesgue measure and atomless, and ''μ''<sub>pp</sub> is a pure point measure.{{sfn|Simon|2005|page=43}}{{sfn | Teschl | 2014 | p=114-119}}

===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}}

===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}}

=== Examples ===
An example of the discrete case is a quantum system that can be in [[two-state quantum system|two possible states]], e.g. the [[light polarization|polarization]] of a [[photon]]. When the polarization is measured, it could be the horizontal state <math>|H\rangle</math> or the vertical state <math>|V\rangle</math>. Until its polarization is measured the photon can be in a [[Quantum superposition|superposition]] of both these states, so its state <math>|\psi\rangle</math> could be written as

:<math>|\psi\rangle = \alpha |H\rangle + \beta|V\rangle</math>,

with <math>\alpha</math> and <math>\beta</math> the probability amplitudes for the states <math>|H\rangle</math> and <math>|V\rangle</math> respectively. When the photon's polarization is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarized is <math>|\alpha|^2</math>, and the probability of being vertically polarized is <math>|\beta|^2</math>.

Hence, a photon in a state <math display="inline">|\psi\rangle = \sqrt{\frac{1}{3}} |H\rangle - i \sqrt{\frac{2}{3}}|V\rangle</math> would have a probability of <math display="inline">\frac{1}{3}</math> to come out horizontally polarized, and a probability of <math display="inline">\frac{2}{3}</math> to come out vertically polarized when an [[statistical ensemble (mathematical physics)|ensemble]] of measurements are made. The order of such results, is, however, completely random.

Another example is quantum spin. If a spin-measuring apparatus is pointing along the z-axis and is therefore able to measure the z-component of the spin (<math display="inline">\sigma_z</math>), the following must be true for the measurement of spin "up" and "down":
:<math>\sigma_z |u\rangle = (+1)|u\rangle </math>
:<math>\sigma_z |d\rangle = (-1)|d\rangle</math>
If one assumes that system is prepared, so that +1 is registered in <math display="inline">\sigma_x</math> and then the apparatus is rotated to measure <math display="inline">\sigma_z</math>, the following holds:
:<math>\begin{align}
\langle r|u \rangle &= \left(\frac{1}{\sqrt{2}}\langle u| + \frac{1}{\sqrt{2}}\langle d|\right) \cdot |u\rangle \\
 &= \left(\frac{1}{\sqrt{2}} \begin{pmatrix}1\\0\end{pmatrix} + \frac{1}{\sqrt{2}} \begin{pmatrix}0\\1\end{pmatrix}\right) \cdot \begin{pmatrix}1\\0\end{pmatrix}  \\
&= \frac{1}{\sqrt{2}}
\end{align}</math>
The probability amplitude of measuring spin up is given by <math display="inline">\langle r|u\rangle</math>, since the system had the initial state <math display="inline"> | r \rangle</math>. The probability of measuring <math display="inline">|u\rangle</math> is given by
:<math>P(|u\rangle) = \langle r|u\rangle\langle u|r\rangle = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}</math>
Which agrees with experiment.