Editing Wave packet (section)

== Gaussian wave packets in quantum mechanics ==
{{Anchor|quantum mechanics}}[[File:Wavepacket1.gif|thumb|right|Superposition of 1D plane waves (blue) that sum to form a Gaussian wave packet (red) that propagates to the right while spreading. Blue dots follow each plane wave's phase velocity while the red line follows the central group velocity.]]

[[File:Wavepacket-a2k4-en (2X speed).gif|300px|thumb|1D Gaussian wave packet, shown in the complex plane, for {{mvar|a}}=2 and {{mvar|k}}=4]]

The above dispersive Gaussian wave packet, unnormalized and just centered at the origin, instead, at {{mvar|t}}=0, can now be written in 3D, now in standard units:<ref>{{Citation|author-link=Wolfgang Pauli|title=Wave Mechanics: Volume 5 of Pauli Lectures on Physics|publisher=[[Dover Publications]]|series=Books on Physics|year=2000|isbn=978-0-486-41462-1|last=Pauli|first=Wolfgang|pp=7-10}}</ref><ref>* {{Citation|title=Quantum Mechanics|first1=E.|last1=Abers|last2=Pearson|first2=Ed|publisher=[[Addison Wesley]], [[Prentice Hall|Prentice-Hall Inc.]]|year=2004|isbn=978-0-13-146100-0|p=51}}</ref>
<math display="block"> \psi(\mathbf{r},0) = e^{-\mathbf{r}\cdot\mathbf{r}/ 2a},</math>
The Fourier transform is also a Gaussian in terms of the wavenumber, the '''k'''-vector, 
<math display="block"> \psi(\mathbf{k},0) = (2\pi a)^{3/2} e^{- a \mathbf{k}\cdot\mathbf{k}/2}.</math>
With {{mvar|a}} and its inverse adhering to the [[uncertainty relation]]
<math display="block">\Delta x \Delta p_x = \hbar/2,</math> 
such that 
<math display="block">a = 2\langle \mathbf r \cdot \mathbf r\rangle/3\langle 1\rangle = 2 (\Delta x)^2,</math>
can be considered the ''square of the width of the wave packet'', whereas its inverse can be written as
<math display="block">1/a = 2\langle\mathbf k\cdot \mathbf k\rangle/3\langle 1\rangle = 2 (\Delta p_x/\hbar)^2.</math>

[[File:Gaussian wavepacket p0=0.webm|thumb|1D Gaussian wave packet, shown in the complex plane, for <math>a = 1, \hbar = 1, k_0 = 0, m = 1</math>. The group velocity is zero. At <math>t=0</math>, the wavefunction has zero phase and minimal width. For <math>t < 0</math>, the wavefunction has quadratic phase, decreasing width. For <math>t > 0</math>, the wavefunction has quadratic phase, increasing width.]]


Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is
{{Equation box 1
|indent =:
|equation = <math> \begin{align}
\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{align}</math>
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[[File:Gaussian wavepacket p0=1.webm|thumb|1D Gaussian wave packet, shown in the complex plane, for <math>a = 1, \hbar = 1, k_0 = 1, m = 1</math>. The overall group velocity is positive, and the wave packet moves as it disperses.]]
The inverse Fourier transform is still a Gaussian, but now the parameter {{mvar|a}} has become complex, and there is an overall normalization factor.
{{Equation box 1
|indent =:
|equation = <math> \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)} }.</math>
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The integral of {{math|Ψ}} over all space is invariant, because it is the inner product of {{math|Ψ}} with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any [[eigenstate|energy eigenstate]] {{math|''η''(''x'')}}, the inner product,
<math display="block">\langle \eta | \psi \rangle = \int \eta(\mathbf{r}) \psi(\mathbf{r})d^3\mathbf{r},</math>
only changes in time in a simple way: its phase rotates with a frequency determined by the energy of {{math|''η''}}. When {{math|''η''}} has zero energy, like the infinite wavelength wave, it doesn't change at all.

For a given <math>t</math>, the phase of the wave function varies with position as <math>\frac{\hbar t/m}{2(a^2 + (\hbar t / m)^2)} \|\mathbf r \|^2 </math>. It varies ''quadratically'' with position, which means that it is different from multiplication by a linear [[phase factor]] <math>e^{i \mathbf k \cdot \mathbf r} </math> 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>-\infty</math> towards <math>0</math> 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>0</math> towards <math>+\infty</math>, as the gaussian wave packet spreads out again.

The integral {{math|∫&thinsp;{{!}}Ψ{{!}}<sup>2</sup>''d''<sup>3</sup>''r''}} is also invariant, which is a statement of the conservation of probability.{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|pp=237-240}} Explicitly,
<math display="block">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}},</math>
where {{math|''r''}} is the distance from the origin, the speed of the particle is zero, and width given by 
<math display="block"> \sqrt{a^2 + (\hbar t/m)^2 \over a},</math>
which is {{math|{{radic|''a''}}}} at (arbitrarily chosen) time {{math|1=''t'' = 0}} while  eventually growing linearly in time, as {{math|''ħt''/(''m''{{radic|''a''}})}}, indicating '''wave-packet spreading'''.<ref>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.</ref>

For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., {{math|10<sup>−10</sup>}} m) then the width of the packet doubles in about {{math|10<sup>−16</sup>}} s. Clearly, particle wave packets spread out very rapidly indeed (in free space):<ref>{{Citation|title=Oscillations and Waves|author=Richard Fitzpatrick|url=https://farside.ph.utexas.edu/teaching/315/Waves/Waveshtml.html}}</ref> For instance, after {{math|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 {{math|1=Δ''x'' = {{radical|''a''/2}}}}, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount {{math|''ħ''/{{radical|2''a''}}}}, a spread in velocity of {{math|''ħ/m''{{radical|2''a''}}}}, and thus in the future position by {{math|''ħt /m''{{radical|2''a''}}}}.  The uncertainty relation is then a strict inequality, very far from saturation, indeed! The initial uncertainty {{math|1=Δ''x''Δ''p'' = ''ħ''/2}} has now increased by a factor of {{math|''ħt/ma''}} (for large {{math|''t''}}).

=== The 2D case ===
[[File:2D Gaussian Quantum Wave Packet.gif|250px|thumb|A 2D gaussian quantum wave packet. The color (yellow green blue) indicates the phase of the wave function <math>\psi</math>, its brightness indicates <math>| \psi | ^2 / | \psi |_{max} ^2</math>. <math>k_{0x}= k_0</math>, <math>k_{0y}=0</math>]]

A gaussian 2D quantum wave function:

<math>\psi(x,y,t)=\psi(x,t) \psi(y,t)</math>
 
<math> \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 ^{i k_0 x} \exp\left[ - \frac{\left(x - \frac{\hbar k_0}{m} t\right) ^2 }{a^2 + \frac{2 i \hbar t}{m}} \right]</math>

where{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|p=59}}

<math> \phi = - \theta - \frac{\hbar k_0 ^2}{2 m} t</math>

<math>\tan (2 \theta) = \frac{2 \hbar t}{m a ^2}</math>