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Wave packet

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From Wikipedia, the free encyclopedia

Short "burst" or "envelope" of restricted wave action that travels as a unit

"Wave train" redirects here. For the mathematics concept, seePeriodic travelling wave.

A looped animation of a wave packet propagating without dispersion: the envelope is maintained even as the phase changes

Inphysics, awave packet (also known as awave train orwave group) is a short burst of localized wave action that travels as a unit, outlined by anenvelope. A wave packet can be analyzed into, or can be synthesized from, a potentially-infinite set of componentsinusoidal waves of differentwavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere.^[1] Any signal of a limited width in time or space requires many frequency components around a center frequency within abandwidth inversely proportional to that width; even agaussian function is considered a wave packet because itsFourier transform is a "packet" of waves of frequencies clustered around a central frequency.^[2] Each componentwave function, and hence the wave packet, are solutions of awave equation. Depending on the wave equation, the wave packet's profile may remain constant (nodispersion) or it may change (dispersion) while propagating.

Historical background

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Ideas related to wave packets –modulation,carrier waves,phase velocity, andgroup velocity – date from the mid-1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed byW.R. Hamilton in 1839, and the first full treatment was byRayleigh in his "Theory of Sound" in 1877.^[3]

Erwin Schrödinger introduced the idea of wave packets just after publishing his famouswave equation.^[4] He solved his wave equation for aquantum harmonic oscillator, introduced thesuperposition principle, and used it to show that a compact state could persist. While this work did result in the important concept ofcoherent states, the wave packet concept did not endure. The year after Schrödinger's paper,Werner Heisenberg published his paper on theuncertainty principle, showing in the process, that Schrödinger's results only applied toquantum harmonic oscillators, not for example toCoulomb potential characteristic of atoms.^[4]^: 829

The following year, 1927,Charles Galton Darwin exploredSchrödinger's equation for an unbound electron in free space, assuming an initialGaussian wave packet.^[5] Darwin showed that at time $t {\displaystyle t}$ later the position $x {\displaystyle x}$ of the packet traveling at velocity $v {\displaystyle v}$ would be

$x_{0}+vt\pm {\sqrt {\sigma ^{2}+(ht/2\pi \sigma m)^{2}}}$

where $\sigma$ is the uncertainty in the initial position.

Later in 1927Paul Ehrenfest showed that the time, $T {\displaystyle T}$ for amatter wave packet of width $\Delta x$ and mass $m {\displaystyle m}$ to spread by a factor of 2 was ${\textstyle T\approx m{\Delta x}^{2}/\hbar }$ . Since $\hbar$ is so small, wave packets on the scale of macroscopic objects, with large width and mass, double only atcosmic time scales.^[6]^: 49

Significance in quantum mechanics

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Quantum mechanics describes the nature of atomic and subatomic systems usingSchrödinger's wave equation. 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".^[4]^: 829

Wave packets and the classical limit

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Schrodinger developed wave packets in hopes of interpreting quantum wave solutions as locally compact wave groups.^[4] Such packets tradeoff position localization for spreading momentum. In the coordinate representation of the wave (such as theCartesian coordinate system), 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 themomentum 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 $\Delta x$ and momentum uncertainty $\Delta p_{x}$ .^[7]^: 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 $\Delta p_{x}$ . The spread is so fast that in the distance of once around an atom the wave packet is unrecognizable.

Wave packets and quantum scattering

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Particle interactions are calledscattering in physics; the wave packet concept plays an important role inquantum scattering approaches. A monochromatic (single momentum) source produces convergence difficulties in the scattering models.^[8]^: 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.^[9]^: 295

Basic behaviors

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Non-dispersive

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A wave packet without dispersion (real or imaginary part)

Without dispersion the wave packet maintains its shape as it propagates.As an example of propagationwithout dispersion, consider wave solutions to the followingwave equation fromclassical physics ${\partial ^{2}u \over \partial t^{2}}=c^{2}\,\nabla ^{2}u,$

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

Using the physics time convention,e^−iωt, the wave equation hasplane-wave solutions $u(\mathbf {x} ,t)=e^{i{(\mathbf {k\cdot x} }-\omega (\mathbf {k} )t)},$

where the relation between theangular frequencyω andangular wave vectork is given by thedispersion relation: $\omega (\mathbf {k} )=\pm |\mathbf {k} |c=\pm {\frac {2\pi c}{\lambda }},$ such that $\omega ^{2}/|\mathbf {k} |^{2}=c^{2}$ . This relation should be valid so that the plane wave is a solution to the wave equation. As the relation islinear, the wave equation is said to benon-dispersive.

To simplify, consider the one-dimensional wave equation withω(k)= ±kc. Then the general solution is $u(x,t)=Ae^{ik(x-ct)}+Be^{ik(x+ct)},$ where the first and second term represent a wave propagating in the positive respectively negativex-direction.

A wave packet is a localized disturbance that results from the sum of many differentwave forms. 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.^[10] From the basic one-dimensional plane-wave solutions, a general form of a wave packet can be expressed as $u(x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }A(k)~e^{i(kx-\omega (k)t)}\,dk.$ where the amplitudeA(k), containing the coefficients of thewave superposition, follows from taking theinverse Fourier transform of a "sufficiently nice" initial waveu(x,t) evaluated att = 0: $A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }u(x,0)~e^{-ikx}\,dx.$ and $1/{\sqrt {2\pi }}$ comes fromFourier transform conventions.

For example, choosing $u(x,0)=e^{-x^{2}+ik_{0}x},$

we obtain $A(k)={\frac {1}{\sqrt {2}}}e^{-{\frac {(k-k_{0})^{2}}{4}}},$

and finally ${\begin{aligned}u(x,t)&=e^{-(x-ct)^{2}+ik_{0}(x-ct)}\\&=e^{-(x-ct)^{2}}\left[\cos \left(2\pi {\frac {x-ct}{\lambda }}\right)+i\sin \left(2\pi {\frac {x-ct}{\lambda }}\right)\right].\end{aligned}}$

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

Dispersive

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A wave packet with dispersion. Notice the wave spreads out and its amplitude reduces.

Position space probability density of an initially Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space.

By contrast, in the case of dispersion, a wave changes shape during propagation. For example, thefree Schrödinger equation , $i\hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi ,$ has plane-wave solutions of the form: $\psi (\mathbf {r} ,t)=Ae^{i{[\mathbf {k\cdot r} }-\omega (\mathbf {k} )t]},$ where $A {\displaystyle A}$ is a constant and the dispersion relation satisfies^[11]^[12] $\omega (\mathbf {k} )={\frac {\hbar \mathbf {k} ^{2}}{2m}}={\frac {\hbar }{2m}}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}),$ with the subscripts denotingunit vector notation. As the dispersion relation is non-linear, the free Schrödinger equation isdispersive.

In this case, the wave packet is given by: $\psi (\mathbf {r} ,t)={\frac {1}{(2\pi )^{3/2}}}\int g(\mathbf {k} )e^{i{[\mathbf {k\cdot r} }-\omega (\mathbf {k} )t]}d^{3}k$ where once again $g(\mathbf {k} )$ is simply the Fourier transform of $\psi (\mathbf {r} ,0)$ . If $\psi (\mathbf {r} ,0)$ (and therefore $g(\mathbf {k} )$ ) is aGaussian function, the wave packet is called aGaussian wave packet.^[13]

For example, the solution to the one-dimensional free Schrödinger equation (with2Δx,m, andħ set equal to one) satisfying the initial condition $\psi (x,0)={\sqrt[{4}]{2/\pi }}\exp \left({-x^{2}+ik_{0}x}\right),$ representing a wave packet localized in space at the origin as a Gaussian function, is seen to be ${\begin{aligned}\psi (x,t)&={\frac {\sqrt[{4}]{2/\pi }}{\sqrt {1+2it}}}e^{-{\frac {1}{4}}k_{0}^{2}}~e^{-{\frac {1}{1+2it}}\left(x-{\frac {ik_{0}}{2}}\right)^{2}}\\&={\frac {\sqrt[{4}]{2/\pi }}{\sqrt {1+2it}}}e^{-{\frac {1}{1+4t^{2}}}(x-k_{0}t)^{2}}~e^{i{\frac {1}{1+4t^{2}}}\left((k_{0}+2tx)x-{\frac {1}{2}}tk_{0}^{2}\right)}~.\end{aligned}}$

An impression of the dispersive behavior of this wave packet is obtained by looking at the probability density: $|\psi (x,t)|^{2}={\frac {\sqrt {2/\pi }}{\sqrt {1+4t^{2}}}}~e^{-{\frac {2(x-k_{0}t)^{2}}{1+4t^{2}}}}~.$ It is evident that this dispersive wave packet, while moving with constant group velocityk_o, is delocalizing rapidly: it has awidth increasing with time as√ 1 + 4t² → 2t, so eventually it diffuses to an unlimited region of space.

Gaussian wave packets in quantum mechanics

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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.

1D Gaussian wave packet, shown in the complex plane, fora=2 andk=4

The above dispersive Gaussian wave packet, unnormalized and just centered at the origin, instead, att=0, can now be written in 3D, now in standard units:^[14]^[15] $\psi (\mathbf {r} ,0)=e^{-\mathbf {r} \cdot \mathbf {r} /2a},$ The Fourier transform is also a Gaussian in terms of the wavenumber, thek-vector, $\psi (\mathbf {k} ,0)=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}.$ Witha and its inverse adhering to theuncertainty relation $\Delta x\Delta p_{x}=\hbar /2,$ such that $a=2\langle \mathbf {r} \cdot \mathbf {r} \rangle /3\langle 1\rangle =2(\Delta x)^{2},$ can be considered thesquare of the width of the wave packet, whereas its inverse can be written as $1/a=2\langle \mathbf {k} \cdot \mathbf {k} \rangle /3\langle 1\rangle =2(\Delta p_{x}/\hbar )^{2}.$

{\displaystyle a=1,\hbar =1,k_{0}=0,m=1} — 1D Gaussian wave packet, shown in the complex plane, for $a=1,\hbar =1,k_{0}=0,m=1$ . The group velocity is zero. At $t=0$ , the wavefunction has zero phase and minimal width. For $t<0$ , the wavefunction has quadratic phase, decreasing width. For $t>0$ , the wavefunction has quadratic phase, increasing width.

{\displaystyle t=0} — 1D Gaussian wave packet, shown in the complex plane, for $a=1,\hbar =1,k_{0}=0,m=1$ . The group velocity is zero. At $t=0$ , the wavefunction has zero phase and minimal width. For $t<0$ , the wavefunction has quadratic phase, decreasing width. For $t>0$ , the wavefunction has quadratic phase, increasing width.

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

${\begin{aligned}\Psi (\mathbf {k} ,t)&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}e^{-iEt/\hbar }\\&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2-i(\hbar ^{2}\mathbf {k} \cdot \mathbf {k} /2m)t/\hbar }\\&=(2\pi a)^{3/2}e^{-(a+i\hbar t/m)\mathbf {k} \cdot \mathbf {k} /2}.\end{aligned}}$

{\displaystyle a=1,\hbar =1,k_{0}=1,m=1} — 1D Gaussian wave packet, shown in the complex plane, for $a=1,\hbar =1,k_{0}=1,m=1$ . 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 parametera has become complex, and there is an overall normalization factor.

$\Psi (\mathbf {r} ,t)=\left({a \over a+i\hbar t/m}\right)^{3/2}e^{-{\mathbf {r} \cdot \mathbf {r} \over 2(a+i\hbar t/m)}}.$

The integral ofΨ over all space is invariant, because it is the inner product ofΨ with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For anyenergy eigenstateη(x), the inner product, $\langle \eta |\psi \rangle =\int \eta (\mathbf {r} )\psi (\mathbf {r} )d^{3}\mathbf {r} ,$ only changes in time in a simple way: its phase rotates with a frequency determined by the energy ofη. Whenη has zero energy, like the infinite wavelength wave, it doesn't change at all.

For a given $t {\displaystyle t}$ , the phase of the wave function varies with position as ${\frac {\hbar t/m}{2(a^{2}+(\hbar t/m)^{2})}}\|\mathbf {r} \|^{2}$ . It variesquadratically with position, which means that it is different from multiplication by a linearphase factor $e^{i\mathbf {k} \cdot \mathbf {r} }$ 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 $-\infty$ towards $0 {\displaystyle 0}$ 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 $0 {\displaystyle 0}$ towards $+\infty$ , as the gaussian wave packet spreads out again.

The integral∫ |Ψ|²d³r is also invariant, which is a statement of the conservation of probability.^[16] Explicitly, $P(r)=|\Psi |^{2}=\Psi ^{*}\Psi =\left({a \over {\sqrt {a^{2}+(\hbar t/m)^{2}}}}\right)^{3}~e^{-{a\,\mathbf {r} \cdot \mathbf {r} \over a^{2}+(\hbar t/m)^{2}}},$ wherer is the distance from the origin, the speed of the particle is zero, and width given by ${\sqrt {a^{2}+(\hbar t/m)^{2} \over a}},$ which is√a at (arbitrarily chosen) timet = 0 while eventually growing linearly in time, asħt/(m√a), indicatingwave-packet spreading.^[17]

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 about10⁻¹⁶ s. Clearly, particle wave packets spread out very rapidly indeed (in free space):^[18] For instance, after1 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 narrowΔx =√a/2, and so has a momentum which is uncertain (according to the uncertainty principle) by the amountħ/√2a, a spread in velocity ofħ/m√2a, and thus in the future position byħt /m√2a. The uncertainty relation is then a strict inequality, very far from saturation, indeed! The initial uncertaintyΔxΔp =ħ/2 has now increased by a factor ofħt/ma (for larget).

The 2D case

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A 2D gaussian quantum wave packet. The color (yellow green blue) indicates the phase of the wave function $\psi$ , its brightness indicates $|\psi |^{2}/|\psi |_{max}^{2}$ . $k_{0x}=k_{0}$ , $k_{0y}=0$

{\displaystyle \psi } — A 2D gaussian quantum wave packet. The color (yellow green blue) indicates the phase of the wave function $\psi$ , its brightness indicates $|\psi |^{2}/|\psi |_{max}^{2}$ . $k_{0x}=k_{0}$ , $k_{0y}=0$

A gaussian 2D quantum wave function:

$\psi (x,y,t)=\psi (x,t)\psi (y,t)$

$\psi (x,t)=\left({\frac {2a^{2}}{\pi }}\right)^{1/4}{\frac {e^{i\phi }}{\left(a^{4}+{\frac {4\hbar ^{2}t^{2}}{m^{2}}}\right)^{1/4}}}e^{ik_{0}x}\exp \left[-{\frac {\left(x-{\frac {\hbar k_{0}}{m}}t\right)^{2}}{a^{2}+{\frac {2i\hbar t}{m}}}}\right]$

where^[19]

$\phi =-\theta -{\frac {\hbar k_{0}^{2}}{2m}}t$

$\tan(2\theta )={\frac {2\hbar t}{ma^{2}}}$

The Airy wave train

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In contrast to the above Gaussian wave packet, which moves at constant group velocity, and always disperses, there exists a wave function based onAiry functions, that propagates freely without envelope dispersion, maintaining its shape, and accelerates in free space:^[20] $\psi =\operatorname {Ai} \left[{\frac {B}{\hbar ^{2/3}}}\left(x-{\frac {B^{3}t^{2}}{4m^{2}}}\right)\right]e^{(iB^{3}t/2m\hbar )[x-(B^{3}t^{2}/6m^{2})]},$ where, for simplicity (andnondimensionalization), choosingħ = 1,m = 1/2, andB an arbitrary constant results in $\psi =\operatorname {Ai} [B(x-B^{3}t^{2})]\,e^{iB^{3}t(x-{\tfrac {2}{3}}B^{3}t^{2})}\,.$

There is no dissonance withEhrenfest's theorem 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.^[21] In higher dimensions, other dispersionless waves are possible.^[22]

The Airy wave train in phase space. Its shape is a series of parabolas with the same axis, but oscillating according to the Airy function. Its time-evolution is a shearing along the $x {\displaystyle x}$ -direction. Each parabola retains its shape under this shearing, and its apex performs a translation along another parabola. Thus, the Airy wave train does not disperse, and the group motion of the wave train undergoes constant acceleration.

Inphase space, this is evident in thepure state Wigner quasiprobability distribution of this wavetrain, whose shape inx andp is invariant as time progresses, but whose features accelerate to the right, in accelerating parabolas. The Wigner function satisfies ${\begin{aligned}W(x,p;t)&=W(x-B^{3}t^{2},p-B^{3}t;0)\\&={\frac {1}{2^{1/3}\pi B}}\,\mathrm {Ai} \left(2^{2/3}\left(B(x-B^{3}t^{2}\right)+\left(p/B-tB^{2})^{2}\right)\right)\\&=W(x-2pt,p;0).\end{aligned}}$ The three equalities demonstrate three facts:

Time-evolution is equivalent to a translation in phase-space by $(B^{3}t^{2},B^{3}t)$ .
The contour lines of the Wigner function are parabolas of form ${\textstyle B\left(x-B^{3}t^{2}\right)+\left(p/B-tB^{2}\right)^{2}=C}$ .
Time-evolution is equivalent to a shearing in phase space along the $x {\displaystyle x}$ -direction at speed $p/m=2p$ .

Note the momentum distribution obtained by integrating over allx is constant. Since this is theprobability density in momentum space, it is evident that the wave function itself is not normalizable.

Free propagator

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The narrow-width limit of the Gaussian wave packet solution discussed is the freepropagator kernelK. For other differential equations, this is usually called theGreen's function,^[23] but in quantum mechanics it is traditional to reserve the name Green's function for the time Fourier transform ofK.

Returning to one dimension for simplicity, withm and ħ set equal to one, whena is the infinitesimal quantityε, the Gaussian initial condition, rescaled so that its integral is one, $\psi _{0}(x)={1 \over {\sqrt {2\pi \varepsilon }}}e^{-{x^{2} \over 2\varepsilon }}\,$ becomes adelta function,δ(x), so that its time evolution, $K_{t}(x)={1 \over {\sqrt {2\pi (it+\varepsilon )}}}e^{-x^{2} \over 2(it+\varepsilon )}\,$ 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 ofx. This might seem strange—the solution goes from being localized at one point to being "everywhere" atall later times, but it is a reflection of the enormousmomentum uncertainty 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 adelta function is divergent in the same way.

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

The propagator is the amplitude for reaching pointx at timet, when starting at the origin,x=0. By translation invariance, the amplitude for reaching a pointx when starting at pointy is the same function, only now translated, $K_{t}(x,y)=K_{t}(x-y)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2} \over 2t}\,.$

In the limit whent is small, the propagator goes to a delta function $\lim _{t\to 0}K_{t}(x-y)=\delta (x-y)~,$ but only in the sense ofdistributions: The integral of this quantity multiplied by an arbitrary differentiabletest function gives the value of the test function at zero.

To see this, note that the integral over all space ofK equals 1 at all times, $\int K_{t}(x)dx=1\,,$ since this integral is the inner-product ofK 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 limitε→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 positiony, $\psi _{0}(x)=\delta (x-y)\,,$ it becomes the oscillatory wave, $\psi _{t}(x)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}\,.$

Now, since every function can be written as a weighted sum of such narrow spikes, $\psi _{0}(x)=\int \psi _{0}(y)\delta (x-y)dy\,,$ the time evolution ofevery functionψ₀ is determined by this propagation kernelK,

$\psi _{t}(x)=\int \psi _{0}(y){1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}dy\,.$

Thus, this is a formal way to express thefundamental solution orgeneral solution. The interpretation of this expression is that the amplitude for a particle to be found at pointx at timet is the amplitude that it started aty, times the amplitude that it went fromy tox,summed over all the possible starting points. In other words, it is aconvolution of the kernelK with the arbitrary initial conditionψ₀, $\psi _{t}=K*\psi _{0}\,.$

Since the amplitude to travel fromx toy after a timet+t' can be considered in two steps, the propagator obeys the composition identity, $\int K(x-y;t)K(y-z;t')dy=K(x-z;t+t')~,$ which can be interpreted as follows: the amplitude to travel fromx toz in timet+t' is the sum of the amplitude to travel fromx toy in timet, multiplied by the amplitude to travel fromy toz in timet', summed overall 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 apath integral.^[24]

Analytic continuation to diffusion

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The spreading of wave packets in quantum mechanics is directly related to the spreading of probability densities indiffusion. For a particle which israndomly walking, the probability density function satisfies thediffusion equation^[25] ${\partial \over \partial t}\rho ={1 \over 2}{\partial ^{2} \over \partial x^{2}}\rho ,$ 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-varyingGaussian function $\rho _{t}(x)={1 \over {\sqrt {2\pi t}}}e^{-x^{2} \over 2t},$ which is a form of theheat kernel. Since the integral ofρ_t is constant while the width is becoming narrow at small times, this function approaches a delta function att=0, $\lim _{t\to 0}\rho _{t}(x)=\delta (x)$ again only in the sense of distributions, so that $\lim _{t\to 0}\int _{x}f(x)\rho _{t}(x)=f(0)$ for anytest functionf.

The time-varying Gaussian is the propagation kernel for the diffusion equation and it obeys theconvolution identity, $K_{t+t'}=K_{t}*K_{t'}\,,$ which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operatorH, $K_{t}(x)=e^{-tH}\,,$ which is the infinitesimal diffusion operator, $H=-{\nabla ^{2} \over 2}\,.$

A matrix has two indices, which in continuous space makes it a function ofx andx'. In this case, because of translation invariance, the matrix elementK 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: $K_{t}(x,x')=K_{t}(x-x')\,.$

Translation invariance means that continuous matrix multiplication, $C(x,x'')=\int _{x'}A(x,x')B(x',x'')\,,$ is essentially convolution, $C(\Delta )=C(x-x'')=\int _{x'}A(x-x')B(x'-x'')=\int _{y}A(\Delta -y)B(y)\,.$

The exponential can be defined over a range ofts which include complex values, so long as integrals over the propagation kernel stay convergent, $K_{z}(x)=e^{-zH}\,.$ As long as the real part ofz is positive, for large values ofx,K is exponentially decreasing, and integrals overK are indeed absolutely convergent.

The limit of this expression forz approaching the pure imaginary axis is the above Schrödinger propagator encountered, $K_{t}^{\rm {Schr}}=K_{it+\varepsilon }=e^{-(it+\varepsilon )H}\,,$ which illustrates the above time evolution of Gaussians.

From the fundamental identity of exponentiation, or path integration, $K_{z}*K_{z'}=K_{z+z'}\,$ holds for all complexz 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 kernelK, $\psi _{0}(x)=K_{a}(x)=K_{a}*\delta (x)\,$ amounts to the time-evolved state, $\psi _{t}=K_{it}*K_{a}=K_{a+it}\,.$

This illustrates the above diffusive form of the complex Gaussian solutions, $\psi _{t}(x)={1 \over {\sqrt {2\pi (a+it)}}}e^{-{x^{2} \over 2(a+it)}}\,.$

Notes

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^Joy Manners (2000),Quantum Physics: An Introduction, CRC Press, pp. 53–56,ISBN 978-0-7503-0720-8
^Schwartz, Matthew."Lecture 11: Wavepackets and dispersion"(PDF).scholar.harvard.edu.Archived(PDF) from the original on 2023-03-18. Retrieved2023-06-22.
^Brillouin, Léon (1960),Wave Propagation and Group Velocity, New York: Academic Press Inc.,OCLC 537250
^^a ^b ^c ^dKragh, Helge (2009)."Wave Packet". In Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.).Compendium of Quantum Physics. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 828–830.doi:10.1007/978-3-540-70626-7_232.ISBN 978-3-540-70622-9.
^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.
^Kragh, Helge; Carazza, Bruno (2000)."Classical Behavior of Macroscopic Bodies from Quantum Principles: Early Discussions".Archive for History of Exact Sciences.55 (1):43–56.ISSN 0003-9519.
^Schiff, Leonard I. (1995).Quantum mechanics. International series in pure and applied physics (3. ed., 29. print ed.). New York: McGraw-Hill.ISBN 978-0-07-055287-6.
^Newton, Roger G. (1982).Scattering theory of waves and particles. Texts and monographs in physics (2 ed.). New York, Heidelberg, Berlin: Springer.ISBN 978-0-387-10950-3.
^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.ISBN 978-0-465-08061-8.
^Jackson 1998, pp. 322–326.
^Hall, Brian C. (2013).Quantum Theory for Mathematicians. New York Heidelberg Dordrecht London: Springer. pp. 91–92.ISBN 978-1-4614-7115-8.
^Cohen-Tannoudji, Diu & Laloë 2019, pp. 13–15.
^Cohen-Tannoudji, Diu & Laloë 2019, pp. 57, 1511.
^Pauli, Wolfgang (2000),Wave Mechanics: Volume 5 of Pauli Lectures on Physics, Books on Physics,Dover Publications, pp. 7–10,ISBN 978-0-486-41462-1
^*Abers, E.; Pearson, Ed (2004),Quantum Mechanics,Addison Wesley,Prentice-Hall Inc., p. 51,ISBN 978-0-13-146100-0
^Cohen-Tannoudji, Diu & Laloë 2019, pp. 237–240.
^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 Character117 (776), 258-293.
^Richard Fitzpatrick,Oscillations and Waves
^Cohen-Tannoudji, Diu & Laloë 2019, p. 59.
^Berry, M. V.; Balazs, N. L. (1979), "Nonspreading wave packets",Am J Phys,47 (3):264–267,Bibcode:1979AmJPh..47..264B,doi:10.1119/1.11855
^Unnikrishnan, K.; Rau, A. R. P. (1996-08-01)."Uniqueness of the Airy packet in quantum mechanics".American Journal of Physics.64 (8):1034–1035.doi:10.1119/1.18322.ISSN 0002-9505.
^Efremidis, Nikolaos K.; Chen, Zhigang; Segev, Mordechai; Christodoulides, Demetrios N. (2019-05-20)."Airy beams and accelerating waves: an overview of recent advances".Optica.6 (5): 686.arXiv:1904.02933.doi:10.1364/OPTICA.6.000686.ISSN 2334-2536.
^Jackson 1998, pp. 38–39.
^Feynman, R. P.; Hibbs, A. R. (1965),Quantum Mechanics and Path Integrals, New York:McGraw-Hill,ISBN 978-0-07-020650-2
^Kozdron 2008, chpt. 3 Albert Einstein's proof of the existence of Brownian motion.

References

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Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019).Quantum Mechanics, Volume 1. Weinheim, Germany: John Wiley & Sons.ISBN 978-3-527-34553-3.
Jackson, John David (1998).Classical Electrodynamics. New York: John Wiley & Sons.ISBN 978-0-471-30932-1.
Kozdron, Michael J. (2008)."Brownian Motion and the Heat Equation"(PDF).University of Regina. RetrievedOctober 29, 2024.