Editing Wave (section)

== Quantum mechanical waves ==
{{Main|Schrödinger equation}}
{{See also|Wave function}}

=== Schrödinger equation ===
The [[Schrödinger equation]] describes the wave-like behavior of [[particle]]s in [[quantum mechanics]]. Solutions of this equation are [[wave function]]s which can be used to describe the probability density of a particle.

=== Dirac equation ===
The [[Dirac equation]] is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-{{frac|1|2}} particles.

[[File:Wave packet (dispersion).gif|class=skin-invert-image|thumb|A propagating wave packet; in general, the ''envelope'' of the wave packet moves at a different speed than the constituent waves.<ref name=Fromhold>{{cite book |title = Quantum Mechanics for Applied Physics and Engineering |author = A.T. Fromhold |chapter = Wave packet solutions |pages = 59 ''ff'' |quote = (p. 61) ...the individual waves move more slowly than the packet and therefore pass back through the packet as it advances |chapter-url = https://books.google.com/books?id=3SOwc6npkIwC&pg=PA59 |isbn = 978-0-486-66741-6 |publisher = Courier Dover Publications |year = 1991 |edition = Reprint of Academic Press 1981 }}</ref>]]

=== de Broglie waves ===
{{Main|Wave packet|Matter wave}}

[[Louis de Broglie]] postulated that all particles with [[momentum]] have a wavelength
: <math>\lambda = \frac{h}{p},</math>
where ''h'' is the [[Planck constant]], and ''p'' is the magnitude of the [[momentum]] of the particle. This hypothesis was at the basis of [[quantum mechanics]]. Nowadays, this wavelength is called the [[de Broglie wavelength]]. For example, the [[electron]]s in a [[cathode-ray tube|CRT]] display have a de Broglie wavelength of about 10<sup>−13</sup>&nbsp;m.

A wave representing such a particle traveling in the ''k''-direction is expressed by the wave function as follows:
: <math>\psi (\mathbf{r}, \, t=0) = A e^{i\mathbf{k \cdot r}} , </math>
where the wavelength is determined by the [[wave vector]] '''k''' as:
: <math> \lambda = \frac {2 \pi}{k} , </math>
and the momentum by:
: <math> \mathbf{p} = \hbar \mathbf{k} . </math>

However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a [[wave packet]],<ref name=Marton>
{{cite book |title = Advances in Electronics and Electron Physics |page = 271 |chapter-url = https://books.google.com/books?id=g5q6tZRwUu4C&pg=PA271 |isbn = 978-0-12-014653-6 |year = 1980 |publisher = Academic Press |volume = 53 |editor1=L. Marton |editor2=Claire Marton |author = Ming Chiang Li |chapter = Electron Interference }}
</ref> a waveform often used in [[quantum mechanics]] to describe the [[wave function]] of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.

In representing the wave function of a localized particle, the [[wave packet]] is often taken to have a [[Gaussian function|Gaussian shape]] and is called a ''Gaussian wave packet''.<ref name=wavepacket>{{cite book |url = https://books.google.com/books?id=7qCMUfwoQcAC&pg=PA60 |title = Quantum Mechanics |author1=Walter Greiner |author2=D. Allan Bromley |page = 60 |isbn = 978-3-540-67458-0 |edition = 2 |year = 2007 |publisher = Springer }}</ref><ref>{{cite book |title = Electronic basis of the strength of materials |author = John Joseph Gilman |url = https://books.google.com/books?id=YWd7zHU0U7UC&pg=PA57 |page = 57 |year = 2003 |isbn = 978-0-521-62005-5 |publisher = Cambridge University Press }}</ref><ref>{{cite book |title = Principles of quantum mechanics |author = Donald D. Fitts |url = https://books.google.com/books?id=8t4DiXKIvRgC&pg=PA17 |page = 17 |isbn = 978-0-521-65841-6 |publisher = Cambridge University Press |year = 1999 }}</ref> Gaussian wave packets also are used to analyze water waves.<ref name=Mei>{{cite book |url = https://books.google.com/books?id=WHMNEL-9lqkC&pg=PA47 |page = 47 |author = Chiang C. Mei |author-link=Chiang C. Mei |title = The applied dynamics of ocean surface waves |isbn = 978-9971-5-0789-3 |year = 1989 |edition = 2nd |publisher = World Scientific }}</ref>

For example, a Gaussian wavefunction ''ψ'' might take the form:<ref name="Bromley">
{{cite book |last1=Greiner |first1=Walter |url=https://books.google.com/books?id=7qCMUfwoQcAC&pg=PA60 |title=Quantum Mechanics |last2=Bromley |first2=D. Allan |publisher=Springer |year=2007 |isbn=978-3-540-67458-0 |edition=2nd |page=60}}
</ref>
: <math> \psi(x,\, t=0) = A \exp \left( -\frac{x^2}{2\sigma^2} + i k_0 x \right) , </math>
at some initial time ''t'' = 0, where the central wavelength is related to the central wave vector ''k''<sub>0</sub> as ''λ''<sub>0</sub> = 2π / ''k''<sub>0</sub>. It is well known from the theory of [[Fourier analysis]],<ref name=Brandt>
{{cite book |page = 23 |url = https://books.google.com/books?id=VM4GFlzHg34C&pg=PA23 |title = The picture book of quantum mechanics |author1=Siegmund Brandt |author2=Hans Dieter Dahmen |isbn = 978-0-387-95141-6 |year = 2001 |edition = 3rd |publisher = Springer }}
</ref> or from the [[Heisenberg uncertainty principle]] (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The [[Fourier transform]] of a Gaussian is itself a Gaussian.<ref name=Gaussian>
{{cite book |title = Modern mathematical methods for physicists and engineers |author = Cyrus D. Cantrell |page = [https://archive.org/details/modernmathematic0000cant/page/677 677] |url = https://archive.org/details/modernmathematic0000cant |url-access = registration |isbn = 978-0-521-59827-9 |publisher = Cambridge University Press |year = 2000 }}
</ref> Given the Gaussian:
: <math>f(x) = e^{-x^2 / \left(2\sigma^2\right)} , </math>
the Fourier transform is:
: <math>\tilde{ f} (k) = \sigma e^{-\sigma^2 k^2 / 2} . </math>

The Gaussian in space therefore is made up of waves:
: <math>f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \ \tilde{f} (k) e^{ikx} \ dk ; </math>
that is, a number of waves of wavelengths ''λ'' such that ''kλ'' = 2 π.

The parameter σ decides the spatial spread of the Gaussian along the ''x''-axis, while the Fourier transform shows a spread in [[wave vector]] ''k'' determined by 1/''σ''. That is, the smaller the extent in space, the larger the extent in ''k'', and hence in ''λ'' = 2π/''k''.

[[File:GravitationalWave CrossPolarization.gif|class=skin-invert-image|thumb|right|Animation showing the effect of a cross-polarized gravitational wave on a ring of [[test particles]]]]