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Wave packet

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</style><div role="note" class="hatnote navigation-not-searchable">"Wave train" redirects here. For the mathematics concept, see <a href="Periodic_travelling_wave" title="Periodic travelling wave">Periodic travelling wave</a>.</div>

In <a href="Physics" title="Physics">physics</a>, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an <a href="Envelope_(waves)" title="Envelope (waves)">envelope</a>. A wave packet can be analyzed into, or can be synthesized from, a potentially-infinite set of component <a href="Sinusoidal_wave" class="mw-redirect" title="Sinusoidal wave">sinusoidal waves</a> of different <a href="Wavenumber" title="Wavenumber">wavenumbers</a>, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere.<a href="#cite_note-1">[1]</a> Any signal of a limited width in time or space requires many frequency components around a center frequency within a <a href="Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a> inversely proportional to that width; even a <a href="Gaussian_function" title="Gaussian function">gaussian function</a> is considered a wave packet because its <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> is a "packet" of waves of frequencies clustered around a central frequency.<a href="#cite_note-2">[2]</a> Each component <a href="Wave_function" title="Wave function">wave function</a>, and hence the wave packet, are solutions of a <a href="Wave_equation" title="Wave equation">wave equation</a>. Depending on the wave equation, the wave packet's profile may remain constant (no <a href="#Non-dispersive">dispersion</a>) or it may change (<a href="#Dispersive">dispersion</a>) while propagating.

<div class="mw-heading mw-heading2"><h2 id="Historical_background">Historical background</h2></div>

Ideas related to wave packets – <a href="Modulation" class="mw-redirect" title="Modulation">modulation</a>, <a href="Carrier_wave" title="Carrier wave">carrier waves</a>, <a href="Phase_velocity" title="Phase velocity">phase velocity</a>, and <a href="Group_velocity" title="Group velocity">group velocity</a> – date from the mid-1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed by <a href="William_Rowan_Hamilton" title="William Rowan Hamilton">W.R. Hamilton</a> in 1839, and the first full treatment was by <a href="John_Strutt%2C_3rd_Baron_Rayleigh" title="John Strutt, 3rd Baron Rayleigh">Rayleigh</a> in his "Theory of Sound" in 1877.<a href="#cite_note-3">[3]</a>

<a href="Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> introduced the idea of wave packets just after publishing his famous <a href="Schrodinger_equation" class="mw-redirect" title="Schrodinger equation">wave equation</a>.<a href="#cite_note-Kragh-4">[4]</a> He solved his wave equation for a <a href="Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a>, introduced the <a href="Superposition_principle" title="Superposition principle">superposition principle</a>, and used it to show that a compact state could persist. While this work did result in the important concept of <a href="Coherent_states" class="mw-redirect" title="Coherent states">coherent states</a>, the wave packet concept did not endure. The year after Schrödinger's paper, <a href="Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> published his paper on the <a href="Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>, showing in the process, that Schrödinger's results only applied to <a href="Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillators</a>, not for example to <a href="Electric_potential" title="Electric potential">Coulomb potential</a> characteristic of atoms.<a href="#cite_note-Kragh-4">[4]</a>: 829

The following year, 1927, <a href="Charles_Galton_Darwin" title="Charles Galton Darwin">Charles Galton Darwin</a> explored <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger's equation</a> for an unbound electron in free space, assuming an initial <a class="mw-selflink-fragment" href="#Gaussian_wave_packets_in_quantum_mechanics">Gaussian wave packet</a>.<a href="#cite_note-5">[5]</a> Darwin showed that at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> later the position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> of the packet traveling at velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> would be

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<annotation encoding="application/x-tex">{\displaystyle x_{0}+vt\pm {\sqrt {\sigma ^{2}+(ht/2\pi \sigma m)^{2}}}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is the uncertainty in the initial position.

Later in 1927 <a href="Paul_Ehrenfest" title="Paul Ehrenfest">Paul Ehrenfest</a> showed that the time, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> for a <a href="Matter_wave" title="Matter wave">matter wave</a> packet of width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation>

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</math><img src="./f3890eb866b6258d7a304fc34c70ee3fb3a81a70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}" loading="lazy"> and mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> to spread by a factor of 2 was <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T\approx m{\Delta x}^{2}/\hbar }">

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<annotation encoding="application/x-tex">{\textstyle T\approx m{\Delta x}^{2}/\hbar }</annotation>

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</math><img src="./86a7e7f45d7700d402d4e406044102a884fafbf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.564ex; height:3.176ex;" alt="{\textstyle T\approx m{\Delta x}^{2}/\hbar }" loading="lazy">. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

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<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

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</math><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"> is so small, wave packets on the scale of macroscopic objects, with large width and mass, double only at <a href="Cosmic_time" title="Cosmic time">cosmic time</a> scales.<a href="#cite_note-6">[6]</a>: 49

<div class="mw-heading mw-heading2"><h2 id="Significance_in_quantum_mechanics">Significance in quantum mechanics</h2></div>

<a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> describes the nature of atomic and subatomic systems using <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger's wave equation</a>. The classical limit of quantum mechanics and many formulations of quantum scattering use wave packets formed from various solutions to this equation.

Quantum wave packet profiles change while propagating; they show dispersion. Physicists have concluded that "wave packets would not do as representations of subatomic particles".<a href="#cite_note-Kragh-4">[4]</a>: 829

<div class="mw-heading mw-heading3"><h3 id="Wave_packets_and_the_classical_limit">Wave packets and the classical limit</h3></div>

Schrodinger developed wave packets in hopes of interpreting quantum wave solutions as locally compact wave groups.<a href="#cite_note-Kragh-4">[4]</a> Such packets tradeoff position localization for spreading momentum. In the coordinate representation of the wave (such as the <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>), the position of the particle's localized probability is specified by the position of the packet solution. The narrower the spatial wave packet, and therefore the better localized the position of the wave packet, the larger the spread in the <a href="Momentum" title="Momentum">momentum</a> of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the Heisenberg uncertainty principle.

One kind of optimal tradeoff minimizes the product of position uncertainty <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation>

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</math><img src="./f3890eb866b6258d7a304fc34c70ee3fb3a81a70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}" loading="lazy"> and momentum uncertainty <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p_{x}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta p_{x}}</annotation>

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</math><img src="./a84011ce3f13232ac79d38f73b7b48c463144cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.278ex; height:2.509ex;" alt="{\displaystyle \Delta p_{x}}" loading="lazy">.<a href="#cite_note-Schiff-7">[7]</a>: 60 If we place such a packet at rest it stays at rest: the average value of the position and momentum match a classical particle. However it spreads out in all directions with a velocity given by the optimal momentum uncertainty <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p_{x}}">

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<div class="mw-heading mw-heading3"><h3 id="Wave_packets_and_quantum_scattering">Wave packets and quantum scattering</h3></div>

Particle interactions are called <a href="Scattering" title="Scattering">scattering</a> in physics; the wave packet concept plays an important role in <a href="Lippmann%E2%80%93Schwinger_equation#Creating_wavepackets" title="Lippmann–Schwinger equation">quantum scattering approaches</a>. A monochromatic (single momentum) source produces convergence difficulties in the scattering models.<a href="#cite_note-Newton-8">[8]</a>: 150 Scattering problems also have classical limits. Whenever the scattering target (for example an atom) has a size much smaller than wave packet, the center of the wave packet follows scattering classical trajectories. In other cases, the wave packet distorts and scatters as it interacts with the target.<a href="#cite_note-Susskind-Friedman-9">[9]</a>: 295

<div class="mw-heading mw-heading2"><h2 id="Basic_behaviors">Basic behaviors</h2></div>

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<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Dispersion_(water_waves)" title="Dispersion (water waves)">Dispersion (water waves)</a> and <a href="Dispersion_(optics)" title="Dispersion (optics)">Dispersion (optics)</a></div>

<div class="mw-heading mw-heading3"><h3 id="Non-dispersive">Non-dispersive</h3></div>

Without dispersion the wave packet maintains its shape as it propagates.

As an example of propagation without dispersion, consider wave solutions to the following <a href="Wave_equation" title="Wave equation">wave equation</a> from <a href="Classical_physics" title="Classical physics">classical physics</a>

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<annotation encoding="application/x-tex">{\displaystyle {\partial ^{2}u \over \partial t^{2}}=c^{2}\,\nabla ^{2}u,}</annotation>

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where c is the speed of the wave's propagation in a given medium.

Using the physics time convention, e−iωt, the wave equation has <a href="Plane-wave" class="mw-redirect" title="Plane-wave">plane-wave</a> solutions

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<annotation encoding="application/x-tex">{\displaystyle u(\mathbf {x} ,t)=e^{i{(\mathbf {k\cdot x} }-\omega (\mathbf {k} )t)},}</annotation>

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where the relation between the <a href="Angular_frequency" title="Angular frequency">angular frequency</a> ω and <a href="Angular_wave_vector" class="mw-redirect" title="Angular wave vector">angular wave vector</a> k is given by the <a href="Dispersion_relation" title="Dispersion relation">dispersion relation</a>:

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<annotation encoding="application/x-tex">{\displaystyle \omega (\mathbf {k} )=\pm |\mathbf {k} |c=\pm {\frac {2\pi c}{\lambda }},}</annotation>

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such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}/|\mathbf {k} |^{2}=c^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega ^{2}/|\mathbf {k} |^{2}=c^{2}}</annotation>

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</math><img src="./65749b708715469e8d485b6a9bde07e0750a9dda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.581ex; height:3.343ex;" alt="{\displaystyle \omega ^{2}/|\mathbf {k} |^{2}=c^{2}}" loading="lazy">. This relation should be valid so that the plane wave is a solution to the wave equation. As the relation is linear, the wave equation is said to be non-dispersive.

To simplify, consider the one-dimensional wave equation with ω(k) = ±kc. Then the general solution is

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<annotation encoding="application/x-tex">{\displaystyle u(x,t)=Ae^{ik(x-ct)}+Be^{ik(x+ct)},}</annotation>

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where the first and second term represent a wave propagating in the positive respectively negative x-direction.

A wave packet is a localized disturbance that results from the sum of many different <a href="Wave_form" class="mw-redirect" title="Wave form">wave forms</a>. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region.<a href="#cite_note-FOOTNOTEJackson1998322–326-10">[10]</a> From the basic one-dimensional plane-wave solutions, a general form of a wave packet can be expressed as

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<annotation encoding="application/x-tex">{\displaystyle u(x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }A(k)~e^{i(kx-\omega (k)t)}\,dk.}</annotation>

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where the amplitude A(k), containing the coefficients of the <a href="Superposition_principle#Wave_superposition" title="Superposition principle">wave superposition</a>, follows from taking the <a href="Fourier_inversion_theorem" title="Fourier inversion theorem">inverse Fourier transform</a> of a "<a href="Fourier_inversion_theorem#Conditions_on_the_function" title="Fourier inversion theorem">sufficiently nice</a>"

initial wave u(x, t) evaluated at t = 0:

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<annotation encoding="application/x-tex">{\displaystyle A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }u(x,0)~e^{-ikx}\,dx.}</annotation>

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and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {2\pi }}}">

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<annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {2\pi }}}</annotation>

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</math><img src="./f7292157fbd787f538eab0aedd2f101441a04aff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.755ex; height:3.176ex;" alt="{\displaystyle 1/{\sqrt {2\pi }}}" loading="lazy"> comes from <a href="Fourier_transform#Other_conventions" title="Fourier transform">Fourier transform conventions</a>.

For example, choosing

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<annotation encoding="application/x-tex">{\displaystyle u(x,0)=e^{-x^{2}+ik_{0}x},}</annotation>

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we obtain

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<annotation encoding="application/x-tex">{\displaystyle A(k)={\frac {1}{\sqrt {2}}}e^{-{\frac {(k-k_{0})^{2}}{4}}},}</annotation>

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and finally

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}u(x,t)&=e^{-(x-ct)^{2}+ik_{0}(x-ct)}\\&=e^{-(x-ct)^{2}}\left[\cos \left(2\pi {\frac {x-ct}{\lambda }}\right)+i\sin \left(2\pi {\frac {x-ct}{\lambda }}\right)\right].\end{aligned}}}</annotation>

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The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.

<div class="mw-heading mw-heading3"><h3 id="Dispersive">Dispersive</h3></div>

By contrast, in the case of dispersion, a wave changes shape during propagation. For example, the <a href="Free_particle#Mathematical_description" title="Free particle">free Schrödinger equation</a> ,

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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi ,}</annotation>

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has plane-wave solutions of the form:

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} ,t)=Ae^{i{[\mathbf {k\cdot r} }-\omega (\mathbf {k} )t]},}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a constant and the dispersion relation satisfies<a href="#cite_note-11">[11]</a><a href="#cite_note-FOOTNOTECohen-TannoudjiDiuLaloë201913–15-12">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle \omega (\mathbf {k} )={\frac {\hbar \mathbf {k} ^{2}}{2m}}={\frac {\hbar }{2m}}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}),}</annotation>

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with the subscripts denoting <a href="Vector_notation#Unit_vector_notation" title="Vector notation">unit vector notation</a>. As the dispersion relation is non-linear, the free Schrödinger equation is dispersive.

In this case, the wave packet is given by:

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} ,t)={\frac {1}{(2\pi )^{3/2}}}\int g(\mathbf {k} )e^{i{[\mathbf {k\cdot r} }-\omega (\mathbf {k} )t]}d^{3}k}</annotation>

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where once again <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\mathbf {k} )}">

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<annotation encoding="application/x-tex">{\displaystyle g(\mathbf {k} )}</annotation>

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</math><img src="./d32e8f2f6210b39f1316d7319c0db3f5caeab559.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.336ex; height:2.843ex;" alt="{\displaystyle g(\mathbf {k} )}" loading="lazy"> is simply the Fourier transform of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {k} ,0)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {k} ,0)}</annotation>

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</math><img src="./4ae3d37497c0c3920567f9cd1662afd7b38e4f47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.93ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {k} ,0)}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {k} ,0)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {k} ,0)}</annotation>

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</math><img src="./4ae3d37497c0c3920567f9cd1662afd7b38e4f47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.93ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {k} ,0)}" loading="lazy"> (and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\mathbf {k} )}">

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<annotation encoding="application/x-tex">{\displaystyle g(\mathbf {k} )}</annotation>

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</math><img src="./d32e8f2f6210b39f1316d7319c0db3f5caeab559.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.336ex; height:2.843ex;" alt="{\displaystyle g(\mathbf {k} )}" loading="lazy">) is a <a href="Gaussian_function" title="Gaussian function">Gaussian function</a>, the wave packet is called a Gaussian wave packet.<a href="#cite_note-FOOTNOTECohen-TannoudjiDiuLaloë201957,_1511-13">[13]</a>

For example, the solution to the one-dimensional free Schrödinger equation (with 2Δx, m, and ħ set equal to one) satisfying the initial condition

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<annotation encoding="application/x-tex">{\displaystyle \psi (x,0)={\sqrt[{4}]{2/\pi }}\exp \left({-x^{2}+ik_{0}x}\right),}</annotation>

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representing a wave packet localized in space at the origin as a Gaussian function, is seen to be

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi (x,t)&={\frac {\sqrt[{4}]{2/\pi }}{\sqrt {1+2it}}}e^{-{\frac {1}{4}}k_{0}^{2}}~e^{-{\frac {1}{1+2it}}\left(x-{\frac {ik_{0}}{2}}\right)^{2}}\\&={\frac {\sqrt[{4}]{2/\pi }}{\sqrt {1+2it}}}e^{-{\frac {1}{1+4t^{2}}}(x-k_{0}t)^{2}}~e^{i{\frac {1}{1+4t^{2}}}\left((k_{0}+2tx)x-{\frac {1}{2}}tk_{0}^{2}\right)}~.\end{aligned}}}</annotation>

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An impression of the dispersive behavior of this wave packet is obtained by looking at the probability density:

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<annotation encoding="application/x-tex">{\displaystyle |\psi (x,t)|^{2}={\frac {\sqrt {2/\pi }}{\sqrt {1+4t^{2}}}}~e^{-{\frac {2(x-k_{0}t)^{2}}{1+4t^{2}}}}~.}</annotation>

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It is evident that this dispersive wave packet, while moving with constant group velocity ko, is delocalizing rapidly: it has a <a href="Gaussian_function" title="Gaussian function">width</a> increasing with time as √ 1 + 4t2 → 2t, so eventually it diffuses to an unlimited region of space.

<div class="mw-heading mw-heading2"><h2 id="Gaussian_wave_packets_in_quantum_mechanics">Gaussian wave packets in quantum mechanics</h2></div>

The above dispersive Gaussian wave packet, unnormalized and just centered at the origin, instead, at t=0, can now be written in 3D, now in standard units:<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} ,0)=e^{-\mathbf {r} \cdot \mathbf {r} /2a},}</annotation>

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The Fourier transform is also a Gaussian in terms of the wavenumber, the k-vector,

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {k} ,0)=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}.}</annotation>

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With a and its inverse adhering to the <a href="Uncertainty_relation" class="mw-redirect" title="Uncertainty relation">uncertainty relation</a>

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<annotation encoding="application/x-tex">{\displaystyle \Delta x\Delta p_{x}=\hbar /2,}</annotation>

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such that

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<annotation encoding="application/x-tex">{\displaystyle a=2\langle \mathbf {r} \cdot \mathbf {r} \rangle /3\langle 1\rangle =2(\Delta x)^{2},}</annotation>

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can be considered the square of the width of the wave packet, whereas its inverse can be written as

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<annotation encoding="application/x-tex">{\displaystyle 1/a=2\langle \mathbf {k} \cdot \mathbf {k} \rangle /3\langle 1\rangle =2(\Delta p_{x}/\hbar )^{2}.}</annotation>

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Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Psi (\mathbf {k} ,t)&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}e^{-iEt/\hbar }\\&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2-i(\hbar ^{2}\mathbf {k} \cdot \mathbf {k} /2m)t/\hbar }\\&=(2\pi a)^{3/2}e^{-(a+i\hbar t/m)\mathbf {k} \cdot \mathbf {k} /2}.\end{aligned}}}</annotation>

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</math><img src="./7a7857fc08d3588de09655127eebff27fe9076d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.421ex; margin-bottom: -0.25ex; width:39.315ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}\Psi (\mathbf {k} ,t)&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}e^{-iEt/\hbar }\\&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2-i(\hbar ^{2}\mathbf {k} \cdot \mathbf {k} /2m)t/\hbar }\\&=(2\pi a)^{3/2}e^{-(a+i\hbar t/m)\mathbf {k} \cdot \mathbf {k} /2}.\end{aligned}}}" loading="lazy">

</div>

The inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor.

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<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} ,t)=\left({a \over a+i\hbar t/m}\right)^{3/2}e^{-{\mathbf {r} \cdot \mathbf {r} \over 2(a+i\hbar t/m)}}.}</annotation>

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</math><img src="./3b2180144357608e2b71cf9f77278f247cbd6c10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.927ex; height:6.843ex;" alt="{\displaystyle \Psi (\mathbf {r} ,t)=\left({a \over a+i\hbar t/m}\right)^{3/2}e^{-{\mathbf {r} \cdot \mathbf {r} \over 2(a+i\hbar t/m)}}.}" loading="lazy">

</div>

The integral of Ψ over all space is invariant, because it is the inner product of Ψ with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any <a href="Eigenstate" class="mw-redirect" title="Eigenstate">energy eigenstate</a> η(x), the inner product,

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<annotation encoding="application/x-tex">{\displaystyle \langle \eta |\psi \rangle =\int \eta (\mathbf {r} )\psi (\mathbf {r} )d^{3}\mathbf {r} ,}</annotation>

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only changes in time in a simple way: its phase rotates with a frequency determined by the energy of η. When η has zero energy, like the infinite wavelength wave, it doesn't change at all.

For a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, the phase of the wave function varies with position as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\hbar t/m}{2(a^{2}+(\hbar t/m)^{2})}}\|\mathbf {r} \|^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\hbar t/m}{2(a^{2}+(\hbar t/m)^{2})}}\|\mathbf {r} \|^{2}}</annotation>

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</math><img src="./11c16f76ea8b0536f250d00957dc1968f6204a7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.626ex; height:6.509ex;" alt="{\displaystyle {\frac {\hbar t/m}{2(a^{2}+(\hbar t/m)^{2})}}\|\mathbf {r} \|^{2}}" loading="lazy">. It varies quadratically with position, which means that it is different from multiplication by a linear <a href="Phase_factor" title="Phase factor">phase factor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }}">

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</math><img src="./79d3a7f7456e17ab60f3a47c5cb3710da3d7bdd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.118ex; height:2.676ex;" alt="{\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }}" loading="lazy"> as is the case of imparting a constant momentum to the wave packet. In general, the phase of a gaussian wave packet has both a linear term and a quadratic term. The coefficient of the quadratic term begins by increasing from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> as the gaussian wave packet becomes sharper, then at the moment of maximum sharpness, the phase of the wave function varies linearly with position. Then the coefficient of the quadratic term increases from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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</math><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy">, as the gaussian wave packet spreads out again.

The integral ∫ |Ψ|2d3r is also invariant, which is a statement of the conservation of probability.<a href="#cite_note-FOOTNOTECohen-TannoudjiDiuLaloë2019237–240-16">[16]</a> Explicitly,

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<annotation encoding="application/x-tex">{\displaystyle P(r)=|\Psi |^{2}=\Psi ^{*}\Psi =\left({a \over {\sqrt {a^{2}+(\hbar t/m)^{2}}}}\right)^{3}~e^{-{a\,\mathbf {r} \cdot \mathbf {r} \over a^{2}+(\hbar t/m)^{2}}},}</annotation>

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where r is the distance from the origin, the speed of the particle is zero, and width given by

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {a^{2}+(\hbar t/m)^{2} \over a}},}</annotation>

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which is √a at (arbitrarily chosen) time t = 0 while eventually growing linearly in time, as ħt/(m√a), indicating wave-packet spreading.<a href="#cite_note-17">[17]</a>

For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., 10−10 m) then the width of the packet doubles in about 10−16 s. Clearly, particle wave packets spread out very rapidly indeed (in free space):<a href="#cite_note-18">[18]</a> For instance, after 1 ms, the width will have grown to about a kilometer.

This linear growth is a reflection of the (time-invariant) momentum uncertainty: the wave packet is confined to a narrow Δx = √a/2, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount ħ/√2a, a spread in velocity of ħ/m√2a, and thus in the future position by ħt /m√2a. The uncertainty relation is then a strict inequality, very far from saturation, indeed! The initial uncertainty ΔxΔp = ħ/2 has now increased by a factor of ħt/ma (for large t).

A gaussian 2D quantum wave function:

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<annotation encoding="application/x-tex">{\displaystyle \psi (x,t)=\left({\frac {2a^{2}}{\pi }}\right)^{1/4}{\frac {e^{i\phi }}{\left(a^{4}+{\frac {4\hbar ^{2}t^{2}}{m^{2}}}\right)^{1/4}}}e^{ik_{0}x}\exp \left[-{\frac {\left(x-{\frac {\hbar k_{0}}{m}}t\right)^{2}}{a^{2}+{\frac {2i\hbar t}{m}}}}\right]}</annotation>

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</math><img src="./a2b31d16e3f612617e46660e90153ee41f1481be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:62.793ex; height:11.176ex;" alt="{\displaystyle \psi (x,t)=\left({\frac {2a^{2}}{\pi }}\right)^{1/4}{\frac {e^{i\phi }}{\left(a^{4}+{\frac {4\hbar ^{2}t^{2}}{m^{2}}}\right)^{1/4}}}e^{ik_{0}x}\exp \left[-{\frac {\left(x-{\frac {\hbar k_{0}}{m}}t\right)^{2}}{a^{2}+{\frac {2i\hbar t}{m}}}}\right]}" loading="lazy">

where<a href="#cite_note-FOOTNOTECohen-TannoudjiDiuLaloë201959-19">[19]</a>

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<annotation encoding="application/x-tex">{\displaystyle \phi =-\theta -{\frac {\hbar k_{0}^{2}}{2m}}t}</annotation>

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</math><img src="./ae4f66919008c2afa164c11eec91b4d0f662396f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.471ex; height:6.009ex;" alt="{\displaystyle \phi =-\theta -{\frac {\hbar k_{0}^{2}}{2m}}t}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \tan(2\theta )={\frac {2\hbar t}{ma^{2}}}}</annotation>

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</math><img src="./3063cf50c73bf397e359f0dbfeca843c8e0548de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.681ex; height:5.676ex;" alt="{\displaystyle \tan(2\theta )={\frac {2\hbar t}{ma^{2}}}}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="The_Airy_wave_train">The Airy wave train</h2></div>

In contrast to the above Gaussian wave packet, which moves at constant group velocity, and always disperses, there exists a wave function based on <a href="Airy_function" title="Airy function">Airy functions</a>, that propagates freely without envelope dispersion, maintaining its shape, and accelerates in free space:<a href="#cite_note-20">[20]</a>

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<annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {Ai} \left[{\frac {B}{\hbar ^{2/3}}}\left(x-{\frac {B^{3}t^{2}}{4m^{2}}}\right)\right]e^{(iB^{3}t/2m\hbar )[x-(B^{3}t^{2}/6m^{2})]},}</annotation>

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where, for simplicity (and <a href="Nondimensionalization" title="Nondimensionalization">nondimensionalization</a>), choosing ħ = 1, m = 1/2, and B an arbitrary constant results in

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<annotation encoding="application/x-tex">{\displaystyle \psi =\operatorname {Ai} [B(x-B^{3}t^{2})]\,e^{iB^{3}t(x-{\tfrac {2}{3}}B^{3}t^{2})}\,.}</annotation>

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There is no dissonance with <a href="Ehrenfest's_theorem" class="mw-redirect" title="Ehrenfest's theorem">Ehrenfest's theorem</a> in this force-free situation, because the state is both non-normalizable and has an undefined (infinite) ⟨x⟩ for all times. (To the extent that it could be defined, ⟨p⟩ = 0 for all times, despite the apparent acceleration of the front.)

The Airy wave train is the only dispersionless wave in one dimensional free space.<a href="#cite_note-21">[21]</a> In higher dimensions, other dispersionless waves are possible.<a href="#cite_note-22">[22]</a>

In <a href="Phase_space_quantum_mechanics" class="mw-redirect" title="Phase space quantum mechanics">phase space</a>, this is evident in the <a href="Pure_state" class="mw-redirect" title="Pure state">pure state</a> <a href="Wigner_quasiprobability_distribution" title="Wigner quasiprobability distribution">Wigner quasiprobability distribution</a> of this wavetrain, whose shape in x and p is invariant as time progresses, but whose features accelerate to the right, in accelerating parabolas. The Wigner function satisfies<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}W(x,p;t)&=W(x-B^{3}t^{2},p-B^{3}t;0)\\&={\frac {1}{2^{1/3}\pi B}}\,\mathrm {Ai} \left(2^{2/3}\left(B(x-B^{3}t^{2}\right)+\left(p/B-tB^{2})^{2}\right)\right)\\&=W(x-2pt,p;0).\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W(x,p;t)&=W(x-B^{3}t^{2},p-B^{3}t;0)\\&={\frac {1}{2^{1/3}\pi B}}\,\mathrm {Ai} \left(2^{2/3}\left(B(x-B^{3}t^{2}\right)+\left(p/B-tB^{2})^{2}\right)\right)\\&=W(x-2pt,p;0).\end{aligned}}}</annotation>

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</math>The three equalities demonstrate three facts:

<ol><li>Time-evolution is equivalent to a translation in phase-space by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B^{3}t^{2},B^{3}t)}">

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<annotation encoding="application/x-tex">{\displaystyle (B^{3}t^{2},B^{3}t)}</annotation>

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</math><img src="./6752c55ad39a68daab6f9d6cb97c3d1f9e129d4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.213ex; height:3.176ex;" alt="{\displaystyle (B^{3}t^{2},B^{3}t)}" loading="lazy">.</li>

<li>The contour lines of the Wigner function are parabolas of form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B\left(x-B^{3}t^{2}\right)+\left(p/B-tB^{2}\right)^{2}=C}">

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<annotation encoding="application/x-tex">{\textstyle B\left(x-B^{3}t^{2}\right)+\left(p/B-tB^{2}\right)^{2}=C}</annotation>

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</math><img src="./a926fcaa30a43cc60ead20c6f389bf43a78fef87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.646ex; height:3.676ex;" alt="{\textstyle B\left(x-B^{3}t^{2}\right)+\left(p/B-tB^{2}\right)^{2}=C}" loading="lazy">.</li>

<li>Time-evolution is equivalent to a shearing in phase space along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-direction at speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/m=2p}">

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<annotation encoding="application/x-tex">{\displaystyle p/m=2p}</annotation>

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</math><img src="./5ff7fc664ac9bb16d238321d7f4c878628f9c36f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.892ex; height:2.843ex;" alt="{\displaystyle p/m=2p}" loading="lazy">.</li></ol>

Note the momentum distribution obtained by integrating over all x is constant. Since this is the <a href="Wigner_quasiprobability_distribution#Mathematical_properties" title="Wigner quasiprobability distribution">probability density in momentum space</a>, it is evident that the wave function itself is not normalizable.

<div class="mw-heading mw-heading2"><h2 id="Free_propagator">Free propagator</h2></div>

The narrow-width limit of the Gaussian wave packet solution discussed is the free <a href="Propagator#Basic_examples:_propagator_of_free_particle_and_harmonic_oscillator" title="Propagator">propagator kernel</a> K. For other differential equations, this is usually called the <a href="Green's_function" title="Green's function">Green's function</a>,<a href="#cite_note-FOOTNOTEJackson199838–39-23">[23]</a> but in quantum mechanics it is traditional to reserve the name Green's function for the time Fourier transform of K.

Returning to one dimension for simplicity, with m and ħ set equal to one, when a is the infinitesimal quantity ε, the Gaussian initial condition, rescaled so that its integral is one,

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)={1 \over {\sqrt {2\pi \varepsilon }}}e^{-{x^{2} \over 2\varepsilon }}\,}</annotation>

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becomes a <a href="Dirac_delta_function" title="Dirac delta function">delta function</a>, δ(x), so that its time evolution,

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<annotation encoding="application/x-tex">{\displaystyle K_{t}(x)={1 \over {\sqrt {2\pi (it+\varepsilon )}}}e^{-x^{2} \over 2(it+\varepsilon )}\,}</annotation>

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yields the propagator.

Note that a very narrow initial wave packet instantly becomes infinitely wide, but with a phase which is more rapidly oscillatory at large values of x. This might seem strange—the solution goes from being localized at one point to being "everywhere" at all later times, but it is a reflection of the enormous <a href="Uncertainty_principle" title="Uncertainty principle">momentum uncertainty</a> of a localized particle, as explained above.

Further note that the norm of the wave function is infinite, which is also correct, since the square of a <a href="Dirac_delta_function" title="Dirac delta function">delta function</a> is divergent in the same way.

The factor involving ε is an infinitesimal quantity which is there to make sure that integrals over K are well defined. In the limit that ε → 0, K becomes purely oscillatory, and integrals of K are not absolutely convergent. In the remainder of this section, it will be set to zero, but in order for all the integrations over intermediate states to be well defined, the limit ε→0 is to be only taken after the final state is calculated.

The propagator is the amplitude for reaching point x at time t, when starting at the origin, x=0. By translation invariance, the amplitude for reaching a point x when starting at point y is the same function, only now translated,

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<annotation encoding="application/x-tex">{\displaystyle K_{t}(x,y)=K_{t}(x-y)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2} \over 2t}\,.}</annotation>

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In the limit when t is small, the propagator goes to a delta function

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<annotation encoding="application/x-tex">{\displaystyle \lim _{t\to 0}K_{t}(x-y)=\delta (x-y)~,}</annotation>

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but only in the sense of <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>: The integral of this quantity multiplied by an arbitrary differentiable <a href="Test_function" class="mw-redirect" title="Test function">test function</a> gives the value of the test function at zero.

To see this, note that the integral over all space of K equals 1 at all times,

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<annotation encoding="application/x-tex">{\displaystyle \int K_{t}(x)dx=1\,,}</annotation>

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since this integral is the inner-product of K with the uniform wave function. But the phase factor in the exponent has a nonzero spatial derivative everywhere except at the origin, and so when the time is small there are fast phase cancellations at all but one point. This is rigorously true when the limit ε→0 is taken at the very end.

So the propagation kernel is the (future) time evolution of a delta function, and it is continuous, in a sense: it goes to the initial delta function at small times. If the initial wave function is an infinitely narrow spike at position y,

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)=\delta (x-y)\,,}</annotation>

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it becomes the oscillatory wave,

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<annotation encoding="application/x-tex">{\displaystyle \psi _{t}(x)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}\,.}</annotation>

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Now, since every function can be written as a weighted sum of such narrow spikes,

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)=\int \psi _{0}(y)\delta (x-y)dy\,,}</annotation>

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the time evolution of every function ψ0 is determined by this propagation kernel K,

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<annotation encoding="application/x-tex">{\displaystyle \psi _{t}(x)=\int \psi _{0}(y){1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}dy\,.}</annotation>

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</math><img src="./498a3e7bb644f8420894c254a6c143c679dc0db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:36.271ex; height:6.176ex;" alt="{\displaystyle \psi _{t}(x)=\int \psi _{0}(y){1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}dy\,.}" loading="lazy">

</div>

Thus, this is a formal way to express the <a href="Fundamental_solution" title="Fundamental solution">fundamental solution</a> or general solution. The interpretation of this expression is that the amplitude for a particle to be found at point x at time t is the amplitude that it started at y, times the amplitude that it went from y to x, summed over all the possible starting points. In other words, it is a <a href="Convolution" title="Convolution">convolution</a> of the kernel K with the arbitrary initial condition ψ0,

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<annotation encoding="application/x-tex">{\displaystyle \psi _{t}=K*\psi _{0}\,.}</annotation>

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Since the amplitude to travel from x to y after a time t+t' can be considered in two steps, the propagator obeys the composition identity,

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<annotation encoding="application/x-tex">{\displaystyle \int K(x-y;t)K(y-z;t')dy=K(x-z;t+t')~,}</annotation>

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which can be interpreted as follows: the amplitude to travel from x to z in time t+t' is the sum of the amplitude to travel from x to y in time t, multiplied by the amplitude to travel from y to z in time t', summed over all possible intermediate states y. This is a property of an arbitrary quantum system, and by subdividing the time into many segments, it allows the time evolution to be expressed as a <a href="Path_integral_formulation" title="Path integral formulation">path integral</a>.<a href="#cite_note-24">[24]</a>

<div class="mw-heading mw-heading2"><h2 id="Analytic_continuation_to_diffusion">Analytic continuation to diffusion</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Heat_equation#Fundamental_solutions" title="Heat equation">Heat equation § Fundamental solutions</a></div>

The spreading of wave packets in quantum mechanics is directly related to the spreading of probability densities in <a href="Diffusion" title="Diffusion">diffusion</a>. For a particle which is <a href="Random_walk" title="Random walk">randomly walking</a>, the probability density function satisfies the <a href="Diffusion_equation" title="Diffusion equation">diffusion equation</a><a href="#cite_note-FOOTNOTEKozdron2008chpt._3_Albert_Einstein's_proof_of_the_existence_of_Brownian_motion-25">[25]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\partial \over \partial t}\rho ={1 \over 2}{\partial ^{2} \over \partial x^{2}}\rho ,}</annotation>

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where the factor of 2, which can be removed by rescaling either time or space, is only for convenience.

A solution of this equation is the time-varying <a href="Gaussian_function" title="Gaussian function">Gaussian function</a>

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<annotation encoding="application/x-tex">{\displaystyle \rho _{t}(x)={1 \over {\sqrt {2\pi t}}}e^{-x^{2} \over 2t},}</annotation>

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which is a form of the <a href="Heat_kernel" title="Heat kernel">heat kernel</a>. Since the integral of ρt is constant while the width is becoming narrow at small times, this function approaches a delta function at t=0,

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again only in the sense of distributions, so that

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for any <a href="Test_function" class="mw-redirect" title="Test function">test function</a> f.

The time-varying Gaussian is the propagation kernel for the diffusion equation and it obeys the <a href="Convolution" title="Convolution">convolution</a> identity,

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<annotation encoding="application/x-tex">{\displaystyle K_{t+t'}=K_{t}*K_{t'}\,,}</annotation>

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which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator H,

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</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{t}(x)=e^{-tH}\,,}</annotation>

</semantics>

</math>

which is the infinitesimal diffusion operator,

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=-{\nabla ^{2} \over 2}\,.}</annotation>

</semantics>

</math>

A matrix has two indices, which in continuous space makes it a function of x and x'. In this case, because of translation invariance, the matrix element K only depend on the difference of the position, and a convenient abuse of notation is to refer to the operator, the matrix elements, and the function of the difference by the same name:

<msub>

</mrow>

</msub>

<msup>

</msup>

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{t}(x,x')=K_{t}(x-x')\,.}</annotation>

</semantics>

</math>

Translation invariance means that continuous matrix multiplication,

<msup>

</msup>

<msub>

<msup>

</msup>

</mrow>

</msub>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(x,x'')=\int _{x'}A(x,x')B(x',x'')\,,}</annotation>

</semantics>

</math>

is essentially convolution,

<msup>

</msup>

<msub>

<msup>

</msup>

</mrow>

</msub>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(\Delta )=C(x-x'')=\int _{x'}A(x-x')B(x'-x'')=\int _{y}A(\Delta -y)B(y)\,.}</annotation>

</semantics>

</math>

The exponential can be defined over a range of ts which include complex values, so long as integrals over the propagation kernel stay convergent,

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{z}(x)=e^{-zH}\,.}</annotation>

</semantics>

</math>

As long as the real part of z is positive, for large values of x, K is exponentially decreasing, and integrals over K are indeed absolutely convergent.

The limit of this expression for z approaching the pure imaginary axis is the above Schrödinger propagator encountered,

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{t}^{\rm {Schr}}=K_{it+\varepsilon }=e^{-(it+\varepsilon )H}\,,}</annotation>

</semantics>

</math>

which illustrates the above time evolution of Gaussians.

From the fundamental identity of exponentiation, or path integration,

<msub>

</mrow>

</msub>

<msub>

<msup>

</msup>

</mrow>

</msub>

<msub>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{z}*K_{z'}=K_{z+z'}\,}</annotation>

</semantics>

</math>

holds for all complex z values, where the integrals are absolutely convergent so that the operators are well defined.

Thus, quantum evolution of a Gaussian, which is the complex diffusion kernel K,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)=K_{a}(x)=K_{a}*\delta (x)\,}</annotation>

</semantics>

</math>

amounts to the time-evolved state,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{t}=K_{it}*K_{a}=K_{a+it}\,.}</annotation>

</semantics>

</math>

This illustrates the above diffusive form of the complex Gaussian solutions,

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{t}(x)={1 \over {\sqrt {2\pi (a+it)}}}e^{-{x^{2} \over 2(a+it)}}\,.}</annotation>

</semantics>

</math>

/* start https://en.wikipedia.org/ */

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<li><a href="Wave_propagation" class="mw-redirect" title="Wave propagation">Wave propagation</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Group_velocity" title="Group velocity">Group velocity</a></li>

<li><a href="Phase_velocity" title="Phase velocity">Phase velocity</a></li>

<li><a href="Free_particle" title="Free particle">Free particle</a></li>

<li><a href="Coherent_states" class="mw-redirect" title="Coherent states">Coherent states</a></li>

<li><a href="Waveform" title="Waveform">Waveform</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>

<li><a href="Matter_wave" title="Matter wave">Matter wave</a></li>

<li><a href="Pulse_(signal_processing)" title="Pulse (signal processing)">Pulse (signal processing)</a></li>

<li><a href="Pulse_(physics)" title="Pulse (physics)">Pulse (physics)</a></li>

<li><a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></li>

<li><a href="Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction to quantum mechanics</a></li>

<li><a href="Soliton" title="Soliton">Soliton</a></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFJoy_Manners2000" class="citation cs2">Joy Manners (2000), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LkDQV7PNJOMC&dq=wave-packet+wavelengths&pg=PA54">Quantum Physics: An Introduction</a>, CRC Press, pp. 53–56, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7503-0720-8</bdi></cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFSchwartz" class="citation web cs1">Schwartz, Matthew. <a rel="nofollow" class="external text" href="https://scholar.harvard.edu/files/schwartz/files/lecture11-wavepackets.pdf">"Lecture 11: Wavepackets and dispersion"</a> (PDF). scholar.harvard.edu. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230318213306/https://scholar.harvard.edu/files/schwartz/files/lecture11-wavepackets.pdf">Archived</a> (PDF) from the original on 2023-03-18. Retrieved 2023-06-22.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFBrillouin1960" class="citation cs2">Brillouin, Léon (1960), Wave Propagation and Group Velocity, New York: Academic Press Inc., <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/537250">537250</a></cite>

</li>

<li id="cite_note-Kragh-4">^ <a href="#cite_ref-Kragh_4-0">a</a> <a href="#cite_ref-Kragh_4-1">b</a> <a href="#cite_ref-Kragh_4-2">c</a> <a href="#cite_ref-Kragh_4-3">d</a> <cite id="CITEREFKragh2009" class="citation book cs1">Kragh, Helge (2009). <a rel="nofollow" class="external text" href="https://link.springer.com/10.1007/978-3-540-70626-7_232">"Wave Packet"</a>. In Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.). Compendium of Quantum Physics. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 828–830. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-70626-7_232">10.1007/978-3-540-70626-7_232</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-70622-9</bdi>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Darwin, Charles Galton. "Free motion in the wave mechanics." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117.776 (1927): 258-293.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFKraghCarazza2000" class="citation journal cs1">Kragh, Helge; Carazza, Bruno (2000). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41134097">"Classical Behavior of Macroscopic Bodies from Quantum Principles: Early Discussions"</a>. Archive for History of Exact Sciences. 55 (1): 43–56. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-9519">0003-9519</a>.</cite>

</li>

<li id="cite_note-Schiff-7"><a href="#cite_ref-Schiff_7-0">^</a> <cite id="CITEREFSchiff1995" class="citation book cs1">Schiff, Leonard I. (1995). Quantum mechanics. International series in pure and applied physics (3. ed., 29. print ed.). New York: McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-055287-6</bdi>.</cite>

</li>

<li id="cite_note-Newton-8"><a href="#cite_ref-Newton_8-0">^</a> <cite id="CITEREFNewton1982" class="citation book cs1">Newton, Roger G. (1982). Scattering theory of waves and particles. Texts and monographs in physics (2 ed.). New York, Heidelberg, Berlin: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-10950-3</bdi>.</cite>

</li>

<li id="cite_note-Susskind-Friedman-9"><a href="#cite_ref-Susskind-Friedman_9-0">^</a> <cite id="CITEREFSusskindFriedmanSusskind2014" class="citation book cs1">Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014). Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics]. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-465-08061-8</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTEJackson1998322–326-10"><a href="#cite_ref-FOOTNOTEJackson1998322–326_10-0">^</a> <a href="#CITEREFJackson1998">Jackson 1998</a>, pp. 322–326.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFHall2013" class="citation book cs1">Hall, Brian C. (2013). Quantum Theory for Mathematicians. New York Heidelberg Dordrecht London: Springer. pp. 91–92. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4614-7115-8</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTECohen-TannoudjiDiuLaloë201913–15-12"><a href="#cite_ref-FOOTNOTECohen-TannoudjiDiuLaloë201913–15_12-0">^</a> <a href="#CITEREFCohen-TannoudjiDiuLaloë2019">Cohen-Tannoudji, Diu & Laloë 2019</a>, pp. 13–15.

</li>

<li id="cite_note-FOOTNOTECohen-TannoudjiDiuLaloë201957,_1511-13"><a href="#cite_ref-FOOTNOTECohen-TannoudjiDiuLaloë201957,_1511_13-0">^</a> <a href="#CITEREFCohen-TannoudjiDiuLaloë2019">Cohen-Tannoudji, Diu & Laloë 2019</a>, pp. 57, 1511.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFPauli2000" class="citation cs2"><a href="Wolfgang_Pauli" title="Wolfgang Pauli">Pauli, Wolfgang</a> (2000), Wave Mechanics: Volume 5 of Pauli Lectures on Physics, Books on Physics, <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, pp. 7–10, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-41462-1</bdi></cite>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> * <cite id="CITEREFAbersPearson2004" class="citation cs2">Abers, E.; Pearson, Ed (2004), Quantum Mechanics, <a href="Addison_Wesley" class="mw-redirect" title="Addison Wesley">Addison Wesley</a>, <a href="Prentice_Hall" title="Prentice Hall">Prentice-Hall Inc.</a>, p. 51, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-146100-0</bdi></cite>

</li>

<li id="cite_note-FOOTNOTECohen-TannoudjiDiuLaloë2019237–240-16"><a href="#cite_ref-FOOTNOTECohen-TannoudjiDiuLaloë2019237–240_16-0">^</a> <a href="#CITEREFCohen-TannoudjiDiuLaloë2019">Cohen-Tannoudji, Diu & Laloë 2019</a>, pp. 237–240.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> Darwin, C. G. (1927). "Free motion in the wave mechanics", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117 (776), 258-293.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFRichard_Fitzpatrick" class="citation cs2">Richard Fitzpatrick, <a rel="nofollow" class="external text" href="https://farside.ph.utexas.edu/teaching/315/Waves/Waveshtml.html">Oscillations and Waves</a></cite>

</li>

<li id="cite_note-FOOTNOTECohen-TannoudjiDiuLaloë201959-19"><a href="#cite_ref-FOOTNOTECohen-TannoudjiDiuLaloë201959_19-0">^</a> <a href="#CITEREFCohen-TannoudjiDiuLaloë2019">Cohen-Tannoudji, Diu & Laloë 2019</a>, p. 59.

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFBerryBalazs1979" class="citation cs2">Berry, M. V.; Balazs, N. L. (1979), "Nonspreading wave packets", Am J Phys, 47 (3): 264–267, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979AmJPh..47..264B">1979AmJPh..47..264B</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.11855">10.1119/1.11855</a></cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFUnnikrishnanRau1996" class="citation journal cs1">Unnikrishnan, K.; Rau, A. R. P. (1996-08-01). <a rel="nofollow" class="external text" href="https://pubs.aip.org/ajp/article/64/8/1034/530435/Uniqueness-of-the-Airy-packet-in-quantum-mechanics">"Uniqueness of the Airy packet in quantum mechanics"</a>. American Journal of Physics. 64 (8): 1034–1035. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.18322">10.1119/1.18322</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9505">0002-9505</a>.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFEfremidisChenSegevChristodoulides2019" class="citation journal cs1">Efremidis, Nikolaos K.; Chen, Zhigang; Segev, Mordechai; Christodoulides, Demetrios N. (2019-05-20). <a rel="nofollow" class="external text" href="https://opg.optica.org/abstract.cfm?URI=optica-6-5-686">"Airy beams and accelerating waves: an overview of recent advances"</a>. Optica. 6 (5): 686. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1904.02933">1904.02933</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FOPTICA.6.000686">10.1364/OPTICA.6.000686</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2334-2536">2334-2536</a>.</cite>

</li>

<li id="cite_note-FOOTNOTEJackson199838–39-23"><a href="#cite_ref-FOOTNOTEJackson199838–39_23-0">^</a> <a href="#CITEREFJackson1998">Jackson 1998</a>, pp. 38–39.

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFFeynmanHibbs1965" class="citation cs2"><a href="Richard_Feynman" title="Richard Feynman">Feynman, R. P.</a>; Hibbs, A. R. (1965), <a rel="nofollow" class="external text" href="https://archive.org/details/quantummechanics0000feyn">Quantum Mechanics and Path Integrals</a>, New York: <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-020650-2</bdi></cite>

</li>

<li id="cite_note-FOOTNOTEKozdron2008chpt._3_Albert_Einstein's_proof_of_the_existence_of_Brownian_motion-25"><a href="#cite_ref-FOOTNOTEKozdron2008chpt._3_Albert_Einstein's_proof_of_the_existence_of_Brownian_motion_25-0">^</a> <a href="#CITEREFKozdron2008">Kozdron 2008</a>, chpt. 3 Albert Einstein's proof of the existence of Brownian motion.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFCohen-TannoudjiDiuLaloë2019" class="citation book cs1">Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019). Quantum Mechanics, Volume 1. Weinheim, Germany: John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-527-34553-3</bdi>.</cite></li>

<li><cite id="CITEREFJackson1998" class="citation book cs1">Jackson, John David (1998). Classical Electrodynamics. New York: John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-30932-1</bdi>.</cite></li>

<li><cite id="CITEREFKozdron2008" class="citation web cs1">Kozdron, Michael J. (2008). <a rel="nofollow" class="external text" href="https://uregina.ca/~kozdron/Research/UgradTalks/BM_and_Heat/heat_and_BM.pdf">"Brownian Motion and the Heat Equation"</a> (PDF). University of Regina. Retrieved October 29, 2024.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li> Learning materials related to <a href="https://en.wikiversity.org/wiki/Topic:Advanced_Classical_Mechanics/The_Eikonal_Approximation_and_Classical_Particle_Motion" class="extiw external" title="v:Topic:Advanced Classical Mechanics/The Eikonal Approximation and Classical Particle Motion">wave packet motion</a> at Wikiversity</li>

<li> The dictionary definition of <a href="https://en.wiktionary.org/wiki/wave_packet" class="extiw external" title="wiktionary:wave packet">wave packet</a> at Wiktionary</li>

<li><a rel="nofollow" class="external text" href="https://www.google.de/?gfe_rd=cr&ei=HwAOVcjcOcyLOvCwgfgG#q=e%5E(-0.25*(x)%5E2)*cos(7*x)%2C+x+is+from+-4+to+4%2C+y+is+from+-2+to+2&safe=off">1d Wave packet plot in Google</a></li>

<li><a rel="nofollow" class="external text" href="https://www.google.de/?gfe_rd=cr&ei=Pv8NVeFsjJE6vI6ByAk#q=e%5E(-0.25*(x)%5E2)*cos(7*x)%2Be%5E(-0.25*(x-7)%5E2)*cos(7*x)%2Be%5E(-0.25*(x%2B7)%5E2)*cos(7*x)%2C+e%5E(-0.25*(x)%5E2)%2Be%5E(-0.25*(x-7)%5E2)+%2Be%5E(-0.25*(x%2B7)%5E2)%2C+x+is+from+-10+to+10%2C+y+is+from+-2+to+2&safe=off">1d Wave train and probability density plot in Google</a></li>

<li><a rel="nofollow" class="external text" href="https://www.google.com/search?q=e%5E(-0.25*(x)%5E2)*cos(5*x)*e%5E(-0.25*(y)%5E2)*cos(5*y)%2C%20x%20is%20from%20-4%20to%204%2C%20y%20is%20from%20-4%20to%204%2C%20z%20is%20from%20-2%20to%202&safe=off&rct=j">2d Wave packet plot in Google</a></li>

<li><a rel="nofollow" class="external text" href="https://www.google.com/search?q=e%5E(-0.25*(x%2B10)%5E2)*cos(5*x)*e%5E(-0.25*(y%2B10)%5E2)*cos(5*y)%2B%20e%5E(-0.25*(x-10)%5E2)*cos(5*x)*e%5E(-0.25*(y-10)%5E2)*cos(5*y)%2Be%5E(-0.25*(x)%5E2)*cos(5*x)*e%5E(-0.25*(y)%5E2)*cos(5*y)%2B%20e%5E(-0.25*(x-5)%5E2)*cos(5*x)*e%5E(-0.25*(y-5)%5E2)*cos(5*y)%2Be%5E(-0.25*(x%2B5)%5E2)*cos(5*x)*e%5E(-0.25*(y%2B5)%5E2)*cos(5*y)%2C%20x%20is%20from%20-10%20to%2010%2C%20y%20is%20from%20-10%20to%2010%2C%20z%20is%20from%20-2%20to%202&safe=off&rct=j">2d Wave train plot in Google</a></li>

<li><a rel="nofollow" class="external text" href="https://www.google.com/search?q=e%5E(-0.25*(x)%5E2)*e%5E(-0.25*(y)%5E2)%2C%20x%20is%20from%20-4%20to%204%2C%20y%20is%20from%20-4%20to%204%2C%20z%20is%20from%20-2%20to%202&safe=off&rct=j">2d probability density plot in Google</a></li>

<li><a rel="nofollow" class="external text" href="https://www.quantum-physics.polytechnique.fr/freeWavepacket.php?lang=1">Quantum physics online : Interactive simulation of a free wavepacket</a></li>

<li><a rel="nofollow" class="external text" href="http://www.nanotechnology.hu/online/web-schroedinger/index.html">Web-Schrödinger</a>: Interactive 2D wave packet dynamics simulation</li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=s8RIqZgFbXA">A simulation of a wave package in 2D (According to FOURIER-Synthesis in 2D)</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-04" href="https://en.wikipedia.org/wiki/?title=Wave_packet&oldid=1304224884">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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