Editing Probability amplitude

{{short description|Complex number whose squared absolute value is a probability}}
{{About|probability amplitude in quantum mechanics|other uses|Amplitude (disambiguation)}}
{{more citations needed|date=January 2014}}
[[File:Hydrogen eigenstate n5 l2 m1.png|thumb|
A [[wave function]] for a single [[electron]] on 5d [[atomic orbital]] of a [[hydrogen atom]]. The solid body shows the places where the electron's [[probability density function|probability density]] is above a certain value (here 0.02 [[nanometre|nm]]<sup>−3</sup>): this is calculated from the probability amplitude. The [[hue]] on the colored surface shows the [[argument (complex analysis)|complex phase]] of the wave function.]]
In [[quantum mechanics]], a '''probability amplitude''' is a [[complex number]] used for describing the behaviour of systems. The square of the [[Absolute value|modulus]] of this quantity at a point in space represents a [[probability density function|probability density]] at that point.

Probability amplitudes provide a relationship between the [[quantum state]] vector of a system and the results of observations of that system, a link that was first proposed by [[Max Born]], in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the [[Copenhagen interpretation]] of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as [[atomic emission spectroscopy|emissions from atoms]] being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 [[Nobel Prize in Physics]] for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and [[quantum measurement]]s, were vigorously contested at the time by the original physicists working on the theory, such as [[Erwin Schrödinger|Schrödinger]] and [[Albert Einstein|Einstein]]. It is the source of the mysterious consequences and philosophical difficulties in the [[interpretations of quantum mechanics]]—topics that continue to be debated even today.

== Physical overview ==
Neglecting some technical complexities, the problem of [[quantum measurement]] is the behaviour of a quantum state, for which the value of the [[observable]] {{mvar|Q}} to be measured is [[uncertainty principle|uncertain]]. Such a state is thought to be a [[quantum superposition|coherent superposition]] of the observable's ''[[eigenstate]]s'', states on which the value of the observable is uniquely defined, for different possible values of the observable.

When a measurement of {{mvar|Q}} is made, the system (under the [[Copenhagen interpretation]]) [[state vector reduction|''jumps'' to one of the eigenstates]], returning the eigenvalue belonging to that eigenstate. The system may always be described by a [[linear combination]] or [[Quantum superposition|superposition]] of these eigenstates with unequal [[weight function|"weights"]]. Intuitively it is clear that eigenstates with heavier "weights" are more "likely" to be produced. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. These numerical weights are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the [[Born rule]].

Clearly, the sum of the probabilities, which equals the sum of the [[Square (algebra)#In complex numbers|absolute squares]] of the probability amplitudes, must equal 1. This is the [[#Normalization|normalization]] requirement.

If the system is known to be in some eigenstate of {{mvar|Q}} (e.g. after an observation of the corresponding eigenvalue of {{mvar|Q}}) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of {{mvar|Q}} (so long as no other important forces act between the measurements). In other words, the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. If the set of eigenstates to which the system can jump upon measurement of {{mvar|Q}} is the same as the set of eigenstates for measurement of {{mvar|R}}, then subsequent measurements of either {{mvar|Q}} or {{mvar|R}} always produce the same values with probability of 1, no matter the order in which they are applied. The probability amplitudes are unaffected by either measurement, and the observables are said to [[commutator|commute]].

By contrast, if the eigenstates of {{mvar|Q}} and {{mvar|R}} are different, then measurement of {{mvar|R}} produces a jump to a state that is not an eigenstate of {{mvar|Q}}. Therefore, if the system is known to be in some eigenstate of {{mvar|Q}} (all probability amplitudes zero except for one eigenstate), then when {{mvar|R}} is observed the probability amplitudes are changed. A second, subsequent observation of {{mvar|Q}} no longer certainly produces the eigenvalue corresponding to the starting state. In other words, the probability amplitudes for the second measurement of {{mvar|Q}} depend on whether it comes before or after a measurement of {{mvar|R}}, and the two observables [[commutator|do not commute]].

==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.

==Normalization==

In the example above, the measurement must give either {{math|{{ket| ''H'' }}}} or {{math|{{ket| ''V'' }}}}, so the total probability of measuring {{math|{{ket| ''H'' }}}} or {{math|{{ket| ''V'' }}}} must be 1. This leads to a constraint that {{math|1=''α''<sup>2</sup> + ''β''<sup>2</sup> = 1}}; more generally '''the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one'''. If to understand "all the possible states" as an [[orthonormal basis]], that makes sense in the discrete case, then this condition is the same as the norm-1 condition explained [[#Mathematical|above]].

One can always divide any non-zero element of a Hilbert space by its norm and obtain a ''normalized'' state vector. Not every wave function belongs to the Hilbert space {{math|''L''<sup>2</sup>(''X'')}}, though. Wave functions that fulfill this constraint are called [[normalizable wave function|normalizable]].

The [[Schrödinger equation]], describing states of quantum particles, has solutions that describe a system and determine precisely how the state [[time evolution operator|changes with time]]. Suppose a [[wave function]] {{math|''&psi;''('''x''', ''t'')}} gives a description of the particle (position {{math|'''x'''}} at a given time {{math|''t''}}). A wave function is [[square integrable]] if 
:<math>\int |\psi(\mathbf x, t)|^2\, \mathrm{d\mathbf x} = a^2 < \infty.</math>
After [[Wave function#Normalization condition|normalization]] the wave function still represents the same state and is therefore equal by definition to{{sfn|Bäuerle|de Kerf|1990|p=330}}<ref>See also [[Wigner's theorem]]</ref>
:<math>\psi(\mathbf{x},t):=\frac{\psi(\mathbf{x},t)}{a}.</math>
Under the standard [[Copenhagen interpretation]], the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, {{math|''ρ''('''x''') {{=}} {{abs|''&psi;''('''x''', ''t'')}}<sup>2</sup>}} is a [[probability density function]] and the probability that the particle is in the volume {{math|''V''}} at fixed time {{math|''t''}} is given by
:<math> P_{\mathbf{x}\in V}(t) = \int_V |\psi(\mathbf {x}, t)|^2\, \mathrm{d\mathbf {x}}=\int_V \rho(\mathbf {x})\, \mathrm{d\mathbf {x}}.</math>
The probability density function does not vary with time as the evolution of the wave function is dictated by the Schrödinger equation and is therefore entirely deterministic.{{sfn | Zwiebach | 2022 | p=170}} This is key to understanding the importance of this interpretation: for a given particle constant [[mass]], initial {{math|''&psi;''('''x''', ''t''<sub>0</sub>)}} and [[potential energy|potential]], the Schrödinger equation fully determines subsequent wavefunctions. The above then gives probabilities of locations of the particle at all subsequent times.

==In the context of the double-slit experiment==
{{main|Double-slit experiment}}
Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic [[double-slit experiment]], electrons are fired randomly at two slits, and the probability distribution of detecting electrons at all parts on a large screen placed behind the slits, is questioned. An intuitive answer is that {{math|1='''P'''(through either slit) = '''P'''(through first slit) + '''P'''(through second slit)}}, where {{math|'''P'''(event)}} is the probability of that event. This is obvious if one assumes that an electron passes through either slit. When no measurement apparatus that determines through which slit the electrons travel is installed, the observed probability distribution on the screen reflects the [[Interference (wave propagation)|interference pattern]] that is common with light waves. If one assumes the above law to be true, then this pattern cannot be explained. The particles cannot be said to go through either slit and the simple explanation does not work. The correct explanation is, however, by the association of probability amplitudes to each event. The complex amplitudes which represent the electron passing each slit ({{math|''&psi;''<sub>first</sub>}} and {{math|''&psi;''<sub>second</sub>}}) follow the law of precisely the form expected: {{math|1=''&psi;''<sub>total</sub> = ''&psi;''<sub>first</sub> + ''&psi;''<sub>second</sub>}}. This is the principle of [[quantum superposition]]. The probability, which is the [[modulus squared]] of the probability amplitude, then, follows the interference pattern under the requirement that amplitudes are complex:
<math display="block">P = \left|\psi_\text{first} + \psi_\text{second}\right|^2 = \left|\psi_\text{first}\right|^2 + \left|\psi_\text{second}\right|^2 + 2 \left|\psi_\text{first}\right| \left|\psi_\text{second}\right| \cos (\varphi_1 - \varphi_2).</math>
Here, <math>\varphi_1</math> and <math>\varphi_2</math>
are the [[Argument (complex analysis)|arguments]] of {{math|''&psi;''<sub>first</sub>}} and {{math|''&psi;''<sub>second</sub>}} respectively. A purely real formulation has too few dimensions to describe the system's state when superposition is taken into account. That is, without the arguments of the amplitudes, we cannot describe the phase-dependent interference. The crucial term <math display="inline"> 2 \left|\psi_\text{first}\right| \left|\psi_\text{second}\right| \cos (\varphi_1 - \varphi_2)</math> is called the "interference term", and this would be missing if we had added the probabilities.

However, one may choose to devise an experiment in which the experimenter observes which slit each electron goes through. Then, due to [[wavefunction collapse]], the interference pattern is not observed on the screen.

One may go further in devising an experiment in which the experimenter gets rid of this "which-path information" by a [[Quantum eraser experiment|"quantum eraser"]]. Then, according to the [[Copenhagen interpretation]], the case A applies again and the interference pattern is restored.<ref>A recent 2013 experiment gives insight regarding the correct physical interpretation of such phenomena. The information can actually be obtained, but then the electron seemingly went through all the possible paths simultaneously. (Certain [[Ensemble interpretation|ensemble-alike]] realistic interpretations of the wavefunction may presume such coexistence in all the points of an orbital.) Cf. {{cite journal |last1=Schmidt|first1=L. Ph. H.| last2=Lower|first2=J.| last3=Jahnke|first3=T. |last4=Schößler|first4=S. |last5=Schöffler|first5=M. S.| last6=Menssen|first6=A. |last7=Lévêque|first7=C. |last8=Sisourat|first8=N. |last9=Taïeb|first9=R. | display-authors=1| year=2013|title=Momentum Transfer to a Free Floating Double Slit: Realization of a Thought Experiment from the Einstein-Bohr Debates |journal=[[Physical Review Letters]]| volume=111|issue=10|pages=103201| doi=10.1103/PhysRevLett.111.103201| first10=H.|last10=Schmidt-Böcking| first11=R.|last11=Dörner| pmid=25166663 |bibcode=2013PhRvL.111j3201S |s2cid=2725093 |url=http://pdfs.semanticscholar.org/e551/f885162ab3b25b16fb7fff48c87dbc1cbd02.pdf | archive-url=https://web.archive.org/web/20190307191633/http://pdfs.semanticscholar.org/e551/f885162ab3b25b16fb7fff48c87dbc1cbd02.pdf |url-status=dead|archive-date=2019-03-07}}</ref>

==Conservation of probabilities and the continuity equation==
{{main|Probability current}}
Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relationship between the change in the probability density of the particle's position and the change in the amplitude at these positions.

Define the [[probability current]] (or flux) {{math|'''j'''}} as
:<math> \mathbf{j} = {\hbar \over m} {1 \over {2 i}} \left( \psi ^{*} \nabla \psi  - \psi \nabla \psi^{*} \right)  = {\hbar \over m} \operatorname{Im} \left( \psi ^{*} \nabla \psi \right),</math>
measured in units of (probability)/(area&nbsp;&times;&nbsp;time).

Then the current satisfies the equation
:<math> \nabla \cdot \mathbf{j} + { \partial \over \partial t} |\psi|^2 = 0.</math>
The probability density is <math>\rho=|\psi|^2</math>, this equation is exactly the [[continuity equation]], appearing in many situations in physics where we need to describe the local conservation of quantities. The best example is in classical electrodynamics, where {{math|'''j'''}} corresponds to current density corresponding to electric charge, and the density is the charge-density. The corresponding continuity equation describes the local [[Charge conservation|conservation of charges]].

==Composite systems==
For two quantum systems with spaces {{math|''L''<sup>2</sup>(''X''<sub>1</sub>)}} and {{math|''L''<sup>2</sup>(''X''<sub>2</sub>)}} and given states {{math|{{ket|&Psi;<sub>1</sub>}}}} and {{math|{{ket|&Psi;<sub>2</sub>}}}} respectively, their combined state {{math|{{ket|&Psi;<sub>1</sub>}} [[outer product|⊗]] {{ket|&Psi;<sub>2</sub>}}}} can be expressed as {{math|''&psi;''<sub>1</sub>(''x''<sub>1</sub>) ''&psi;''<sub>2</sub>(''x''<sub>2</sub>)}} a function on {{math|''X''<sub>1</sub> [[direct product|×]] ''X''<sub>2</sub>}}, that gives the
[[product measure|product of respective probability measures]]. In other words, amplitudes of a non-[[quantum entanglement|entangled]] composite state are [[multiplication|products]] of original amplitudes, and [[#convenient|respective observables]] on the systems 1 and 2 behave on these states as [[independent random variables]]. This strengthens the probabilistic interpretation explicated [[#The laws of calculating probabilities of events|above]] .

==Amplitudes in operators==
The concept of amplitudes is also used in the context of [[scattering theory]], notably in the form of [[S-matrix|S-matrices]]. Whereas moduli of vector components squared, for a given vector, give a fixed probability distribution, moduli of [[matrix element (physics)|matrix element]]s squared are interpreted as [[transition probabilities]] just as in a random process. Like a finite-dimensional [[unit vector]] specifies a finite probability distribution, a finite-dimensional [[unitary matrix]] specifies transition probabilities between a finite number of states.

The "transitional" interpretation may be applied to {{math|''L''<sup>2</sup>}}s on non-discrete spaces as well.{{clarify|reason= [applied to ..] as opposed to what? On what "non-discrete space"?|date=November 2023}}

==See also==
*[[Expectation value (quantum mechanics)]]
*[[Free particle]]
*[[Finite potential barrier]]
*[[Matter wave]]
*[[Phase space formulation]]
*[[Uncertainty principle]]
*[[John Clive Ward|Ward's probability amplitude]]
*[[Wave packet]]

==Notes==
{{reflist}}

==References==
* {{cite book | last=Bäuerle | first=Gerard G. A. | last2=de Kerf | first2=Eddy A. | title=Lie Algebras, Part 1: Finite and Infinite Dimensional Lie Algebras and Applications in Physics |series= Studies in Mathematical Physics| publisher=North Holland | publication-place=Amsterdam | date=1990 | isbn=0-444-88776-8}}
*{{cite thesis |last=de la Madrid Modino |first= R. |date=2001 |title= Quantum mechanics in rigged Hilbert space language|url=https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en |degree=  PhD |publisher= Universidad de Valladolid}}
*{{cite book |last1=Feynman |first1=R. P. |last2=Leighton |first2=R. B. |last3=Sands |first3=M. |title=[[The Feynman Lectures on Physics]] |volume=3 |location=Redwood City |publisher=Addison-Wesley |year=1989 |isbn=0-201-51005-7 |chapter=Probability Amplitudes |chapter-url=https://feynmanlectures.caltech.edu/III_03.html}}
*{{cite book |first=Stanley P. |last=Gudder |title=Quantum Probability |location=San Diego |publisher=Academic Press |year=1988 |isbn=0-12-305340-4 }}
* {{cite book | last1=Simon | first1=Barry | author1-link=Barry Simon | title=Orthogonal polynomials on the unit circle. Part 1. Classical theory | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-3446-6 | mr=2105088 |year=2005 | volume=54}}
* {{cite book | last=Teschl | first=G. | title=Mathematical Methods in Quantum Mechanics | publisher=American Mathematical Soc. | publication-place=Providence (R.I) | date=2014 | isbn=978-1-4704-1704-8}}
* {{cite book | last=Zwiebach | first=Barton | title=Mastering Quantum Mechanics | publisher=MIT Press | publication-place=Cambridge, Mass | date=2022 | isbn=978-0-262-04613-8}}

{{Quantum mechanics topics}}

[[Category:Quantum measurement]]
[[Category:Physical quantities]]
[[Category:Particle statistics]]