38:
3236:
10793:
5400:
2814:
10766:
10754:
4257:
2371:
634:. A monochromatic (single momentum) source produces convergence difficulties in the scattering models. 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.
3885:
4263:
3433:
6918:
2809:{\displaystyle {\begin{aligned}\psi (x,t)&={\frac {\sqrt{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{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}}}
6525:
515:. 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".
4252:{\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}}}
10783:
5132:
6583:
3215:
5956:
4514:
1213:. 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. From the basic solutions in one dimension, a general form of a wave packet can be expressed as
4266:
3436:
4270:
4269:
4265:
4264:
3440:
3439:
3435:
3434:
4271:
3441:
9501:
7814:
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
2990:
6528:
2073:
3711:
6532:
6531:
6527:
6526:
1348:
73:, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Any signal of a limited width in time or space requires many frequency components around a center frequency within a
6533:
4625:
4268:
3438:
8653:
10788:
4919:
7424:
528:), 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
3428:
7293:
6557:-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.
523:
Schrodinger developed wave packets in hopes of interpreting quantum wave solutions as locally compact wave groups. Such packets tradeoff position localization for spreading momentum. In the coordinate representation of the wave (such as the
10758:
8240:
1623:
7822:
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
7647:
3875:
10072:
9330:
6913:{\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}}}
6494:
3042:
10798:
2198:
588:. 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
6530:
3230:
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
4812:
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
9692:
4766:
3330:
8753:
7995:
1802:
755:
2366:
857:
338:
7093:
6330:
5664:
4358:
1020:
8485:
1156:
8086:
4856:
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
6027:
2821:
8920:
1809:
3603:
2256:
9863:
5257:
10331:
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.
7738:
6096:
658:
8831:
9215:
7184:. For other differential equations, this is usually called the Green's function, but in quantum mechanics it is traditional to reserve the name Green's function for the time Fourier transform of
924:
6588:
4810:
3890:
2376:
2086:
1814:
1216:
9945:
9007:
9765:
5658:
4544:
478:
8572:
3764:
7313:
5494:
7886:
3343:
9117:
7206:
9133:
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:
2094:
1705:
8344:
4342:
3512:
6537:
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
4267:
3437:
10778:
1512:
7808:
9566:
9070:
6981:
9335:
7508:
3773:
3227:
5127:{\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}}},}
1470:
9952:
6357:
6529:
5537:
2129:
616:
586:
9594:
5573:
4898:
4834:
556:
414:
360:
8108:
7153:
3590:
3564:
498:
5421:
3258:
8661:
7891:
3538:
1712:
7116:
6555:
4875:
4854:
4660:
679:
434:
387:
244:
224:
204:
781:
251:
6112:
10240:
8531:. 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
4665:
1046:
8002:
7429:
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
7164:
6105:
In contrast to the above
Gaussian wave packet, which moves at constant group velocity, and always disperses, there exists a wave function based on
532:
of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the
Heisenberg uncertainty principle.
8836:
6521:
The Airy wave train is the only dispersionless wave in one dimensional free space. In higher dimensions, other dispersionless waves are possible.
2097:
Position space probability density of an initially
Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space.
1487:
contains the coefficients of the linear superposition of the plane-wave solutions. These coefficients can in turn be expressed as a function of
3210:{\displaystyle j=\rho v={\frac {1}{2i}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})=\rho \left(k_{0}+{\frac {4t(x-k_{0}t)}{1+4t^{2}}}\right).}
2203:
1043:
To simplify, consider only waves propagating in one dimension (extension to three dimensions is straightforward). Then the general solution is
9781:
5191:
2263:
7658:
5951:{\displaystyle \psi (x,t)=({\frac {2a^{2}}{\pi }})^{1/4}{\frac {e^{i\phi }}{(a^{4}+{\frac {4\hbar ^{2}t^{2}}{m^{2}}})^{1/4}}}e^{ik_{0}x}\exp}
9222:
8367:
8769:
4509:{\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)}}.}
6989:
864:
9868:
10592:
929:
631:
6580:
is invariant as time progresses, but whose features accelerate to the right, in accelerating parabolas. The Wigner function satisfies
3716:
5965:
7830:
9075:
9136:
1630:
10177:
in dispersive equations, such as for the quantum harmonic oscillator, may result in the emergence of envelope-non-dispersive,
8939:
8299:
2260:
Once again, restricting attention to one dimension, the solution to the Schrödinger equation satisfying the initial condition
10734:
10699:
10493:
10460:
10428:
10403:
10378:
10353:
10316:
10213:
9700:
6033:
500:
is so small, wave packets on the scale of macroscopic objects, with large width and mass, double only at cosmic time scales.
10203:
7176:
6569:
7758:
9512:
9016:
5300:
This linear growth is a reflection of the (time-invariant) momentum uncertainty: the wave packet is confined to a narrow
4775:
41:
A looped animation of a wave packet propagating without dispersion: the envelope is maintained even as the phase changes
5583:
10178:
2985:{\displaystyle |\psi (x,t)|^{2}={\frac {\sqrt {2/\pi }}{\sqrt {1+4t^{2}}}}~e^{-{\frac {2(x-k_{0}t)^{2}}{1+4t^{2}}}}~.}
10148:
2077:
The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.
439:
8553:
The spreading of wave packets in quantum mechanics is directly related to the spreading of probability densities in
125:– date from the mid-1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed by
5426:
5363:. The uncertainty relation is then a strict inequality, very far from saturation, indeed! The initial uncertainty
2068:{\displaystyle {\begin{aligned}u(x,t)&=e^{-(x-ct)^{2}+ik_{0}(x-ct)}\\&=e^{-(x-ct)^{2}}\left.\end{aligned}}}
37:
3706:{\displaystyle 1/a=2\langle \mathbf {k} \cdot \mathbf {k} \rangle /3\langle 1\rangle =2(\Delta p_{x}/\hbar )^{2},}
151:, and used it to show that a compact state could persist. While this work did result in the important concept of
130:
10509:
Proceedings of the Royal
Society of London. Series A, Containing Papers of a Mathematical and Physical Character
2818:
An impression of the dispersive behavior of this wave packet is obtained by looking at the probability density:
10679:
4527:
with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any
4278:
3448:
10229:
7478:
be set to zero, but in order for all the integrations over intermediate states to be well defined, the limit
6562:
74:
9009:
which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator
8762:
is constant while the width is becoming narrow at small times, this function approaches a delta function at
6109:, that propagates freely without envelope dispersion, maintaining its shape, and accelerates in free space:
1343:{\displaystyle u(x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }A(k)~e^{i(kx-\omega (k)t)}\,dk.}
10181:—see coherent states: Such "minimum uncertainty states" do saturate the uncertainty principle permanently.
7444:
Further note that the norm of the wave function is infinite, which is also correct, since the square of a
5403:
A 2D gaussian quantum wave packet. The color (yellow green blue) indicates the phase of the wave function
5182:
The width of the
Gaussian is the interesting quantity which can be read off from the probability density,
3879:
Each separate wave only phase-rotates in time, so that the time dependent
Fourier-transformed solution is
65:. A wave packet can be analyzed into, or can be synthesized from, a potentially-infinite set of component
10803:
6927:
6502:
in this force-free situation, because the state is both non-normalizable and has an undefined (infinite)
4620:{\displaystyle \langle \eta |\psi \rangle =\int \eta (\mathbf {r} )\psi (\mathbf {r} )d^{3}\mathbf {r} ,}
525:
164:
144:
10348:. International series in pure and applied physics (3. ed., 29. print ed.). New York: McGraw-Hill.
8655:
where the factor of 2, which can be removed by rescaling either time or space, is only for convenience.
8648:{\displaystyle {\partial \over \partial t}\rho ={1 \over 2}{\partial ^{2} \over \partial x^{2}}\rho ~,}
10834:
7741:
7433:. This might seem strange—the solution goes from being localized at one point to being "everywhere" at
5285:
For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e.,
1473:
1440:
618:. The spread is so fast that in the distance of once around an atom the wave packet is unrecognizable.
17:
7419:{\displaystyle K_{t}(x)={1 \over {\sqrt {2\pi (it+\varepsilon )}}}e^{-x^{2} \over 2it+\varepsilon }\,}
5293:
s. Clearly, particle wave packets spread out very rapidly indeed (in free space): For instance, after
2108:
665:
Without dispersion the wave packet maintains its shape as it propagates. As an example of propagation
10630:
Efremidis, Nikolaos K.; Chen, Zhigang; Segev, Mordechai; Christodoulides, Demetrios N. (2019-05-20).
10133:
8532:
4627:
only changes in time in a simple way: its phase rotates with a frequency determined by the energy of
3423:{\displaystyle a=2\langle \mathbf {r} \cdot \mathbf {r} \rangle /3\langle 1\rangle =2(\Delta x)^{2}.}
3251:
The above dispersive
Gaussian wave packet, unnormalized and just centered at the origin, instead, at
7288:{\displaystyle \psi _{0}(x)={1 \over {\sqrt {2\pi \varepsilon }}}e^{-{x^{2} \over 2\varepsilon }}\,}
8544:
643:
31:
10808:
10525:
9509:
s which include complex values, so long as integrals over the propagation kernel stay convergent,
8255:. The interpretation of this expression is that the amplitude for a particle to be found at point
5499:
5317:, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount
591:
561:
10829:
148:
10143:
512:
179:
8235:{\displaystyle \psi _{t}(x)=\int \psi _{0}(y){1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}dy\,.}
6499:
1618:{\displaystyle A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }u(x,0)~e^{-ikx}\,dx.}
126:
10398:. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books.
9771:
values, where the integrals are absolutely convergent so that the operators are well defined.
5542:
4880:
4816:
538:
396:
8932:
The spreading
Gaussian is the propagation kernel for the diffusion equation and it obeys the
7438:
6333:
4772:
with position, which means that it is different from multiplication by a linear phase factor
345:
175:
160:
7121:
3569:
3543:
483:
10691:
10566:
9496:{\displaystyle C(\Delta )=C(x-x'')=\int _{x'}A(x-x')B(x'-x'')=\int _{y}A(\Delta -y)B(y)\,.}
8246:
7642:{\displaystyle K_{t}(x,y)=K_{t}(x-y)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2} \over 2t}\,.}
7445:
7296:
5406:
3870:{\displaystyle \psi (\mathbf {k} ,0)=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}.}
3767:
140:
85:
is a "packet" of waves of frequencies clustered around a central frequency. Each component
10296:
2992:
It is evident that this dispersive wave packet, while moving with constant group velocity
1036:
should be valid so that the plane wave is a solution to the wave equation. It is called a
136:
8:
10631:
10067:{\displaystyle \psi _{t}(x)={1 \over {\sqrt {2\pi (a+it)}}}e^{-{x^{2} \over 2(a+it)}}\,.}
3517:
3036:
2103:
1037:
647:
10770:
10570:
10373:. Texts and monographs in physics (2 ed.). New York, Heidelberg, Berlin: Springer.
5172:
is the distance from the origin; the speed of the particle is zero; and the time origin
4913:
is also invariant, which is a statement of the conservation of probability. Explicitly,
10719:
10684:
10643:
10452:
8562:
7101:
6540:
4860:
4839:
4645:
419:
372:
229:
209:
189:
168:
6489:{\displaystyle \psi =\operatorname {Ai} \,e^{iB^{3}t(x-{\tfrac {2}{3}}B^{3}t^{2})}\,.}
10730:
10695:
10661:
10612:
10489:
10456:
10424:
10399:
10374:
10349:
10312:
10269:
10209:
9591:
approaching the pure imaginary axis is the above Schrödinger propagator encountered,
3002:
2193:{\displaystyle i{\partial \psi \over \partial t}=-{\frac {1}{2}}{\nabla ^{2}\psi },}
2089:
A wave packet with dispersion. Notice the wave spreads out and its amplitude reduces.
1209:
A wave packet is a localized disturbance that results from the sum of many different
674:
508:
156:
82:
78:
4344:. The overall group velocity is positive, and the wave packet moves as it disperses.
93:. Depending on the wave equation, the wave packet's profile may remain constant (no
10653:
10604:
10574:
10304:
10093:
10088:
62:
61:) is a short burst of localized wave action that travels as a unit, outlined by an
10308:
7999:
Now, since every function can be written as a weighted sum of such narrow spikes,
7175:
The narrow-width limit of the Gaussian wave packet solution discussed is the free
5399:
10714:
10138:
10113:
9687:{\displaystyle K_{t}^{\rm {Schr}}=K_{it+\varepsilon }=e^{-(it+\varepsilon )H}\,,}
8364:' can be considered in two steps, the propagator obeys the composition identity,
4761:{\displaystyle {\frac {\hbar t/m}{2(a^{2}+(\hbar t/m)^{2})}}\|\mathbf {r} \|^{2}}
2107:
where a wave changes shape during propagation, consider instead solutions to the
152:
66:
3235:
10481:
10444:
10103:
10098:
366:
155:, the wave packet concept did not endure. The year after Schrödinger's paper,
122:
118:
10814:
4639:
has zero energy, like the infinite wavelength wave, it doesn't change at all.
657:
10823:
10665:
10616:
10485:
10108:
9949:
This illustrates the above diffusive form of the complex Gaussian solutions,
9774:
Thus, quantum evolution of a Gaussian, which is the complex diffusion kernel
8923:
8566:
7745:
7455:
is an infinitesimal quantity which is there to make sure that integrals over
6106:
3325:{\displaystyle \psi (\mathbf {r} ,0)=e^{-\mathbf {r} \cdot \mathbf {r} /2a},}
2368:, representing a wave packet localized in space at the origin, is seen to be
670:
90:
86:
10657:
8748:{\displaystyle \rho _{t}(x)={1 \over {\sqrt {2\pi t}}}e^{-x^{2} \over 2t}~,}
7990:{\displaystyle \psi _{t}(x)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}\,.}
27:
Short "burst" or "envelope" of restricted wave action that travels as a unit
10815:
A simulation of a wave package in 2D (According to FOURIER-Synthesis in 2D)
9121:
A matrix has two indices, which in continuous space makes it a function of
2085:
1797:{\displaystyle A(k)={\frac {1}{\sqrt {2}}}e^{-{\frac {(k-k_{0})^{2}}{4}}},}
535:
One kind of optimal tradeoff minimizes the product of position uncertainty
114:
10273:
7744:: The integral of this quantity multiplied by an arbitrary differentiable
750:{\displaystyle {\partial ^{2}u \over \partial t^{2}}=c^{2}\,\nabla ^{2}u,}
10804:
Quantum physics online : Interactive simulation of a free wavepacket
10726:
10128:
8933:
8558:
8548:
8280:
4348:
The inverse Fourier transform is still a Gaussian, but now the parameter
3596:
The Fourier transform is also a Gaussian in terms of the wavenumber, the
2361:{\textstyle \psi (x,0)={\sqrt{2/\pi }}\exp \left({-x^{2}+ik_{0}x}\right)}
390:
10765:
10753:
7203:, the Gaussian initial condition, rescaled so that its integral is one,
852:{\displaystyle u(\mathbf {x} ,t)=e^{i{(\mathbf {k\cdot x} }-\omega t)},}
333:{\displaystyle x_{0}+vt\pm {\sqrt {\sigma ^{2}+(ht/2\pi \sigma m)^{2}}}}
9129:'. In this case, because of translation invariance, the matrix element
6566:
4528:
775:
627:
110:
70:
9697:
From the fundamental identity of exponentiation, or path integration,
2093:
10299:. In Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.).
8554:
7088:{\textstyle B\left(x-B^{3}t^{2}\right)+\left(p/B-tB^{2}\right)^{2}=C}
6325:{\displaystyle \psi =\operatorname {Ai} \lefte^{(iB^{3}t/2m\hbar )},}
1210:
163:, showing in the process, that Schrödinger's results only applied to
139:
introduced the idea of wave packets just after publishing his famous
10608:
10578:
10303:. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 828–830.
9219:
Translation invariance means that continuous matrix multiplication,
7474:
are not absolutely convergent. In the remainder of this section, it
7098:
Time-evolution is equivalent to a shearing in phase space along the
10648:
10632:"Airy beams and accelerating waves: an overview of recent advances"
10118:
7167:, it is evident that the wave function itself is not normalizable.
1352:
As in the plane-wave case the wave packet travels to the right for
529:
10423:. New York Heidelberg Dordrecht London: Springer. pp. 91–92.
8487:
which can be interpreted as follows: the amplitude to travel from
7497:=0. By translation invariance, the amplitude for reaching a point
4352:
has become complex, and there is an overall normalization factor.
10629:
10557:
Berry, M. V.; Balazs, N. L. (1979), "Nonspreading wave packets",
10153:
10123:
1015:{\displaystyle |\mathbf {k} |^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}.}
46:
7159:
Note the momentum distribution obtained by integrating over all
6924:
Time-evolution is equivalent to a translation in phase-space by
6518:
for all times, despite the apparent acceleration of the front.)
4523:
over all space is invariant, because it is the inner product of
1151:{\displaystyle u(x,t)=Ae^{i(kx-\omega t)}+Be^{-i(kx+\omega t)},}
8081:{\displaystyle \psi _{0}(x)=\int \psi _{0}(y)\delta (x-y)dy\,,}
6986:
The contour lines of the Wigner function are parabolas of form
1168:. The first term represents a wave propagating in the positive
183:
8561:, the probability density function at any point satisfies the
3566:, the wavefunction has quadratic phase, decreasing width. For
3226:
630:
in physics; wave packet mathematics play an important role in
30:"Wave train" redirects here. For the mathematics concept, see
6022:{\displaystyle \phi =-\theta -{\frac {\hbar k_{0}^{2}}{2m}}t}
3219:
3021:, so eventually it diffuses to an unlimited region of space.
10394:
Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014).
8915:{\displaystyle \lim _{t\to 0}\int _{x}f(x)\rho _{t}(x)=f(0)}
7482:→0 is to be only taken after the final state is calculated.
10507:
Darwin, C. G. (1927). "Free motion in the wave mechanics",
10083:
511:
describes the nature of atomic and subatomic systems using
182:
for an unbound electron in free space, assuming an initial
765:
is the speed of the wave's propagation in a given medium.
9694:
which illustrates the above time evolution of Gaussians.
4662:, the phase of the wave function varies with position as
4275:
1D Gaussian wave packet, shown in the complex plane, for
3592:, the wavefunction has quadratic phase, increasing width.
3540:, the wavefunction has zero phase and minimal width. For
3445:
1D Gaussian wave packet, shown in the complex plane, for
3239:
1D Gaussian wave packet, shown in the complex plane, for
2251:{\displaystyle \omega ={\frac {1}{2}}|\mathbf {k} |^{2}.}
661:
A wave packet without dispersion (real or imaginary part)
10393:
9858:{\displaystyle \psi _{0}(x)=K_{a}(x)=K_{a}*\delta (x)\,}
6510:
for all times. (To the extent that it could be defined,
5252:{\displaystyle {\sqrt {a^{2}+(\hbar t/m)^{2} \over a}}.}
8658:
A solution of this equation is the spreading Gaussian,
7733:{\displaystyle \lim _{t\to 0}K_{t}(x-y)=\delta (x-y)~,}
6091:{\displaystyle tg(2\theta )={\frac {2\hbar t}{ma^{2}}}}
518:
9325:{\displaystyle C(x,x'')=\int _{x'}A(x,x')B(x',x'')\,,}
8480:{\displaystyle \int K(x-y;t)K(y-z;t')dy=K(x-z;t+t')~,}
7751:
To see this, note that the integral over all space of
6992:
6446:
2266:
621:
442:
10449:
Wave Mechanics: Volume 5 of Pauli Lectures on Physics
9955:
9871:
9784:
9703:
9597:
9515:
9338:
9225:
9139:
9078:
9019:
8942:
8839:
8826:{\displaystyle \lim _{t\to 0}\rho _{t}(x)=\delta (x)}
8772:
8664:
8575:
8538:
8370:
8302:
8111:
8005:
7894:
7833:
7761:
7661:
7511:
7316:
7209:
7124:
7104:
6930:
6586:
6543:
6360:
6115:
6036:
5968:
5667:
5586:
5545:
5502:
5429:
5409:
5194:
4922:
4883:
4863:
4842:
4819:
4778:
4668:
4648:
4547:
4361:
4281:
3888:
3776:
3719:
3606:
3572:
3546:
3520:
3451:
3346:
3261:
3255:=0, can now be written in 3D, now in standard units:
3045:
2824:
2374:
2206:
2132:
1812:
1715:
1633:
1515:
1443:
1219:
1049:
932:
867:
784:
682:
594:
564:
541:
486:
422:
399:
375:
348:
254:
232:
212:
192:
10784:
1d Wave train and probability density plot in Google
10593:"Uniqueness of the Airy packet in quantum mechanics"
5297:
ms, the width will have grown to about a kilometer.
919:{\displaystyle \omega ^{2}=|\mathbf {k} |^{2}c^{2},}
503:
7485:The propagator is the amplitude for reaching point
4805:{\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }}
1509:by inverting the Fourier transform relation above:
10718:
10683:
10066:
9940:{\displaystyle \psi _{t}=K_{it}*K_{a}=K_{a+it}\,.}
9939:
9857:
9759:
9686:
9560:
9495:
9324:
9209:
9111:
9064:
9001:
8914:
8833:again only in the sense of distributions, so that
8825:
8747:
8647:
8479:
8338:
8234:
8080:
7989:
7880:
7802:
7732:
7655:is small, the propagator goes to a delta function
7641:
7418:
7287:
7147:
7110:
7087:
6975:
6912:
6549:
6488:
6324:
6090:
6021:
5950:
5652:
5567:
5531:
5488:
5415:
5251:
5126:
4892:
4869:
4848:
4828:
4804:
4760:
4654:
4619:
4508:
4336:
4251:
3869:
3758:
3705:
3584:
3558:
3532:
3506:
3422:
3324:
3209:
2984:
2808:
2360:
2250:
2192:
2067:
1796:
1699:
1617:
1464:
1342:
1150:
1014:
918:
851:
749:
610:
580:
550:
492:
472:
428:
408:
381:
354:
332:
238:
218:
198:
5653:{\displaystyle \psi (x,y,t)=\psi (x,t)\psi (y,t)}
5289:m) then the width of the packet doubles in about
5259:This width eventually grows linearly in time, as
4900:, as the gaussian wave packet spreads out again.
10821:
10811:: Interactive 2D wave packet dynamics simulation
10387:
9580:is exponentially decreasing, and integrals over
8841:
8774:
7663:
7191:Returning to one dimension for simplicity, with
1197:, represents a wave propagating in the negative
473:{\textstyle T\approx \Delta x{\sqrt {m/\hbar }}}
89:, and hence the wave packet, are solutions of a
10590:
10523:
9505:The exponential can be defined over a range of
9072:which is the infinitesimal diffusion operator,
10591:Unnikrishnan, K.; Rau, A. R. P. (1996-08-01).
7748:gives the value of the test function at zero.
6920:The three equalities demonstrate three facts:
3759:{\displaystyle \Delta x\Delta p_{x}=\hbar /2,}
8515:, multiplied by the amplitude to travel from
8503:' is the sum of the amplitude to travel from
7470:becomes purely oscillatory, and integrals of
7441:of a localized particle, as explained above.
5489:{\displaystyle |\psi |^{2}/|\psi |_{max}^{2}}
129:in 1839, and the first full treatment was by
77:inversely proportional to that width; even a
10713:
10475:
10396:Quantum mechanics: the theoretical minimum;
8277:summed over all the possible starting points
7881:{\displaystyle \psi _{0}(x)=\delta (x-y)\,,}
7810:since this integral is the inner-product of
4749:
4740:
4562:
4548:
3657:
3651:
3640:
3624:
3389:
3383:
3372:
3356:
362:is the uncertainty in the initial position.
10556:
10201:
9112:{\displaystyle H=-{\nabla ^{2} \over 2}\,.}
7505:is the same function, only now translated,
1187:only; the second term, being a function of
669:, consider wave solutions to the following
9210:{\displaystyle K_{t}(x,x')=K_{t}(x-x')\,.}
8545:Heat equation § Fundamental solutions
8245:Thus, this is a formal way to express the
3220:Gaussian wave packets in quantum mechanics
1700:{\displaystyle u(x,0)=e^{-x^{2}+ik_{0}x},}
10647:
10263:
10060:
9933:
9854:
9756:
9680:
9554:
9489:
9318:
9203:
9105:
9058:
8995:
8332:
8228:
8098:is determined by this propagation kernel
8074:
7983:
7874:
7796:
7635:
7437:, but it is a reflection of the enormous
7415:
7284:
6728:
6482:
6414:
5060:
1605:
1559:
1330:
1269:
730:
10371:Scattering theory of waves and particles
10339:
10337:
10230:"Lecture 11: Wavepackets and dispersion"
9002:{\displaystyle K_{t+t'}=K_{t}*K_{t'}\,,}
8339:{\displaystyle \psi _{t}=K*\psi _{0}\,.}
8267:, times the amplitude that it went from
6523:
5398:
4337:{\displaystyle a=1,\hbar =1,k_{0}=1,m=1}
4261:
3507:{\displaystyle a=1,\hbar =1,k_{0}=0,m=1}
3431:
3234:
3225:
2092:
2084:
656:
104:
81:is considered a wave packet because its
36:
10678:
10290:
10288:
10286:
10284:
10282:
9760:{\displaystyle K_{z}*K_{z'}=K_{z+z'}\,}
14:
10822:
10368:
10343:
6100:
3338:square of the width of the wave packet
10799:2d probability density plot in Google
10443:
10362:
10334:
10294:
8287:with the arbitrary initial condition
7165:probability density in momentum space
5578:A gaussian 2D quantum wave function:
5348:, and thus in the future position by
10721:Quantum Mechanics and Path Integrals
10418:
10279:
10227:
8263:is the amplitude that it started at
7803:{\displaystyle \int K_{t}(x)dx=1\,,}
7459:are well defined. In the limit that
6570:Wigner quasiprobability distribution
3001:, is delocalizing rapidly: it has a
519:Wave packets and the classical limit
226:of the packet traveling at velocity
143:. He solved his wave equation for a
10266:Wave Propagation and Group Velocity
9865:amounts to the time-evolved state,
9561:{\displaystyle K_{z}(x)=e^{-zH}\,.}
9065:{\displaystyle K_{t}(x)=e^{-tH}\,,}
8348:Since the amplitude to travel from
6976:{\displaystyle (B^{3}t^{2},B^{3}t)}
768:Using the physics time convention,
622:Wave packets and quantum scattering
94:
24:
10539:Cohen-Tannoudji, Diu & Laloë,
9618:
9615:
9612:
9609:
9584:are indeed absolutely convergent.
9465:
9345:
9091:
8620:
8610:
8582:
8578:
8539:Analytic continuation to diffusion
8529:all possible intermediate states y
7170:
6733:
6730:
6572:of this wavetrain, whose shape in
4971:
4962:
4943:
4887:
4823:
4362:
3893:
3726:
3720:
3669:
3401:
3101:
3089:
2174:
2147:
2139:
1560:
1554:
1270:
1264:
732:
701:
687:
637:
595:
565:
542:
449:
400:
133:in his "Theory of Sound" in 1877.
25:
10846:
10746:
10421:Quantum Theory for Mathematicians
10268:, New York: Academic Press Inc.,
10173:By contrast, the introduction of
10149:Introduction to quantum mechanics
9587:The limit of this expression for
9572:is positive, for large values of
7888:it becomes the oscillatory wave,
6354:an arbitrary constant results in
6255:
6064:
5930:
5379:has now increased by a factor of
5215:
5092:
5005:
4710:
4481:
4405:
4294:
4202:
4137:
4006:
3742:
3687:
3514:. The group velocity is zero. At
3464:
2200:yielding the dispersion relation
1465:{\displaystyle 1/{\sqrt {2\pi }}}
652:
626:Particle interactions are called
504:Significance in quantum mechanics
465:
10764:
10752:
10205:Quantum Physics: An Introduction
5070:
5062:
4796:
4788:
4744:
4610:
4592:
4578:
4460:
4452:
4369:
4228:
4220:
4111:
4103:
4071:
4063:
3970:
3962:
3900:
3850:
3842:
3784:
3636:
3628:
3368:
3360:
3302:
3294:
3269:
2229:
939:
887:
827:
821:
792:
109:Ideas related to wave packets –
10707:
10672:
10623:
10584:
10550:
10533:
10517:
10501:
10476:Abers, E.; Pearson, Ed (2004),
10468:
10437:
10246:from the original on 2023-03-18
10167:
7493:, when starting at the origin,
7163:is constant. Since this is the
3336:is a positive real number, the
436:to spread by a factor of 2 was
98:
10757:Learning materials related to
10412:
10325:
10257:
10221:
10195:
10052:
10037:
10006:
9991:
9972:
9966:
9851:
9845:
9823:
9817:
9801:
9795:
9672:
9657:
9532:
9526:
9486:
9480:
9474:
9462:
9443:
9421:
9415:
9398:
9374:
9357:
9348:
9342:
9315:
9293:
9287:
9270:
9246:
9229:
9200:
9183:
9167:
9150:
9036:
9030:
8909:
8903:
8894:
8888:
8875:
8869:
8848:
8820:
8814:
8805:
8799:
8781:
8681:
8675:
8468:
8439:
8424:
8401:
8395:
8377:
8200:
8187:
8153:
8147:
8128:
8122:
8065:
8053:
8047:
8041:
8022:
8016:
7961:
7948:
7911:
7905:
7871:
7859:
7850:
7844:
7781:
7775:
7721:
7709:
7700:
7688:
7670:
7614:
7601:
7562:
7550:
7534:
7522:
7448:is divergent in the same way.
7367:
7352:
7333:
7327:
7310:, so that its time evolution,
7226:
7220:
7199:is the infinitesimal quantity
6970:
6931:
6900:
6870:
6838:
6768:
6682:
6625:
6612:
6594:
6477:
6436:
6411:
6408:
6379:
6373:
6314:
6311:
6270:
6261:
6258:
6225:
6052:
6043:
5945:
5897:
5862:
5853:
5804:
5748:
5715:
5689:
5683:
5671:
5647:
5635:
5629:
5617:
5608:
5590:
5465:
5456:
5440:
5431:
5394:
5230:
5212:
5107:
5089:
5020:
5002:
4948:
4939:
4932:
4926:
4734:
4725:
4707:
4691:
4596:
4588:
4582:
4574:
4555:
4495:
4469:
4379:
4365:
4216:
4190:
4165:
4152:
4126:
4089:
4034:
4021:
3933:
3920:
3910:
3896:
3813:
3800:
3794:
3780:
3691:
3666:
3600:-vector, (with inverse width,
3408:
3398:
3279:
3265:
3172:
3150:
3114:
3076:
2941:
2918:
2850:
2845:
2833:
2826:
2752:
2727:
2672:
2649:
2394:
2382:
2282:
2270:
2235:
2224:
2101:By contrast, as an example of
1945:
1929:
1906:
1891:
1866:
1850:
1832:
1820:
1774:
1754:
1725:
1719:
1649:
1637:
1580:
1568:
1525:
1519:
1325:
1319:
1313:
1298:
1284:
1278:
1235:
1223:
1140:
1122:
1100:
1082:
1065:
1053:
945:
934:
893:
882:
841:
817:
802:
788:
319:
292:
13:
1:
10789:2d Wave packet plot in Google
10779:1d Wave packet plot in Google
10769:The dictionary definition of
10309:10.1007/978-3-540-70626-7_232
10301:Compendium of Quantum Physics
10208:, CRC Press, pp. 53–56,
10188:
7819:→0 is taken at the very end.
7195:and ħ set equal to one, when
2080:
1474:Fourier transform conventions
632:quantum scattering approaches
186:. Darwin showed that at time
10794:2d Wave train plot in Google
9568:As long as the real part of
9332:is essentially convolution,
6498:There is no dissonance with
5532:{\displaystyle k_{0x}=k_{0}}
611:{\displaystyle \Delta p_{x}}
581:{\displaystyle \Delta p_{x}}
165:quantum harmonic oscillators
7:
10597:American Journal of Physics
10344:Schiff, Leonard I. (1995).
10179:classical-looking solutions
10076:
8755:and, since the integral of
8279:. In other words, it is a
6332:where, for simplicity (and
5423:, its brightness indicates
5179:can be chosen arbitrarily.
526:Cartesian coordinate system
513:Schrödinger's wave equation
159:published his paper on the
145:quantum harmonic oscillator
10:
10851:
10690:(2nd ed.), New York:
10160:
8557:. For a particle which is
8542:
5333:, a spread in velocity of
1177:since it is a function of
641:
174:The following year, 1927,
29:
10686:Classical Electrodynamics
10369:Newton, Roger G. (1982).
10134:Pulse (signal processing)
7740:but only in the sense of
6565:, this is evident in the
2111:(nondimensionalized with
2109:free Schrödinger equation
558:and momentum uncertainty
171:characteristic of atoms.
10264:Brillouin, Léon (1960),
5568:{\displaystyle k_{0y}=0}
4893:{\displaystyle +\infty }
4829:{\displaystyle -\infty }
3005:increasing with time as
774:, the wave equation has
644:Dispersion (water waves)
551:{\displaystyle \Delta x}
409:{\displaystyle \Delta x}
32:Periodic travelling wave
10717:; Hibbs, A. R. (1965),
10658:10.1364/OPTICA.6.000686
10419:Hall, Brian C. (2013).
7755:equals 1 at all times,
7501:when starting at point
7426:yields the propagator.
3766:i.e., it satisfies the
3035:remains invariant. The
1627:For instance, choosing
355:{\displaystyle \sigma }
149:superposition principle
10527:Oscillations and Waves
10068:
9941:
9859:
9767:holds for all complex
9761:
9688:
9562:
9497:
9326:
9211:
9113:
9066:
9003:
8916:
8827:
8749:
8649:
8481:
8340:
8236:
8088:the time evolution of
8082:
7991:
7882:
7804:
7734:
7643:
7420:
7289:
7149:
7148:{\displaystyle p/m=2p}
7112:
7089:
6977:
6914:
6558:
6551:
6490:
6326:
6092:
6023:
5952:
5654:
5575:
5569:
5533:
5490:
5417:
5253:
5128:
4894:
4871:
4850:
4830:
4806:
4762:
4656:
4621:
4510:
4345:
4338:
4253:
3871:
3760:
3707:
3593:
3586:
3585:{\displaystyle t>0}
3560:
3559:{\displaystyle t<0}
3534:
3508:
3424:
3326:
3248:
3232:
3211:
2986:
2810:
2362:
2252:
2194:
2098:
2090:
2069:
1798:
1701:
1619:
1466:
1393:, and to the left for
1344:
1152:
1024:This relation between
1016:
920:
853:
751:
662:
612:
582:
552:
494:
493:{\displaystyle \hbar }
474:
430:
410:
383:
369:showed that the time,
356:
334:
240:
220:
200:
180:Schrödinger's equation
42:
10692:John Wiley & Sons
10524:Richard Fitzpatrick,
10295:Kragh, Helge (2009).
10069:
9942:
9860:
9762:
9689:
9563:
9498:
9327:
9212:
9114:
9067:
9004:
8917:
8828:
8750:
8650:
8482:
8341:
8237:
8083:
7992:
7883:
7805:
7735:
7644:
7451:The factor involving
7421:
7290:
7150:
7113:
7090:
6978:
6915:
6552:
6536:
6491:
6334:nondimensionalization
6327:
6093:
6024:
5953:
5655:
5570:
5534:
5491:
5418:
5416:{\displaystyle \psi }
5402:
5280:wave-packet spreading
5254:
5129:
4895:
4872:
4851:
4831:
4807:
4763:
4657:
4622:
4541:, the inner product,
4511:
4339:
4274:
4254:
3872:
3761:
3708:
3587:
3561:
3535:
3509:
3444:
3425:
3327:
3238:
3229:
3212:
3024:The momentum profile
2987:
2811:
2363:
2253:
2195:
2096:
2088:
2070:
1799:
1702:
1620:
1467:
1345:
1158:in which we may take
1153:
1017:
921:
854:
752:
660:
613:
583:
553:
495:
475:
431:
411:
384:
357:
335:
241:
221:
201:
176:Charles Galton Darwin
167:, not for example to
161:uncertainty principle
105:Historical background
101:) while propagating.
40:
10451:, Books on Physics,
10202:Joy Manners (2000),
10144:Schrödinger equation
9953:
9869:
9782:
9701:
9595:
9513:
9336:
9223:
9137:
9076:
9017:
8940:
8837:
8770:
8662:
8573:
8368:
8300:
8247:fundamental solution
8109:
8003:
7892:
7831:
7759:
7659:
7509:
7439:momentum uncertainty
7314:
7207:
7122:
7118:-direction at speed
7102:
6990:
6928:
6584:
6541:
6358:
6113:
6034:
5966:
5665:
5584:
5543:
5500:
5427:
5407:
5192:
4920:
4881:
4861:
4840:
4817:
4776:
4666:
4646:
4545:
4359:
4279:
3886:
3774:
3768:uncertainty relation
3717:
3604:
3570:
3544:
3518:
3449:
3344:
3259:
3043:
2822:
2372:
2264:
2204:
2130:
1810:
1713:
1631:
1513:
1441:
1217:
1047:
930:
865:
782:
680:
592:
562:
539:
484:
440:
420:
397:
373:
346:
252:
230:
210:
190:
184:Gaussian wave packet
97:) or it may change (
10571:1979AmJPh..47..264B
10237:scholar.harvard.edu
10228:Schwartz, Matthew.
9623:
6500:Ehrenfest's theorem
6101:The Airy wave train
6004:
5485:
3533:{\displaystyle t=0}
3037:probability current
2788:
2477:
2126:set equal to one),
1564:
1274:
1038:dispersion relation
1008:
990:
972:
648:Dispersion (optics)
206:later the position
10759:wave packet motion
10541:Quantum Mechanics,
10486:Prentice-Hall Inc.
10453:Dover Publications
10064:
9937:
9855:
9757:
9684:
9598:
9558:
9493:
9322:
9207:
9109:
9062:
8999:
8912:
8855:
8823:
8788:
8745:
8645:
8563:diffusion equation
8477:
8336:
8232:
8078:
7987:
7878:
7800:
7730:
7677:
7651:In the limit when
7639:
7416:
7285:
7145:
7108:
7085:
6973:
6910:
6908:
6559:
6547:
6486:
6455:
6322:
6088:
6019:
5990:
5948:
5650:
5576:
5565:
5529:
5486:
5463:
5413:
5249:
5124:
4890:
4867:
4846:
4826:
4802:
4758:
4652:
4617:
4506:
4346:
4334:
4249:
4247:
3867:
3756:
3703:
3594:
3582:
3556:
3530:
3504:
3420:
3322:
3249:
3233:
3207:
2982:
2806:
2804:
2774:
2463:
2358:
2248:
2190:
2099:
2091:
2065:
2063:
1794:
1697:
1615:
1546:
1462:
1340:
1256:
1148:
1012:
994:
976:
958:
916:
849:
747:
667:without dispersion
663:
608:
578:
548:
490:
470:
426:
406:
379:
352:
330:
236:
216:
196:
43:
10835:Quantum mechanics
10736:978-0-07-020650-2
10701:978-0-471-43132-9
10495:978-0-13-146100-0
10478:Quantum Mechanics
10462:978-0-486-41462-1
10430:978-1-4614-7115-8
10405:978-0-465-08061-8
10380:978-0-387-10950-3
10355:978-0-07-055287-6
10346:Quantum mechanics
10318:978-3-540-70622-9
10215:978-0-7503-0720-8
10175:interaction terms
10056:
10011:
10009:
9103:
8840:
8773:
8741:
8736:
8705:
8703:
8641:
8634:
8605:
8589:
8473:
8177:
8175:
7938:
7936:
7726:
7662:
7632:
7589:
7587:
7412:
7372:
7370:
7280:
7250:
7248:
7178:propagator kernel
7111:{\displaystyle x}
6726:
6550:{\displaystyle x}
6534:
6454:
6208:
6156:
6086:
6014:
5943:
5940:
5891:
5822:
5801:
5712:
5244:
5243:
5117:
5045:
5031:
5029:
4870:{\displaystyle 0}
4849:{\displaystyle 0}
4738:
4655:{\displaystyle t}
4529:energy eigenstate
4499:
4420:
4272:
3442:
3197:
3074:
2978:
2972:
2903:
2899:
2898:
2877:
2798:
2769:
2720:
2685:
2647:
2611:
2610:
2593:
2547:
2513:
2482:
2461:
2443:
2442:
2425:
2308:
2221:
2170:
2154:
2046:
1997:
1787:
1741:
1740:
1585:
1544:
1543:
1460:
1289:
1254:
1253:
715:
675:classical physics
509:Quantum mechanics
468:
429:{\displaystyle m}
382:{\displaystyle T}
328:
239:{\displaystyle v}
219:{\displaystyle x}
199:{\displaystyle t}
169:Coulomb potential
157:Werner Heisenberg
147:, introduced the
137:Erwin Schrödinger
83:Fourier transform
79:gaussian function
53:(also known as a
16:(Redirected from
10842:
10768:
10756:
10740:
10739:
10724:
10711:
10705:
10704:
10689:
10676:
10670:
10669:
10651:
10627:
10621:
10620:
10603:(8): 1034–1035.
10588:
10582:
10581:
10554:
10548:
10537:
10531:
10530:
10521:
10515:
10505:
10499:
10498:
10472:
10466:
10465:
10441:
10435:
10434:
10416:
10410:
10409:
10391:
10385:
10384:
10366:
10360:
10359:
10341:
10332:
10329:
10323:
10322:
10292:
10277:
10276:
10261:
10255:
10254:
10252:
10251:
10245:
10234:
10225:
10219:
10218:
10199:
10182:
10171:
10094:Fourier analysis
10089:Wave propagation
10073:
10071:
10070:
10065:
10059:
10058:
10057:
10055:
10032:
10031:
10022:
10012:
10010:
9984:
9979:
9965:
9964:
9946:
9944:
9943:
9938:
9932:
9931:
9910:
9909:
9897:
9896:
9881:
9880:
9864:
9862:
9861:
9856:
9838:
9837:
9816:
9815:
9794:
9793:
9766:
9764:
9763:
9758:
9755:
9754:
9753:
9731:
9730:
9729:
9713:
9712:
9693:
9691:
9690:
9685:
9679:
9678:
9645:
9644:
9622:
9621:
9606:
9590:
9583:
9579:
9575:
9571:
9567:
9565:
9564:
9559:
9553:
9552:
9525:
9524:
9502:
9500:
9499:
9494:
9458:
9457:
9442:
9431:
9414:
9394:
9393:
9392:
9373:
9331:
9329:
9328:
9323:
9314:
9303:
9286:
9266:
9265:
9264:
9245:
9216:
9214:
9213:
9208:
9199:
9182:
9181:
9166:
9149:
9148:
9132:
9128:
9124:
9118:
9116:
9115:
9110:
9104:
9099:
9098:
9089:
9071:
9069:
9068:
9063:
9057:
9056:
9029:
9028:
9012:
9008:
9006:
9005:
9000:
8994:
8993:
8992:
8976:
8975:
8963:
8962:
8961:
8928:
8921:
8919:
8918:
8913:
8887:
8886:
8865:
8864:
8854:
8832:
8830:
8829:
8824:
8798:
8797:
8787:
8754:
8752:
8751:
8746:
8739:
8738:
8737:
8735:
8727:
8726:
8725:
8712:
8706:
8704:
8693:
8688:
8674:
8673:
8654:
8652:
8651:
8646:
8639:
8635:
8633:
8632:
8631:
8618:
8617:
8608:
8606:
8598:
8590:
8588:
8577:
8559:randomly walking
8526:
8522:
8518:
8514:
8510:
8506:
8502:
8498:
8494:
8490:
8486:
8484:
8483:
8478:
8471:
8467:
8423:
8363:
8359:
8355:
8351:
8345:
8343:
8342:
8337:
8331:
8330:
8312:
8311:
8295:
8286:
8274:
8270:
8266:
8262:
8258:
8252:general solution
8241:
8239:
8238:
8233:
8221:
8220:
8213:
8208:
8207:
8178:
8176:
8162:
8157:
8146:
8145:
8121:
8120:
8101:
8094:
8087:
8085:
8084:
8079:
8040:
8039:
8015:
8014:
7996:
7994:
7993:
7988:
7982:
7981:
7974:
7969:
7968:
7939:
7937:
7923:
7918:
7904:
7903:
7887:
7885:
7884:
7879:
7843:
7842:
7826:
7809:
7807:
7806:
7801:
7774:
7773:
7754:
7739:
7737:
7736:
7731:
7724:
7687:
7686:
7676:
7648:
7646:
7645:
7640:
7634:
7633:
7631:
7623:
7622:
7621:
7596:
7590:
7588:
7574:
7569:
7549:
7548:
7521:
7520:
7473:
7469:
7465:
7458:
7454:
7425:
7423:
7422:
7417:
7414:
7413:
7411:
7394:
7393:
7392:
7379:
7373:
7371:
7345:
7340:
7326:
7325:
7309:
7294:
7292:
7291:
7286:
7283:
7282:
7281:
7279:
7271:
7270:
7261:
7251:
7249:
7238:
7233:
7219:
7218:
7202:
7198:
7187:
7183:
7162:
7154:
7152:
7151:
7146:
7132:
7117:
7115:
7114:
7109:
7094:
7092:
7091:
7086:
7078:
7077:
7072:
7068:
7067:
7066:
7048:
7031:
7027:
7026:
7025:
7016:
7015:
6982:
6980:
6979:
6974:
6966:
6965:
6953:
6952:
6943:
6942:
6919:
6917:
6916:
6911:
6909:
6860:
6856:
6852:
6851:
6847:
6846:
6845:
6836:
6835:
6817:
6801:
6797:
6796:
6795:
6786:
6785:
6759:
6758:
6754:
6736:
6727:
6725:
6718:
6717:
6713:
6696:
6688:
6672:
6671:
6653:
6652:
6643:
6642:
6556:
6554:
6553:
6548:
6535:
6517:
6509:
6495:
6493:
6492:
6487:
6481:
6480:
6476:
6475:
6466:
6465:
6456:
6447:
6432:
6431:
6407:
6406:
6397:
6396:
6349:
6342:
6331:
6329:
6328:
6323:
6318:
6317:
6310:
6309:
6297:
6292:
6291:
6282:
6281:
6248:
6240:
6239:
6219:
6215:
6214:
6210:
6209:
6207:
6206:
6205:
6192:
6191:
6190:
6181:
6180:
6170:
6157:
6155:
6154:
6150:
6134:
6097:
6095:
6094:
6089:
6087:
6085:
6084:
6083:
6070:
6059:
6028:
6026:
6025:
6020:
6015:
6013:
6005:
6003:
5998:
5985:
5957:
5955:
5954:
5949:
5944:
5942:
5941:
5936:
5922:
5917:
5916:
5906:
5905:
5904:
5892:
5887:
5886:
5885:
5872:
5860:
5846:
5845:
5841:
5840:
5823:
5821:
5820:
5819:
5815:
5802:
5800:
5799:
5790:
5789:
5788:
5779:
5778:
5765:
5760:
5759:
5746:
5745:
5733:
5731:
5730:
5726:
5713:
5708:
5707:
5706:
5693:
5659:
5657:
5656:
5651:
5574:
5572:
5571:
5566:
5558:
5557:
5538:
5536:
5535:
5530:
5528:
5527:
5515:
5514:
5495:
5493:
5492:
5487:
5484:
5479:
5468:
5459:
5454:
5449:
5448:
5443:
5434:
5422:
5420:
5419:
5414:
5390:
5384:
5378:
5362:
5361:
5360:
5347:
5346:
5345:
5332:
5331:
5330:
5316:
5315:
5314:
5296:
5292:
5288:
5277:
5275:
5274:
5258:
5256:
5255:
5250:
5245:
5239:
5238:
5237:
5225:
5208:
5207:
5197:
5196:
5185:
5178:
5171:
5165:
5158:
5148:is the width of
5147:
5146:
5145:
5133:
5131:
5130:
5125:
5120:
5119:
5118:
5116:
5115:
5114:
5102:
5085:
5084:
5074:
5073:
5065:
5055:
5043:
5042:
5041:
5036:
5032:
5030:
5028:
5027:
5015:
4998:
4997:
4988:
4983:
4970:
4969:
4957:
4956:
4951:
4942:
4912:
4899:
4897:
4896:
4891:
4876:
4874:
4873:
4868:
4855:
4853:
4852:
4847:
4835:
4833:
4832:
4827:
4811:
4809:
4808:
4803:
4801:
4800:
4799:
4791:
4767:
4765:
4764:
4759:
4757:
4756:
4747:
4739:
4737:
4733:
4732:
4720:
4703:
4702:
4686:
4682:
4670:
4661:
4659:
4658:
4653:
4638:
4632:
4626:
4624:
4623:
4618:
4613:
4608:
4607:
4595:
4581:
4558:
4540:
4526:
4522:
4519:The integral of
4515:
4513:
4512:
4507:
4502:
4501:
4500:
4498:
4491:
4464:
4463:
4455:
4449:
4439:
4438:
4434:
4425:
4421:
4419:
4415:
4391:
4372:
4351:
4343:
4341:
4340:
4335:
4315:
4314:
4273:
4258:
4256:
4255:
4250:
4248:
4241:
4240:
4236:
4231:
4223:
4212:
4181:
4180:
4176:
4145:
4141:
4140:
4136:
4119:
4114:
4106:
4101:
4100:
4079:
4074:
4066:
4050:
4049:
4045:
4014:
4010:
4009:
4005:
3983:
3982:
3978:
3973:
3965:
3949:
3948:
3944:
3903:
3876:
3874:
3873:
3868:
3863:
3862:
3858:
3853:
3845:
3829:
3828:
3824:
3787:
3765:
3763:
3762:
3757:
3749:
3738:
3737:
3712:
3710:
3709:
3704:
3699:
3698:
3686:
3681:
3680:
3647:
3639:
3631:
3614:
3591:
3589:
3588:
3583:
3565:
3563:
3562:
3557:
3539:
3537:
3536:
3531:
3513:
3511:
3510:
3505:
3485:
3484:
3443:
3429:
3427:
3426:
3421:
3416:
3415:
3379:
3371:
3363:
3335:
3331:
3329:
3328:
3323:
3318:
3317:
3310:
3305:
3297:
3272:
3254:
3246:
3242:
3216:
3214:
3213:
3208:
3203:
3199:
3198:
3196:
3195:
3194:
3175:
3168:
3167:
3142:
3137:
3136:
3113:
3112:
3088:
3087:
3075:
3073:
3062:
3034:
3020:
3015:
3014:
3000:
2991:
2989:
2988:
2983:
2976:
2975:
2974:
2973:
2971:
2970:
2969:
2950:
2949:
2948:
2936:
2935:
2913:
2901:
2900:
2897:
2896:
2878:
2873:
2865:
2864:
2859:
2858:
2853:
2829:
2815:
2813:
2812:
2807:
2805:
2796:
2795:
2794:
2793:
2789:
2787:
2782:
2770:
2762:
2739:
2738:
2721:
2719:
2718:
2717:
2695:
2683:
2682:
2681:
2680:
2679:
2667:
2666:
2648:
2646:
2645:
2644:
2622:
2612:
2594:
2592:
2587:
2583:
2574:
2573:
2565:
2561:
2560:
2559:
2558:
2553:
2549:
2548:
2543:
2542:
2541:
2528:
2514:
2512:
2492:
2480:
2479:
2478:
2476:
2471:
2462:
2454:
2444:
2426:
2424:
2419:
2415:
2406:
2405:
2367:
2365:
2364:
2359:
2357:
2353:
2349:
2348:
2333:
2332:
2309:
2307:
2302:
2298:
2289:
2257:
2255:
2254:
2249:
2244:
2243:
2238:
2232:
2227:
2222:
2214:
2199:
2197:
2196:
2191:
2186:
2182:
2181:
2171:
2163:
2155:
2153:
2145:
2137:
2121:
2117:
2074:
2072:
2071:
2066:
2064:
2057:
2053:
2052:
2048:
2047:
2042:
2028:
2003:
1999:
1998:
1993:
1979:
1955:
1954:
1953:
1952:
1914:
1910:
1909:
1890:
1889:
1874:
1873:
1803:
1801:
1800:
1795:
1790:
1789:
1788:
1783:
1782:
1781:
1772:
1771:
1752:
1742:
1736:
1732:
1706:
1704:
1703:
1698:
1693:
1692:
1688:
1687:
1672:
1671:
1624:
1622:
1621:
1616:
1604:
1603:
1583:
1563:
1557:
1545:
1536:
1532:
1508:
1501:
1486:
1476:. The amplitude
1471:
1469:
1468:
1463:
1461:
1453:
1451:
1433:
1406:
1392:
1365:
1349:
1347:
1346:
1341:
1329:
1328:
1287:
1273:
1267:
1255:
1246:
1242:
1205:
1203:
1196:
1186:
1176:
1174:
1167:
1157:
1155:
1154:
1149:
1144:
1143:
1104:
1103:
1035:
1029:
1021:
1019:
1018:
1013:
1007:
1002:
989:
984:
971:
966:
954:
953:
948:
942:
937:
925:
923:
922:
917:
912:
911:
902:
901:
896:
890:
885:
877:
876:
858:
856:
855:
850:
845:
844:
831:
830:
795:
773:
764:
756:
754:
753:
748:
740:
739:
729:
728:
716:
714:
713:
712:
699:
695:
694:
684:
617:
615:
614:
609:
607:
606:
587:
585:
584:
579:
577:
576:
557:
555:
554:
549:
499:
497:
496:
491:
479:
477:
476:
471:
469:
464:
456:
435:
433:
432:
427:
415:
413:
412:
407:
393:packet of width
388:
386:
385:
380:
361:
359:
358:
353:
339:
337:
336:
331:
329:
327:
326:
305:
288:
287:
278:
264:
263:
245:
243:
242:
237:
225:
223:
222:
217:
205:
203:
202:
197:
67:sinusoidal waves
21:
10850:
10849:
10845:
10844:
10843:
10841:
10840:
10839:
10820:
10819:
10809:Web-Schrödinger
10749:
10744:
10743:
10737:
10712:
10708:
10702:
10677:
10673:
10628:
10624:
10609:10.1119/1.18322
10589:
10585:
10579:10.1119/1.11855
10555:
10551:
10546:
10538:
10534:
10522:
10518:
10514:(776), 258-293.
10506:
10502:
10496:
10473:
10469:
10463:
10445:Pauli, Wolfgang
10442:
10438:
10431:
10417:
10413:
10406:
10392:
10388:
10381:
10367:
10363:
10356:
10342:
10335:
10330:
10326:
10319:
10293:
10280:
10262:
10258:
10249:
10247:
10243:
10232:
10226:
10222:
10216:
10200:
10196:
10191:
10186:
10185:
10172:
10168:
10163:
10158:
10139:Pulse (physics)
10114:Coherent states
10079:
10033:
10027:
10023:
10021:
10017:
10013:
9983:
9978:
9960:
9956:
9954:
9951:
9950:
9918:
9914:
9905:
9901:
9889:
9885:
9876:
9872:
9870:
9867:
9866:
9833:
9829:
9811:
9807:
9789:
9785:
9783:
9780:
9779:
9746:
9739:
9735:
9722:
9721:
9717:
9708:
9704:
9702:
9699:
9698:
9653:
9649:
9631:
9627:
9608:
9607:
9602:
9596:
9593:
9592:
9588:
9581:
9577:
9573:
9569:
9542:
9538:
9520:
9516:
9514:
9511:
9510:
9453:
9449:
9435:
9424:
9407:
9385:
9384:
9380:
9366:
9337:
9334:
9333:
9307:
9296:
9279:
9257:
9256:
9252:
9238:
9224:
9221:
9220:
9192:
9177:
9173:
9159:
9144:
9140:
9138:
9135:
9134:
9130:
9126:
9122:
9094:
9090:
9088:
9077:
9074:
9073:
9046:
9042:
9024:
9020:
9018:
9015:
9014:
9010:
8985:
8984:
8980:
8971:
8967:
8954:
8947:
8943:
8941:
8938:
8937:
8926:
8922:for any smooth
8882:
8878:
8860:
8856:
8844:
8838:
8835:
8834:
8793:
8789:
8777:
8771:
8768:
8767:
8760:
8728:
8721:
8717:
8713:
8711:
8707:
8692:
8687:
8669:
8665:
8663:
8660:
8659:
8627:
8623:
8619:
8613:
8609:
8607:
8597:
8581:
8576:
8574:
8571:
8570:
8551:
8541:
8527:', summed over
8524:
8520:
8516:
8512:
8508:
8504:
8500:
8496:
8492:
8488:
8460:
8416:
8369:
8366:
8365:
8361:
8357:
8353:
8349:
8326:
8322:
8307:
8303:
8301:
8298:
8297:
8294:
8288:
8284:
8272:
8268:
8264:
8260:
8256:
8243:
8209:
8203:
8199:
8183:
8179:
8161:
8156:
8141:
8137:
8116:
8112:
8110:
8107:
8106:
8099:
8097:
8092:
8035:
8031:
8010:
8006:
8004:
8001:
8000:
7970:
7964:
7960:
7944:
7940:
7922:
7917:
7899:
7895:
7893:
7890:
7889:
7838:
7834:
7832:
7829:
7828:
7824:
7769:
7765:
7760:
7757:
7756:
7752:
7682:
7678:
7666:
7660:
7657:
7656:
7624:
7617:
7613:
7597:
7595:
7591:
7573:
7568:
7544:
7540:
7516:
7512:
7510:
7507:
7506:
7471:
7467:
7460:
7456:
7452:
7435:all later times
7395:
7388:
7384:
7380:
7378:
7374:
7344:
7339:
7321:
7317:
7315:
7312:
7311:
7300:
7272:
7266:
7262:
7260:
7256:
7252:
7237:
7232:
7214:
7210:
7208:
7205:
7204:
7200:
7196:
7185:
7181:
7173:
7171:Free propagator
7160:
7128:
7123:
7120:
7119:
7103:
7100:
7099:
7073:
7062:
7058:
7044:
7040:
7036:
7035:
7021:
7017:
7011:
7007:
7000:
6996:
6991:
6988:
6987:
6961:
6957:
6948:
6944:
6938:
6934:
6929:
6926:
6925:
6907:
6906:
6858:
6857:
6841:
6837:
6831:
6827:
6813:
6809:
6805:
6791:
6787:
6781:
6777:
6764:
6760:
6750:
6746:
6742:
6741:
6737:
6729:
6709:
6705:
6701:
6700:
6695:
6686:
6685:
6667:
6663:
6648:
6644:
6638:
6634:
6615:
6587:
6585:
6582:
6581:
6542:
6539:
6538:
6524:
6511:
6503:
6471:
6467:
6461:
6457:
6445:
6427:
6423:
6419:
6415:
6402:
6398:
6392:
6388:
6359:
6356:
6355:
6344:
6337:
6305:
6301:
6293:
6287:
6283:
6277:
6273:
6244:
6235:
6231:
6224:
6220:
6201:
6197:
6193:
6186:
6182:
6176:
6172:
6171:
6169:
6162:
6158:
6146:
6142:
6138:
6133:
6132:
6128:
6114:
6111:
6110:
6103:
6079:
6075:
6071:
6060:
6058:
6035:
6032:
6031:
6006:
5999:
5994:
5986:
5984:
5967:
5964:
5963:
5923:
5921:
5912:
5908:
5907:
5900:
5896:
5881:
5877:
5873:
5871:
5861:
5859:
5836:
5832:
5828:
5824:
5811:
5807:
5803:
5795:
5791:
5784:
5780:
5774:
5770:
5766:
5764:
5755:
5751:
5747:
5738:
5734:
5732:
5722:
5718:
5714:
5702:
5698:
5694:
5692:
5666:
5663:
5662:
5585:
5582:
5581:
5550:
5546:
5544:
5541:
5540:
5523:
5519:
5507:
5503:
5501:
5498:
5497:
5480:
5469:
5464:
5455:
5450:
5444:
5439:
5438:
5430:
5428:
5425:
5424:
5408:
5405:
5404:
5397:
5386:
5380:
5364:
5355:
5353:
5349:
5340:
5338:
5334:
5325:
5323:
5318:
5309:
5307:
5301:
5294:
5290:
5286:
5270:
5268:
5260:
5233:
5229:
5221:
5203:
5199:
5198:
5195:
5193:
5190:
5189:
5183:
5173:
5167:
5160:
5149:
5141:
5139:
5138:
5135:
5110:
5106:
5098:
5080:
5076:
5075:
5069:
5061:
5056:
5054:
5050:
5046:
5037:
5023:
5019:
5011:
4993:
4989:
4987:
4982:
4978:
4977:
4965:
4961:
4952:
4947:
4946:
4938:
4921:
4918:
4917:
4904:
4882:
4879:
4878:
4862:
4859:
4858:
4841:
4838:
4837:
4818:
4815:
4814:
4795:
4787:
4783:
4779:
4777:
4774:
4773:
4752:
4748:
4743:
4728:
4724:
4716:
4698:
4694:
4687:
4678:
4671:
4669:
4667:
4664:
4663:
4647:
4644:
4643:
4634:
4628:
4609:
4603:
4599:
4591:
4577:
4554:
4546:
4543:
4542:
4531:
4524:
4520:
4517:
4487:
4465:
4459:
4451:
4450:
4448:
4444:
4440:
4430:
4426:
4411:
4395:
4390:
4386:
4385:
4368:
4360:
4357:
4356:
4349:
4310:
4306:
4280:
4277:
4276:
4262:
4260:
4246:
4245:
4232:
4227:
4219:
4208:
4186:
4182:
4172:
4168:
4164:
4143:
4142:
4132:
4115:
4110:
4102:
4096:
4092:
4075:
4070:
4062:
4055:
4051:
4041:
4037:
4033:
4012:
4011:
4001:
3988:
3984:
3974:
3969:
3961:
3954:
3950:
3940:
3936:
3932:
3913:
3899:
3889:
3887:
3884:
3883:
3854:
3849:
3841:
3834:
3830:
3820:
3816:
3812:
3783:
3775:
3772:
3771:
3745:
3733:
3729:
3718:
3715:
3714:
3694:
3690:
3682:
3676:
3672:
3643:
3635:
3627:
3610:
3605:
3602:
3601:
3571:
3568:
3567:
3545:
3542:
3541:
3519:
3516:
3515:
3480:
3476:
3450:
3447:
3446:
3432:
3411:
3407:
3375:
3367:
3359:
3345:
3342:
3341:
3333:
3306:
3301:
3293:
3289:
3285:
3268:
3260:
3257:
3256:
3252:
3244:
3240:
3222:
3190:
3186:
3176:
3163:
3159:
3143:
3141:
3132:
3128:
3127:
3123:
3108:
3104:
3083:
3079:
3066:
3061:
3044:
3041:
3040:
3025:
3009:
3007:
3006:
2998:
2993:
2965:
2961:
2951:
2944:
2940:
2931:
2927:
2914:
2912:
2908:
2904:
2892:
2888:
2869:
2863:
2854:
2849:
2848:
2825:
2823:
2820:
2819:
2803:
2802:
2783:
2778:
2761:
2734:
2730:
2726:
2722:
2713:
2709:
2699:
2694:
2690:
2686:
2675:
2671:
2662:
2658:
2640:
2636:
2626:
2621:
2617:
2613:
2588:
2579:
2575:
2572:
2563:
2562:
2554:
2537:
2533:
2529:
2527:
2520:
2516:
2515:
2496:
2491:
2487:
2483:
2472:
2467:
2453:
2449:
2445:
2420:
2411:
2407:
2404:
2397:
2375:
2373:
2370:
2369:
2344:
2340:
2328:
2324:
2320:
2316:
2303:
2294:
2290:
2288:
2265:
2262:
2261:
2239:
2234:
2233:
2228:
2223:
2213:
2205:
2202:
2201:
2177:
2173:
2172:
2162:
2146:
2138:
2136:
2131:
2128:
2127:
2119:
2112:
2083:
2062:
2061:
2029:
2027:
2020:
2016:
1980:
1978:
1971:
1967:
1960:
1956:
1948:
1944:
1925:
1921:
1912:
1911:
1885:
1881:
1869:
1865:
1846:
1842:
1835:
1813:
1811:
1808:
1807:
1777:
1773:
1767:
1763:
1753:
1751:
1747:
1743:
1731:
1714:
1711:
1710:
1683:
1679:
1667:
1663:
1659:
1655:
1632:
1629:
1628:
1590:
1586:
1558:
1550:
1531:
1514:
1511:
1510:
1503:
1488:
1477:
1452:
1447:
1442:
1439:
1438:
1408:
1394:
1367:
1353:
1294:
1290:
1268:
1260:
1241:
1218:
1215:
1214:
1199:
1198:
1188:
1178:
1170:
1169:
1159:
1115:
1111:
1078:
1074:
1048:
1045:
1044:
1031:
1025:
1003:
998:
985:
980:
967:
962:
949:
944:
943:
938:
933:
931:
928:
927:
907:
903:
897:
892:
891:
886:
881:
872:
868:
866:
863:
862:
820:
816:
812:
808:
791:
783:
780:
779:
769:
760:
735:
731:
724:
720:
708:
704:
700:
690:
686:
685:
683:
681:
678:
677:
655:
650:
640:
638:Basic behaviors
624:
602:
598:
593:
590:
589:
572:
568:
563:
560:
559:
540:
537:
536:
521:
506:
485:
482:
481:
460:
455:
441:
438:
437:
421:
418:
417:
398:
395:
394:
374:
371:
370:
347:
344:
343:
322:
318:
301:
283:
279:
277:
259:
255:
253:
250:
249:
231:
228:
227:
211:
208:
207:
191:
188:
187:
153:coherent states
107:
35:
28:
23:
22:
15:
12:
11:
5:
10848:
10838:
10837:
10832:
10830:Wave mechanics
10818:
10817:
10812:
10806:
10801:
10796:
10791:
10786:
10781:
10776:
10762:
10761:at Wikiversity
10748:
10747:External links
10745:
10742:
10741:
10735:
10715:Feynman, R. P.
10706:
10700:
10680:Jackson, J. D.
10671:
10622:
10583:
10565:(3): 264–267,
10549:
10544:
10532:
10516:
10500:
10494:
10482:Addison Wesley
10467:
10461:
10436:
10429:
10411:
10404:
10386:
10379:
10361:
10354:
10333:
10324:
10317:
10278:
10256:
10220:
10214:
10193:
10192:
10190:
10187:
10184:
10183:
10165:
10164:
10162:
10159:
10157:
10156:
10151:
10146:
10141:
10136:
10131:
10126:
10121:
10116:
10111:
10106:
10104:Phase velocity
10101:
10099:Group velocity
10096:
10091:
10086:
10080:
10078:
10075:
10063:
10054:
10051:
10048:
10045:
10042:
10039:
10036:
10030:
10026:
10020:
10016:
10008:
10005:
10002:
9999:
9996:
9993:
9990:
9987:
9982:
9977:
9974:
9971:
9968:
9963:
9959:
9936:
9930:
9927:
9924:
9921:
9917:
9913:
9908:
9904:
9900:
9895:
9892:
9888:
9884:
9879:
9875:
9853:
9850:
9847:
9844:
9841:
9836:
9832:
9828:
9825:
9822:
9819:
9814:
9810:
9806:
9803:
9800:
9797:
9792:
9788:
9752:
9749:
9745:
9742:
9738:
9734:
9728:
9725:
9720:
9716:
9711:
9707:
9683:
9677:
9674:
9671:
9668:
9665:
9662:
9659:
9656:
9652:
9648:
9643:
9640:
9637:
9634:
9630:
9626:
9620:
9617:
9614:
9611:
9605:
9601:
9557:
9551:
9548:
9545:
9541:
9537:
9534:
9531:
9528:
9523:
9519:
9492:
9488:
9485:
9482:
9479:
9476:
9473:
9470:
9467:
9464:
9461:
9456:
9452:
9448:
9445:
9441:
9438:
9434:
9430:
9427:
9423:
9420:
9417:
9413:
9410:
9406:
9403:
9400:
9397:
9391:
9388:
9383:
9379:
9376:
9372:
9369:
9365:
9362:
9359:
9356:
9353:
9350:
9347:
9344:
9341:
9321:
9317:
9313:
9310:
9306:
9302:
9299:
9295:
9292:
9289:
9285:
9282:
9278:
9275:
9272:
9269:
9263:
9260:
9255:
9251:
9248:
9244:
9241:
9237:
9234:
9231:
9228:
9206:
9202:
9198:
9195:
9191:
9188:
9185:
9180:
9176:
9172:
9169:
9165:
9162:
9158:
9155:
9152:
9147:
9143:
9108:
9102:
9097:
9093:
9087:
9084:
9081:
9061:
9055:
9052:
9049:
9045:
9041:
9038:
9035:
9032:
9027:
9023:
8998:
8991:
8988:
8983:
8979:
8974:
8970:
8966:
8960:
8957:
8953:
8950:
8946:
8911:
8908:
8905:
8902:
8899:
8896:
8893:
8890:
8885:
8881:
8877:
8874:
8871:
8868:
8863:
8859:
8853:
8850:
8847:
8843:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8796:
8792:
8786:
8783:
8780:
8776:
8758:
8744:
8734:
8731:
8724:
8720:
8716:
8710:
8702:
8699:
8696:
8691:
8686:
8683:
8680:
8677:
8672:
8668:
8644:
8638:
8630:
8626:
8622:
8616:
8612:
8604:
8601:
8596:
8593:
8587:
8584:
8580:
8565:(also see the
8540:
8537:
8476:
8470:
8466:
8463:
8459:
8456:
8453:
8450:
8447:
8444:
8441:
8438:
8435:
8432:
8429:
8426:
8422:
8419:
8415:
8412:
8409:
8406:
8403:
8400:
8397:
8394:
8391:
8388:
8385:
8382:
8379:
8376:
8373:
8335:
8329:
8325:
8321:
8318:
8315:
8310:
8306:
8292:
8283:of the kernel
8231:
8227:
8224:
8219:
8216:
8212:
8206:
8202:
8198:
8195:
8192:
8189:
8186:
8182:
8174:
8171:
8168:
8165:
8160:
8155:
8152:
8149:
8144:
8140:
8136:
8133:
8130:
8127:
8124:
8119:
8115:
8104:
8095:
8090:every function
8077:
8073:
8070:
8067:
8064:
8061:
8058:
8055:
8052:
8049:
8046:
8043:
8038:
8034:
8030:
8027:
8024:
8021:
8018:
8013:
8009:
7986:
7980:
7977:
7973:
7967:
7963:
7959:
7956:
7953:
7950:
7947:
7943:
7935:
7932:
7929:
7926:
7921:
7916:
7913:
7910:
7907:
7902:
7898:
7877:
7873:
7870:
7867:
7864:
7861:
7858:
7855:
7852:
7849:
7846:
7841:
7837:
7799:
7795:
7792:
7789:
7786:
7783:
7780:
7777:
7772:
7768:
7764:
7729:
7723:
7720:
7717:
7714:
7711:
7708:
7705:
7702:
7699:
7696:
7693:
7690:
7685:
7681:
7675:
7672:
7669:
7665:
7638:
7630:
7627:
7620:
7616:
7612:
7609:
7606:
7603:
7600:
7594:
7586:
7583:
7580:
7577:
7572:
7567:
7564:
7561:
7558:
7555:
7552:
7547:
7543:
7539:
7536:
7533:
7530:
7527:
7524:
7519:
7515:
7446:delta function
7410:
7407:
7404:
7401:
7398:
7391:
7387:
7383:
7377:
7369:
7366:
7363:
7360:
7357:
7354:
7351:
7348:
7343:
7338:
7335:
7332:
7329:
7324:
7320:
7297:delta function
7278:
7275:
7269:
7265:
7259:
7255:
7247:
7244:
7241:
7236:
7231:
7228:
7225:
7222:
7217:
7213:
7172:
7169:
7157:
7156:
7144:
7141:
7138:
7135:
7131:
7127:
7107:
7096:
7084:
7081:
7076:
7071:
7065:
7061:
7057:
7054:
7051:
7047:
7043:
7039:
7034:
7030:
7024:
7020:
7014:
7010:
7006:
7003:
6999:
6995:
6984:
6972:
6969:
6964:
6960:
6956:
6951:
6947:
6941:
6937:
6933:
6905:
6902:
6899:
6896:
6893:
6890:
6887:
6884:
6881:
6878:
6875:
6872:
6869:
6866:
6863:
6861:
6859:
6855:
6850:
6844:
6840:
6834:
6830:
6826:
6823:
6820:
6816:
6812:
6808:
6804:
6800:
6794:
6790:
6784:
6780:
6776:
6773:
6770:
6767:
6763:
6757:
6753:
6749:
6745:
6740:
6735:
6732:
6724:
6721:
6716:
6712:
6708:
6704:
6699:
6694:
6691:
6689:
6687:
6684:
6681:
6678:
6675:
6670:
6666:
6662:
6659:
6656:
6651:
6647:
6641:
6637:
6633:
6630:
6627:
6624:
6621:
6618:
6616:
6614:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6590:
6589:
6546:
6485:
6479:
6474:
6470:
6464:
6460:
6453:
6450:
6444:
6441:
6438:
6435:
6430:
6426:
6422:
6418:
6413:
6410:
6405:
6401:
6395:
6391:
6387:
6384:
6381:
6378:
6375:
6372:
6369:
6366:
6363:
6321:
6316:
6313:
6308:
6304:
6300:
6296:
6290:
6286:
6280:
6276:
6272:
6269:
6266:
6263:
6260:
6257:
6254:
6251:
6247:
6243:
6238:
6234:
6230:
6227:
6223:
6218:
6213:
6204:
6200:
6196:
6189:
6185:
6179:
6175:
6168:
6165:
6161:
6153:
6149:
6145:
6141:
6137:
6131:
6127:
6124:
6121:
6118:
6107:Airy functions
6102:
6099:
6082:
6078:
6074:
6069:
6066:
6063:
6057:
6054:
6051:
6048:
6045:
6042:
6039:
6018:
6012:
6009:
6002:
5997:
5993:
5989:
5983:
5980:
5977:
5974:
5971:
5947:
5939:
5935:
5932:
5929:
5926:
5920:
5915:
5911:
5903:
5899:
5895:
5890:
5884:
5880:
5876:
5870:
5867:
5864:
5858:
5855:
5852:
5849:
5844:
5839:
5835:
5831:
5827:
5818:
5814:
5810:
5806:
5798:
5794:
5787:
5783:
5777:
5773:
5769:
5763:
5758:
5754:
5750:
5744:
5741:
5737:
5729:
5725:
5721:
5717:
5711:
5705:
5701:
5697:
5691:
5688:
5685:
5682:
5679:
5676:
5673:
5670:
5649:
5646:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5564:
5561:
5556:
5553:
5549:
5526:
5522:
5518:
5513:
5510:
5506:
5483:
5478:
5475:
5472:
5467:
5462:
5458:
5453:
5447:
5442:
5437:
5433:
5412:
5396:
5393:
5248:
5242:
5236:
5232:
5228:
5224:
5220:
5217:
5214:
5211:
5206:
5202:
5123:
5113:
5109:
5105:
5101:
5097:
5094:
5091:
5088:
5083:
5079:
5072:
5068:
5064:
5059:
5053:
5049:
5040:
5035:
5026:
5022:
5018:
5014:
5010:
5007:
5004:
5001:
4996:
4992:
4986:
4981:
4976:
4973:
4968:
4964:
4960:
4955:
4950:
4945:
4941:
4937:
4934:
4931:
4928:
4925:
4915:
4889:
4886:
4866:
4845:
4825:
4822:
4798:
4794:
4790:
4786:
4782:
4755:
4751:
4746:
4742:
4736:
4731:
4727:
4723:
4719:
4715:
4712:
4709:
4706:
4701:
4697:
4693:
4690:
4685:
4681:
4677:
4674:
4651:
4616:
4612:
4606:
4602:
4598:
4594:
4590:
4587:
4584:
4580:
4576:
4573:
4570:
4567:
4564:
4561:
4557:
4553:
4550:
4505:
4497:
4494:
4490:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4462:
4458:
4454:
4447:
4443:
4437:
4433:
4429:
4424:
4418:
4414:
4410:
4407:
4404:
4401:
4398:
4394:
4389:
4384:
4381:
4378:
4375:
4371:
4367:
4364:
4354:
4333:
4330:
4327:
4324:
4321:
4318:
4313:
4309:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4244:
4239:
4235:
4230:
4226:
4222:
4218:
4215:
4211:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4185:
4179:
4175:
4171:
4167:
4163:
4160:
4157:
4154:
4151:
4148:
4146:
4144:
4139:
4135:
4131:
4128:
4125:
4122:
4118:
4113:
4109:
4105:
4099:
4095:
4091:
4088:
4085:
4082:
4078:
4073:
4069:
4065:
4061:
4058:
4054:
4048:
4044:
4040:
4036:
4032:
4029:
4026:
4023:
4020:
4017:
4015:
4013:
4008:
4004:
4000:
3997:
3994:
3991:
3987:
3981:
3977:
3972:
3968:
3964:
3960:
3957:
3953:
3947:
3943:
3939:
3935:
3931:
3928:
3925:
3922:
3919:
3916:
3914:
3912:
3909:
3906:
3902:
3898:
3895:
3892:
3891:
3881:
3866:
3861:
3857:
3852:
3848:
3844:
3840:
3837:
3833:
3827:
3823:
3819:
3815:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3786:
3782:
3779:
3755:
3752:
3748:
3744:
3741:
3736:
3732:
3728:
3725:
3722:
3702:
3697:
3693:
3689:
3685:
3679:
3675:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3646:
3642:
3638:
3634:
3630:
3626:
3623:
3620:
3617:
3613:
3609:
3581:
3578:
3575:
3555:
3552:
3549:
3529:
3526:
3523:
3503:
3500:
3497:
3494:
3491:
3488:
3483:
3479:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3419:
3414:
3410:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3378:
3374:
3370:
3366:
3362:
3358:
3355:
3352:
3349:
3321:
3316:
3313:
3309:
3304:
3300:
3296:
3292:
3288:
3284:
3281:
3278:
3275:
3271:
3267:
3264:
3221:
3218:
3206:
3202:
3193:
3189:
3185:
3182:
3179:
3174:
3171:
3166:
3162:
3158:
3155:
3152:
3149:
3146:
3140:
3135:
3131:
3126:
3122:
3119:
3116:
3111:
3107:
3103:
3100:
3097:
3094:
3091:
3086:
3082:
3078:
3072:
3069:
3065:
3060:
3057:
3054:
3051:
3048:
2996:
2981:
2968:
2964:
2960:
2957:
2954:
2947:
2943:
2939:
2934:
2930:
2926:
2923:
2920:
2917:
2911:
2907:
2895:
2891:
2887:
2884:
2881:
2876:
2872:
2868:
2862:
2857:
2852:
2847:
2844:
2841:
2838:
2835:
2832:
2828:
2801:
2792:
2786:
2781:
2777:
2773:
2768:
2765:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2737:
2733:
2729:
2725:
2716:
2712:
2708:
2705:
2702:
2698:
2693:
2689:
2678:
2674:
2670:
2665:
2661:
2657:
2654:
2651:
2643:
2639:
2635:
2632:
2629:
2625:
2620:
2616:
2609:
2606:
2603:
2600:
2597:
2591:
2586:
2582:
2578:
2571:
2568:
2566:
2564:
2557:
2552:
2546:
2540:
2536:
2532:
2526:
2523:
2519:
2511:
2508:
2505:
2502:
2499:
2495:
2490:
2486:
2475:
2470:
2466:
2460:
2457:
2452:
2448:
2441:
2438:
2435:
2432:
2429:
2423:
2418:
2414:
2410:
2403:
2400:
2398:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2377:
2356:
2352:
2347:
2343:
2339:
2336:
2331:
2327:
2323:
2319:
2315:
2312:
2306:
2301:
2297:
2293:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2247:
2242:
2237:
2231:
2226:
2220:
2217:
2212:
2209:
2189:
2185:
2180:
2176:
2169:
2166:
2161:
2158:
2152:
2149:
2144:
2141:
2135:
2082:
2079:
2060:
2056:
2051:
2045:
2041:
2038:
2035:
2032:
2026:
2023:
2019:
2015:
2012:
2009:
2006:
2002:
1996:
1992:
1989:
1986:
1983:
1977:
1974:
1970:
1966:
1963:
1959:
1951:
1947:
1943:
1940:
1937:
1934:
1931:
1928:
1924:
1920:
1917:
1915:
1913:
1908:
1905:
1902:
1899:
1896:
1893:
1888:
1884:
1880:
1877:
1872:
1868:
1864:
1861:
1858:
1855:
1852:
1849:
1845:
1841:
1838:
1836:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1815:
1793:
1786:
1780:
1776:
1770:
1766:
1762:
1759:
1756:
1750:
1746:
1739:
1735:
1730:
1727:
1724:
1721:
1718:
1696:
1691:
1686:
1682:
1678:
1675:
1670:
1666:
1662:
1658:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1614:
1611:
1608:
1602:
1599:
1596:
1593:
1589:
1582:
1579:
1576:
1573:
1570:
1567:
1562:
1556:
1553:
1549:
1542:
1539:
1535:
1530:
1527:
1524:
1521:
1518:
1459:
1456:
1450:
1446:
1339:
1336:
1333:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1293:
1286:
1283:
1280:
1277:
1272:
1266:
1263:
1259:
1252:
1249:
1245:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1147:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1114:
1110:
1107:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1077:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1011:
1006:
1001:
997:
993:
988:
983:
979:
975:
970:
965:
961:
957:
952:
947:
941:
936:
915:
910:
906:
900:
895:
889:
884:
880:
875:
871:
848:
843:
840:
837:
834:
829:
826:
823:
819:
815:
811:
807:
804:
801:
798:
794:
790:
787:
746:
743:
738:
734:
727:
723:
719:
711:
707:
703:
698:
693:
689:
654:
653:Non-dispersive
651:
639:
636:
623:
620:
605:
601:
597:
575:
571:
567:
547:
544:
520:
517:
505:
502:
489:
467:
463:
459:
454:
451:
448:
445:
425:
405:
402:
378:
367:Paul Ehrenfest
365:Later in 1927
351:
325:
321:
317:
314:
311:
308:
304:
300:
297:
294:
291:
286:
282:
276:
273:
270:
267:
262:
258:
235:
215:
195:
123:group velocity
119:phase velocity
106:
103:
26:
9:
6:
4:
3:
2:
10847:
10836:
10833:
10831:
10828:
10827:
10825:
10816:
10813:
10810:
10807:
10805:
10802:
10800:
10797:
10795:
10792:
10790:
10787:
10785:
10782:
10780:
10777:
10775:at Wiktionary
10774:
10773:
10767:
10763:
10760:
10755:
10751:
10750:
10738:
10732:
10728:
10723:
10722:
10716:
10710:
10703:
10697:
10693:
10688:
10687:
10681:
10675:
10667:
10663:
10659:
10655:
10650:
10645:
10641:
10637:
10633:
10626:
10618:
10614:
10610:
10606:
10602:
10598:
10594:
10587:
10580:
10576:
10572:
10568:
10564:
10560:
10553:
10542:
10536:
10529:
10528:
10520:
10513:
10510:
10504:
10497:
10491:
10487:
10483:
10479:
10471:
10464:
10458:
10454:
10450:
10446:
10440:
10432:
10426:
10422:
10415:
10407:
10401:
10397:
10390:
10382:
10376:
10372:
10365:
10357:
10351:
10347:
10340:
10338:
10328:
10320:
10314:
10310:
10306:
10302:
10298:
10297:"Wave Packet"
10291:
10289:
10287:
10285:
10283:
10275:
10271:
10267:
10260:
10242:
10238:
10231:
10224:
10217:
10211:
10207:
10206:
10198:
10194:
10180:
10176:
10170:
10166:
10155:
10152:
10150:
10147:
10145:
10142:
10140:
10137:
10135:
10132:
10130:
10127:
10125:
10122:
10120:
10117:
10115:
10112:
10110:
10109:Free particle
10107:
10105:
10102:
10100:
10097:
10095:
10092:
10090:
10087:
10085:
10082:
10081:
10074:
10061:
10049:
10046:
10043:
10040:
10034:
10028:
10024:
10018:
10014:
10003:
10000:
9997:
9994:
9988:
9985:
9980:
9975:
9969:
9961:
9957:
9947:
9934:
9928:
9925:
9922:
9919:
9915:
9911:
9906:
9902:
9898:
9893:
9890:
9886:
9882:
9877:
9873:
9848:
9842:
9839:
9834:
9830:
9826:
9820:
9812:
9808:
9804:
9798:
9790:
9786:
9777:
9772:
9770:
9750:
9747:
9743:
9740:
9736:
9732:
9726:
9723:
9718:
9714:
9709:
9705:
9695:
9681:
9675:
9669:
9666:
9663:
9660:
9654:
9650:
9646:
9641:
9638:
9635:
9632:
9628:
9624:
9603:
9599:
9585:
9555:
9549:
9546:
9543:
9539:
9535:
9529:
9521:
9517:
9508:
9503:
9490:
9483:
9477:
9471:
9468:
9459:
9454:
9450:
9446:
9439:
9436:
9432:
9428:
9425:
9418:
9411:
9408:
9404:
9401:
9395:
9389:
9386:
9381:
9377:
9370:
9367:
9363:
9360:
9354:
9351:
9339:
9319:
9311:
9308:
9304:
9300:
9297:
9290:
9283:
9280:
9276:
9273:
9267:
9261:
9258:
9253:
9249:
9242:
9239:
9235:
9232:
9226:
9217:
9204:
9196:
9193:
9189:
9186:
9178:
9174:
9170:
9163:
9160:
9156:
9153:
9145:
9141:
9119:
9106:
9100:
9095:
9085:
9082:
9079:
9059:
9053:
9050:
9047:
9043:
9039:
9033:
9025:
9021:
8996:
8989:
8986:
8981:
8977:
8972:
8968:
8964:
8958:
8955:
8951:
8948:
8944:
8935:
8930:
8925:
8924:test function
8906:
8900:
8897:
8891:
8883:
8879:
8872:
8866:
8861:
8857:
8851:
8845:
8817:
8811:
8808:
8802:
8794:
8790:
8784:
8778:
8765:
8761:
8742:
8732:
8729:
8722:
8718:
8714:
8708:
8700:
8697:
8694:
8689:
8684:
8678:
8670:
8666:
8656:
8642:
8636:
8628:
8624:
8614:
8602:
8599:
8594:
8591:
8585:
8568:
8567:heat equation
8564:
8560:
8556:
8550:
8546:
8536:
8534:
8533:path integral
8530:
8474:
8464:
8461:
8457:
8454:
8451:
8448:
8445:
8442:
8436:
8433:
8430:
8427:
8420:
8417:
8413:
8410:
8407:
8404:
8398:
8392:
8389:
8386:
8383:
8380:
8374:
8371:
8356:after a time
8346:
8333:
8327:
8323:
8319:
8316:
8313:
8308:
8304:
8291:
8282:
8278:
8254:
8253:
8248:
8242:
8229:
8225:
8222:
8217:
8214:
8210:
8204:
8196:
8193:
8190:
8184:
8180:
8172:
8169:
8166:
8163:
8158:
8150:
8142:
8138:
8134:
8131:
8125:
8117:
8113:
8103:
8091:
8075:
8071:
8068:
8062:
8059:
8056:
8050:
8044:
8036:
8032:
8028:
8025:
8019:
8011:
8007:
7997:
7984:
7978:
7975:
7971:
7965:
7957:
7954:
7951:
7945:
7941:
7933:
7930:
7927:
7924:
7919:
7914:
7908:
7900:
7896:
7875:
7868:
7865:
7862:
7856:
7853:
7847:
7839:
7835:
7820:
7818:
7813:
7797:
7793:
7790:
7787:
7784:
7778:
7770:
7766:
7762:
7749:
7747:
7746:test function
7743:
7742:distributions
7727:
7718:
7715:
7712:
7706:
7703:
7697:
7694:
7691:
7683:
7679:
7673:
7667:
7654:
7649:
7636:
7628:
7625:
7618:
7610:
7607:
7604:
7598:
7592:
7584:
7581:
7578:
7575:
7570:
7565:
7559:
7556:
7553:
7545:
7541:
7537:
7531:
7528:
7525:
7517:
7513:
7504:
7500:
7496:
7492:
7488:
7483:
7481:
7477:
7463:
7449:
7447:
7442:
7440:
7436:
7432:
7427:
7408:
7405:
7402:
7399:
7396:
7389:
7385:
7381:
7375:
7364:
7361:
7358:
7355:
7349:
7346:
7341:
7336:
7330:
7322:
7318:
7307:
7303:
7298:
7276:
7273:
7267:
7263:
7257:
7253:
7245:
7242:
7239:
7234:
7229:
7223:
7215:
7211:
7194:
7189:
7180:
7179:
7168:
7166:
7142:
7139:
7136:
7133:
7129:
7125:
7105:
7097:
7082:
7079:
7074:
7069:
7063:
7059:
7055:
7052:
7049:
7045:
7041:
7037:
7032:
7028:
7022:
7018:
7012:
7008:
7004:
7001:
6997:
6993:
6985:
6967:
6962:
6958:
6954:
6949:
6945:
6939:
6935:
6923:
6922:
6921:
6903:
6897:
6894:
6891:
6888:
6885:
6882:
6879:
6876:
6873:
6867:
6864:
6862:
6853:
6848:
6842:
6832:
6828:
6824:
6821:
6818:
6814:
6810:
6806:
6802:
6798:
6792:
6788:
6782:
6778:
6774:
6771:
6765:
6761:
6755:
6751:
6747:
6743:
6738:
6722:
6719:
6714:
6710:
6706:
6702:
6697:
6692:
6690:
6679:
6676:
6673:
6668:
6664:
6660:
6657:
6654:
6649:
6645:
6639:
6635:
6631:
6628:
6622:
6619:
6617:
6609:
6606:
6603:
6600:
6597:
6591:
6579:
6575:
6571:
6568:
6564:
6544:
6522:
6519:
6515:
6507:
6501:
6496:
6483:
6472:
6468:
6462:
6458:
6451:
6448:
6442:
6439:
6433:
6428:
6424:
6420:
6416:
6403:
6399:
6393:
6389:
6385:
6382:
6376:
6370:
6367:
6364:
6361:
6353:
6347:
6340:
6335:
6319:
6306:
6302:
6298:
6294:
6288:
6284:
6278:
6274:
6267:
6264:
6252:
6249:
6245:
6241:
6236:
6232:
6228:
6221:
6216:
6211:
6202:
6198:
6194:
6187:
6183:
6177:
6173:
6166:
6163:
6159:
6151:
6147:
6143:
6139:
6135:
6129:
6125:
6122:
6119:
6116:
6108:
6098:
6080:
6076:
6072:
6067:
6061:
6055:
6049:
6046:
6040:
6037:
6029:
6016:
6010:
6007:
6000:
5995:
5991:
5987:
5981:
5978:
5975:
5972:
5969:
5961:
5958:
5937:
5933:
5927:
5924:
5918:
5913:
5909:
5901:
5893:
5888:
5882:
5878:
5874:
5868:
5865:
5856:
5850:
5847:
5842:
5837:
5833:
5829:
5825:
5816:
5812:
5808:
5796:
5792:
5785:
5781:
5775:
5771:
5767:
5761:
5756:
5752:
5742:
5739:
5735:
5727:
5723:
5719:
5709:
5703:
5699:
5695:
5686:
5680:
5677:
5674:
5668:
5660:
5644:
5641:
5638:
5632:
5626:
5623:
5620:
5614:
5611:
5605:
5602:
5599:
5596:
5593:
5587:
5579:
5562:
5559:
5554:
5551:
5547:
5524:
5520:
5516:
5511:
5508:
5504:
5481:
5476:
5473:
5470:
5460:
5451:
5445:
5435:
5410:
5401:
5392:
5389:
5383:
5376:
5372:
5368:
5359:
5352:
5344:
5337:
5329:
5321:
5312:
5305:
5298:
5283:
5281:
5278:, indicating
5273:
5267:
5263:
5246:
5240:
5234:
5226:
5222:
5218:
5209:
5204:
5200:
5187:
5180:
5176:
5170:
5163:
5156:
5152:
5144:
5134:
5121:
5111:
5103:
5099:
5095:
5086:
5081:
5077:
5066:
5057:
5051:
5047:
5038:
5033:
5024:
5016:
5012:
5008:
4999:
4994:
4990:
4984:
4979:
4974:
4966:
4958:
4953:
4935:
4929:
4923:
4914:
4911:
4908:
4903:The integral
4901:
4884:
4864:
4843:
4820:
4792:
4784:
4780:
4771:
4770:quadratically
4753:
4729:
4721:
4717:
4713:
4704:
4699:
4695:
4688:
4683:
4679:
4675:
4672:
4649:
4640:
4637:
4631:
4614:
4604:
4600:
4585:
4571:
4568:
4565:
4559:
4551:
4538:
4534:
4530:
4516:
4503:
4492:
4488:
4484:
4478:
4475:
4472:
4466:
4456:
4445:
4441:
4435:
4431:
4427:
4422:
4416:
4412:
4408:
4402:
4399:
4396:
4392:
4387:
4382:
4376:
4373:
4353:
4331:
4328:
4325:
4322:
4319:
4316:
4311:
4307:
4303:
4300:
4297:
4291:
4288:
4285:
4282:
4259:
4242:
4237:
4233:
4224:
4213:
4209:
4205:
4199:
4196:
4193:
4187:
4183:
4177:
4173:
4169:
4161:
4158:
4155:
4149:
4147:
4133:
4129:
4123:
4120:
4116:
4107:
4097:
4093:
4086:
4083:
4080:
4076:
4067:
4059:
4056:
4052:
4046:
4042:
4038:
4030:
4027:
4024:
4018:
4016:
4002:
3998:
3995:
3992:
3989:
3985:
3979:
3975:
3966:
3958:
3955:
3951:
3945:
3941:
3937:
3929:
3926:
3923:
3917:
3915:
3907:
3904:
3880:
3877:
3864:
3859:
3855:
3846:
3838:
3835:
3831:
3825:
3821:
3817:
3809:
3806:
3803:
3797:
3791:
3788:
3777:
3769:
3753:
3750:
3746:
3739:
3734:
3730:
3723:
3700:
3695:
3683:
3677:
3673:
3663:
3660:
3654:
3648:
3644:
3632:
3621:
3618:
3615:
3611:
3607:
3599:
3579:
3576:
3573:
3553:
3550:
3547:
3527:
3524:
3521:
3501:
3498:
3495:
3492:
3489:
3486:
3481:
3477:
3473:
3470:
3467:
3461:
3458:
3455:
3452:
3430:
3417:
3412:
3404:
3395:
3392:
3386:
3380:
3376:
3364:
3353:
3350:
3347:
3339:
3319:
3314:
3311:
3307:
3298:
3290:
3286:
3282:
3276:
3273:
3262:
3237:
3228:
3224:
3217:
3204:
3200:
3191:
3187:
3183:
3180:
3177:
3169:
3164:
3160:
3156:
3153:
3147:
3144:
3138:
3133:
3129:
3124:
3120:
3117:
3109:
3105:
3098:
3095:
3092:
3084:
3080:
3070:
3067:
3063:
3058:
3055:
3052:
3049:
3046:
3038:
3032:
3028:
3022:
3019:
3013:
3004:
2999:
2979:
2966:
2962:
2958:
2955:
2952:
2945:
2937:
2932:
2928:
2924:
2921:
2915:
2909:
2905:
2893:
2889:
2885:
2882:
2879:
2874:
2870:
2866:
2860:
2855:
2842:
2839:
2836:
2830:
2816:
2799:
2790:
2784:
2779:
2775:
2771:
2766:
2763:
2758:
2755:
2749:
2746:
2743:
2740:
2735:
2731:
2723:
2714:
2710:
2706:
2703:
2700:
2696:
2691:
2687:
2676:
2668:
2663:
2659:
2655:
2652:
2641:
2637:
2633:
2630:
2627:
2623:
2618:
2614:
2607:
2604:
2601:
2598:
2595:
2589:
2584:
2580:
2576:
2569:
2567:
2555:
2550:
2544:
2538:
2534:
2530:
2524:
2521:
2517:
2509:
2506:
2503:
2500:
2497:
2493:
2488:
2484:
2473:
2468:
2464:
2458:
2455:
2450:
2446:
2439:
2436:
2433:
2430:
2427:
2421:
2416:
2412:
2408:
2401:
2399:
2391:
2388:
2385:
2379:
2354:
2350:
2345:
2341:
2337:
2334:
2329:
2325:
2321:
2317:
2313:
2310:
2304:
2299:
2295:
2291:
2285:
2279:
2276:
2273:
2267:
2258:
2245:
2240:
2218:
2215:
2210:
2207:
2187:
2183:
2178:
2167:
2164:
2159:
2156:
2150:
2142:
2133:
2125:
2116:
2110:
2106:
2105:
2095:
2087:
2078:
2075:
2058:
2054:
2049:
2043:
2039:
2036:
2033:
2030:
2024:
2021:
2017:
2013:
2010:
2007:
2004:
2000:
1994:
1990:
1987:
1984:
1981:
1975:
1972:
1968:
1964:
1961:
1957:
1949:
1941:
1938:
1935:
1932:
1926:
1922:
1918:
1916:
1903:
1900:
1897:
1894:
1886:
1882:
1878:
1875:
1870:
1862:
1859:
1856:
1853:
1847:
1843:
1839:
1837:
1829:
1826:
1823:
1817:
1804:
1791:
1784:
1778:
1768:
1764:
1760:
1757:
1748:
1744:
1737:
1733:
1728:
1722:
1716:
1707:
1694:
1689:
1684:
1680:
1676:
1673:
1668:
1664:
1660:
1656:
1652:
1646:
1643:
1640:
1634:
1625:
1612:
1609:
1606:
1600:
1597:
1594:
1591:
1587:
1577:
1574:
1571:
1565:
1551:
1547:
1540:
1537:
1533:
1528:
1522:
1516:
1506:
1502:evaluated at
1499:
1495:
1491:
1484:
1480:
1475:
1457:
1454:
1448:
1444:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1405:
1401:
1397:
1390:
1386:
1382:
1378:
1374:
1370:
1364:
1360:
1356:
1350:
1337:
1334:
1331:
1322:
1316:
1310:
1307:
1304:
1301:
1295:
1291:
1281:
1275:
1261:
1257:
1250:
1247:
1243:
1238:
1232:
1229:
1226:
1220:
1212:
1207:
1202:
1195:
1191:
1185:
1181:
1173:
1166:
1162:
1145:
1137:
1134:
1131:
1128:
1125:
1119:
1116:
1112:
1108:
1105:
1097:
1094:
1091:
1088:
1085:
1079:
1075:
1071:
1068:
1062:
1059:
1056:
1050:
1041:
1039:
1034:
1028:
1022:
1009:
1004:
999:
995:
991:
986:
981:
977:
973:
968:
963:
959:
955:
950:
913:
908:
904:
898:
878:
873:
869:
859:
846:
838:
835:
832:
824:
813:
809:
805:
799:
796:
785:
777:
772:
766:
763:
757:
744:
741:
736:
725:
721:
717:
709:
705:
696:
691:
676:
672:
671:wave equation
668:
659:
649:
645:
635:
633:
629:
619:
603:
599:
573:
569:
545:
533:
531:
527:
516:
514:
510:
501:
487:
461:
457:
452:
446:
443:
423:
403:
392:
376:
368:
363:
349:
340:
323:
315:
312:
309:
306:
302:
298:
295:
289:
284:
280:
274:
271:
268:
265:
260:
256:
247:
233:
213:
193:
185:
181:
177:
172:
170:
166:
162:
158:
154:
150:
146:
142:
141:wave equation
138:
134:
132:
128:
127:W.R. Hamilton
124:
120:
116:
115:carrier waves
112:
102:
100:
96:
92:
91:wave equation
88:
87:wave function
84:
80:
76:
72:
69:of different
68:
64:
60:
56:
52:
48:
39:
33:
19:
10771:
10725:, New York:
10720:
10709:
10685:
10674:
10639:
10635:
10625:
10600:
10596:
10586:
10562:
10558:
10552:
10543:complement G
10540:
10535:
10526:
10519:
10511:
10508:
10503:
10477:
10470:
10448:
10439:
10420:
10414:
10395:
10389:
10370:
10364:
10345:
10327:
10300:
10265:
10259:
10248:. Retrieved
10236:
10223:
10204:
10197:
10174:
10169:
9948:
9775:
9773:
9768:
9696:
9586:
9506:
9504:
9218:
9120:
8931:
8763:
8756:
8657:
8552:
8528:
8347:
8289:
8276:
8251:
8250:
8244:
8105:
8089:
7998:
7821:
7816:
7811:
7750:
7652:
7650:
7502:
7498:
7494:
7490:
7486:
7484:
7479:
7475:
7461:
7450:
7443:
7434:
7430:
7428:
7305:
7301:
7192:
7190:
7177:
7174:
7158:
6577:
6573:
6560:
6520:
6513:
6505:
6497:
6351:
6345:
6338:
6104:
6030:
5962:
5959:
5661:
5580:
5577:
5387:
5381:
5374:
5370:
5366:
5357:
5350:
5342:
5335:
5327:
5319:
5310:
5303:
5299:
5284:
5279:
5271:
5265:
5261:
5188:
5181:
5174:
5168:
5161:
5154:
5150:
5142:
5136:
4916:
4909:
4906:
4902:
4769:
4768:. It varies
4642:For a given
4641:
4635:
4629:
4536:
4532:
4518:
4355:
4347:
3882:
3878:
3597:
3595:
3337:
3250:
3223:
3030:
3026:
3023:
3017:
3011:
2994:
2817:
2259:
2123:
2114:
2102:
2100:
2076:
1806:and finally
1805:
1708:
1626:
1504:
1497:
1493:
1489:
1482:
1478:
1436:
1429:
1425:
1421:
1417:
1413:
1409:
1403:
1399:
1395:
1388:
1384:
1380:
1376:
1372:
1368:
1362:
1358:
1354:
1351:
1208:
1200:
1193:
1189:
1183:
1179:
1171:
1164:
1160:
1042:
1032:
1026:
1023:
860:
770:
767:
761:
758:
666:
664:
625:
534:
522:
507:
364:
341:
248:
173:
135:
108:
58:
54:
50:
44:
10772:wave packet
10727:McGraw-Hill
10129:Matter wave
8934:convolution
8549:Heat kernel
8281:convolution
6563:phase space
6336:), chosing
5395:The 2D case
5385:(for large
4905:∫ |Ψ|
1472:comes from
1437:The factor
391:matter wave
71:wavenumbers
51:wave packet
10824:Categories
10649:1904.02933
10642:(5): 686.
10250:2023-06-22
10189:References
8936:identity,
8543:See also:
7295:becomes a
6567:pure state
2104:dispersion
2081:Dispersive
1709:we obtain
1211:wave forms
1204:-direction
1175:-direction
778:solutions
776:plane-wave
642:See also:
628:scattering
111:modulation
99:dispersion
95:dispersion
59:wave group
55:wave train
18:Wave train
10666:2334-2536
10617:0002-9505
10559:Am J Phys
10019:−
9989:π
9958:ψ
9899:∗
9874:ψ
9843:δ
9840:∗
9787:ψ
9715:∗
9670:ε
9655:−
9642:ε
9544:−
9469:−
9466:Δ
9451:∫
9433:−
9405:−
9382:∫
9364:−
9346:Δ
9254:∫
9190:−
9092:∇
9086:−
9048:−
8978:∗
8880:ρ
8858:∫
8849:→
8812:δ
8791:ρ
8782:→
8715:−
8698:π
8667:ρ
8637:ρ
8621:∂
8611:∂
8592:ρ
8583:∂
8579:∂
8555:diffusion
8446:−
8408:−
8384:−
8372:∫
8324:ψ
8320:∗
8305:ψ
8194:−
8167:π
8139:ψ
8135:∫
8114:ψ
8060:−
8051:δ
8033:ψ
8029:∫
8008:ψ
7955:−
7928:π
7897:ψ
7866:−
7857:δ
7836:ψ
7763:∫
7716:−
7707:δ
7695:−
7671:→
7608:−
7579:π
7557:−
7409:ε
7382:−
7365:ε
7350:π
7277:ε
7258:−
7246:ε
7243:π
7212:ψ
7053:−
7005:−
6877:−
6822:−
6775:−
6720:π
6661:−
6632:−
6443:−
6386:−
6371:
6362:ψ
6268:−
6256:ℏ
6167:−
6140:ℏ
6126:
6117:ψ
6065:ℏ
6050:θ
5988:ℏ
5982:−
5979:θ
5976:−
5970:ϕ
5931:ℏ
5875:ℏ
5869:−
5857:−
5851:
5772:ℏ
5743:ϕ
5710:π
5669:ψ
5633:ψ
5615:ψ
5588:ψ
5461:ψ
5436:ψ
5411:ψ
5216:ℏ
5137:in which
5093:ℏ
5067:⋅
5052:−
5006:ℏ
4972:Ψ
4967:∗
4963:Ψ
4944:Ψ
4888:∞
4824:∞
4821:−
4793:⋅
4750:‖
4741:‖
4711:ℏ
4673:ℏ
4586:ψ
4572:η
4569:∫
4563:⟩
4560:ψ
4552:η
4549:⟨
4482:ℏ
4457:⋅
4446:−
4406:ℏ
4363:Ψ
4295:ℏ
4225:⋅
4203:ℏ
4188:−
4159:π
4138:ℏ
4108:⋅
4094:ℏ
4084:−
4068:⋅
4057:−
4028:π
4007:ℏ
3990:−
3967:⋅
3956:−
3927:π
3894:Ψ
3847:⋅
3836:−
3807:π
3778:ψ
3743:ℏ
3727:Δ
3721:Δ
3688:ℏ
3670:Δ
3658:⟩
3652:⟨
3641:⟩
3633:⋅
3625:⟨
3465:ℏ
3402:Δ
3390:⟩
3384:⟨
3373:⟩
3365:⋅
3357:⟨
3299:⋅
3291:−
3263:ψ
3231:velocity.
3157:−
3121:ρ
3110:∗
3106:ψ
3102:∇
3099:ψ
3096:−
3093:ψ
3090:∇
3085:∗
3081:ψ
3053:ρ
2925:−
2910:−
2875:π
2831:ψ
2759:−
2656:−
2619:−
2585:π
2525:−
2489:−
2451:−
2417:π
2380:ψ
2322:−
2314:
2300:π
2268:ψ
2208:ω
2184:ψ
2175:∇
2160:−
2148:∂
2143:ψ
2140:∂
2044:λ
2034:−
2025:π
2014:
1995:λ
1985:−
1976:π
1965:
1936:−
1927:−
1898:−
1857:−
1848:−
1761:−
1749:−
1661:−
1592:−
1561:∞
1555:∞
1552:−
1548:∫
1541:π
1458:π
1311:ω
1308:−
1271:∞
1265:∞
1262:−
1258:∫
1251:π
1135:ω
1117:−
1095:ω
1092:−
870:ω
836:ω
833:−
825:⋅
733:∇
702:∂
688:∂
596:Δ
566:Δ
543:Δ
488:ℏ
466:ℏ
450:Δ
447:≈
416:and mass
401:Δ
350:σ
313:σ
310:π
281:σ
275:±
246:would be
178:explored
75:bandwidth
10694:, Inc.,
10682:(1975),
10447:(2000),
10241:Archived
10119:Waveform
10077:See also
9751:′
9727:′
9440:″
9429:′
9412:′
9390:′
9371:″
9312:″
9301:′
9284:′
9262:′
9243:″
9197:′
9164:′
8990:′
8959:′
8523:in time
8511:in time
8495:in time
8465:′
8421:′
8259:at time
7489:at time
4877:towards
4836:towards
3713:so that
1407:, since
1366:, since
530:momentum
480:. Since
131:Rayleigh
63:envelope
10567:Bibcode
10161:Remarks
10154:Soliton
10124:Wavelet
5354:√
5339:√
5324:√
5308:√
5269:√
5140:√
4633:. When
3243:=2 and
3008:√
47:physics
10733:
10698:
10664:
10636:Optica
10615:
10547:, §3-a
10492:
10459:
10427:
10402:
10377:
10352:
10315:
10274:537250
10272:
10212:
8740:
8640:
8547:, and
8472:
7725:
6350:, and
5960:where
5044:
3332:where
2977:
2902:
2797:
2684:
2481:
2122:, and
1584:
1288:
861:where
759:where
389:for a
342:where
121:, and
10644:arXiv
10244:(PDF)
10233:(PDF)
6516:⟩ = 0
6348:= 1/2
5382:ħt/ma
5351:ħt /m
3010:1 + 4
3003:width
1402:) = −
673:from
10731:ISBN
10696:ISBN
10662:ISSN
10613:ISSN
10490:ISBN
10457:ISBN
10425:ISBN
10400:ISBN
10375:ISBN
10350:ISBN
10313:ISBN
10270:OCLC
10210:ISBN
10084:Wave
9125:and
8766:=0,
7476:will
6576:and
3577:>
3551:<
1420:) =
1379:) =
1361:) =
1030:and
926:and
646:and
49:, a
10654:doi
10605:doi
10575:doi
10512:117
10305:doi
8842:lim
8775:lim
8569:),
8519:to
8507:to
8491:to
8352:to
8271:to
8249:or
7664:lim
7464:→ 0
6561:In
6341:= 1
5848:exp
5391:).
5336:ħ/m
5184:|Ψ|
5177:= 0
5164:= 0
5159:at
3770:),
3039:is
3016:→ 2
2311:exp
2011:sin
1962:cos
1507:= 0
57:or
45:In
10826::
10729:,
10660:.
10652:.
10638:.
10634:.
10611:.
10601:64
10599:.
10595:.
10573:,
10563:47
10561:,
10488:,
10484:,
10480:,
10474:*
10455:,
10336:^
10311:.
10281:^
10239:.
10235:.
9778:,
9576:,
9013:,
8929:.
8535:.
8296:,
8275:,
8102:,
7827:,
7466:,
7299:,
7188:.
6368:Ai
6343:,
6123:Ai
5539:,
5496:.
5377:/2
5373:=
5313:/2
5306:=
5291:10
5287:10
5282:.
5264:/(
5262:ħt
5186:,
5166:;
3340:,
3247:=4
2118:,
2113:2Δ
1496:,
1434:.
1430:ct
1428:+
1416:,
1404:kc
1389:ct
1387:−
1375:,
1363:kc
1206:.
1194:ct
1192:+
1184:ct
1180:x
1165:kc
1161:ω
1040:.
117:,
113:,
10668:.
10656::
10646::
10640:6
10619:.
10607::
10577::
10569::
10545:I
10433:.
10408:.
10383:.
10358:.
10321:.
10307::
10253:.
10062:.
10053:)
10050:t
10047:i
10044:+
10041:a
10038:(
10035:2
10029:2
10025:x
10015:e
10007:)
10004:t
10001:i
9998:+
9995:a
9992:(
9986:2
9981:1
9976:=
9973:)
9970:x
9967:(
9962:t
9935:.
9929:t
9926:i
9923:+
9920:a
9916:K
9912:=
9907:a
9903:K
9894:t
9891:i
9887:K
9883:=
9878:t
9852:)
9849:x
9846:(
9835:a
9831:K
9827:=
9824:)
9821:x
9818:(
9813:a
9809:K
9805:=
9802:)
9799:x
9796:(
9791:0
9776:K
9769:z
9748:z
9744:+
9741:z
9737:K
9733:=
9724:z
9719:K
9710:z
9706:K
9682:,
9676:H
9673:)
9667:+
9664:t
9661:i
9658:(
9651:e
9647:=
9639:+
9636:t
9633:i
9629:K
9625:=
9619:r
9616:h
9613:c
9610:S
9604:t
9600:K
9589:z
9582:K
9578:K
9574:x
9570:z
9556:.
9550:H
9547:z
9540:e
9536:=
9533:)
9530:x
9527:(
9522:z
9518:K
9507:t
9491:.
9487:)
9484:y
9481:(
9478:B
9475:)
9472:y
9463:(
9460:A
9455:y
9447:=
9444:)
9437:x
9426:x
9422:(
9419:B
9416:)
9409:x
9402:x
9399:(
9396:A
9387:x
9378:=
9375:)
9368:x
9361:x
9358:(
9355:C
9352:=
9349:)
9343:(
9340:C
9320:,
9316:)
9309:x
9305:,
9298:x
9294:(
9291:B
9288:)
9281:x
9277:,
9274:x
9271:(
9268:A
9259:x
9250:=
9247:)
9240:x
9236:,
9233:x
9230:(
9227:C
9205:.
9201:)
9194:x
9187:x
9184:(
9179:t
9175:K
9171:=
9168:)
9161:x
9157:,
9154:x
9151:(
9146:t
9142:K
9131:K
9127:x
9123:x
9107:.
9101:2
9096:2
9083:=
9080:H
9060:,
9054:H
9051:t
9044:e
9040:=
9037:)
9034:x
9031:(
9026:t
9022:K
9011:H
8997:,
8987:t
8982:K
8973:t
8969:K
8965:=
8956:t
8952:+
8949:t
8945:K
8927:f
8910:)
8907:0
8904:(
8901:f
8898:=
8895:)
8892:x
8889:(
8884:t
8876:)
8873:x
8870:(
8867:f
8862:x
8852:0
8846:t
8821:)
8818:x
8815:(
8809:=
8806:)
8803:x
8800:(
8795:t
8785:0
8779:t
8764:t
8759:t
8757:ρ
8743:,
8733:t
8730:2
8723:2
8719:x
8709:e
8701:t
8695:2
8690:1
8685:=
8682:)
8679:x
8676:(
8671:t
8643:,
8629:2
8625:x
8615:2
8603:2
8600:1
8595:=
8586:t
8525:t
8521:z
8517:y
8513:t
8509:y
8505:x
8501:t
8499:+
8497:t
8493:z
8489:x
8475:,
8469:)
8462:t
8458:+
8455:t
8452:;
8449:z
8443:x
8440:(
8437:K
8434:=
8431:y
8428:d
8425:)
8418:t
8414:;
8411:z
8405:y
8402:(
8399:K
8396:)
8393:t
8390:;
8387:y
8381:x
8378:(
8375:K
8362:t
8360:+
8358:t
8354:y
8350:x
8334:.
8328:0
8317:K
8314:=
8309:t
8293:0
8290:ψ
8285:K
8273:x
8269:y
8265:y
8261:t
8257:x
8230:.
8226:y
8223:d
8218:t
8215:2
8211:/
8205:2
8201:)
8197:y
8191:x
8188:(
8185:i
8181:e
8173:t
8170:i
8164:2
8159:1
8154:)
8151:y
8148:(
8143:0
8132:=
8129:)
8126:x
8123:(
8118:t
8100:K
8096:0
8093:ψ
8076:,
8072:y
8069:d
8066:)
8063:y
8057:x
8054:(
8048:)
8045:y
8042:(
8037:0
8026:=
8023:)
8020:x
8017:(
8012:0
7985:.
7979:t
7976:2
7972:/
7966:2
7962:)
7958:y
7952:x
7949:(
7946:i
7942:e
7934:t
7931:i
7925:2
7920:1
7915:=
7912:)
7909:x
7906:(
7901:t
7876:,
7872:)
7869:y
7863:x
7860:(
7854:=
7851:)
7848:x
7845:(
7840:0
7825:y
7817:ε
7812:K
7798:,
7794:1
7791:=
7788:x
7785:d
7782:)
7779:x
7776:(
7771:t
7767:K
7753:K
7728:,
7722:)
7719:y
7713:x
7710:(
7704:=
7701:)
7698:y
7692:x
7689:(
7684:t
7680:K
7674:0
7668:t
7653:t
7637:.
7629:t
7626:2
7619:2
7615:)
7611:y
7605:x
7602:(
7599:i
7593:e
7585:t
7582:i
7576:2
7571:1
7566:=
7563:)
7560:y
7554:x
7551:(
7546:t
7542:K
7538:=
7535:)
7532:y
7529:,
7526:x
7523:(
7518:t
7514:K
7503:y
7499:x
7495:x
7491:t
7487:x
7480:ε
7472:K
7468:K
7462:ε
7457:K
7453:ε
7431:x
7406:+
7403:t
7400:i
7397:2
7390:2
7386:x
7376:e
7368:)
7362:+
7359:t
7356:i
7353:(
7347:2
7342:1
7337:=
7334:)
7331:x
7328:(
7323:t
7319:K
7308:)
7306:x
7304:(
7302:δ
7274:2
7268:2
7264:x
7254:e
7240:2
7235:1
7230:=
7227:)
7224:x
7221:(
7216:0
7201:ε
7197:a
7193:m
7186:K
7182:K
7161:x
7155:.
7143:p
7140:2
7137:=
7134:m
7130:/
7126:p
7106:x
7095:.
7083:C
7080:=
7075:2
7070:)
7064:2
7060:B
7056:t
7050:B
7046:/
7042:p
7038:(
7033:+
7029:)
7023:2
7019:t
7013:3
7009:B
7002:x
6998:(
6994:B
6983:.
6971:)
6968:t
6963:3
6959:B
6955:,
6950:2
6946:t
6940:3
6936:B
6932:(
6904:.
6901:)
6898:0
6895:;
6892:p
6889:,
6886:t
6883:p
6880:2
6874:x
6871:(
6868:W
6865:=
6854:)
6849:)
6843:2
6839:)
6833:2
6829:B
6825:t
6819:B
6815:/
6811:p
6807:(
6803:+
6799:)
6793:2
6789:t
6783:3
6779:B
6772:x
6769:(
6766:B
6762:(
6756:3
6752:/
6748:2
6744:2
6739:(
6734:i
6731:A
6723:B
6715:3
6711:/
6707:1
6703:2
6698:1
6693:=
6683:)
6680:0
6677:;
6674:t
6669:3
6665:B
6658:p
6655:,
6650:2
6646:t
6640:3
6636:B
6629:x
6626:(
6623:W
6620:=
6613:)
6610:t
6607:;
6604:p
6601:,
6598:x
6595:(
6592:W
6578:p
6574:x
6545:x
6514:p
6512:⟨
6508:⟩
6506:x
6504:⟨
6484:.
6478:)
6473:2
6469:t
6463:3
6459:B
6452:3
6449:2
6440:x
6437:(
6434:t
6429:3
6425:B
6421:i
6417:e
6412:]
6409:)
6404:2
6400:t
6394:3
6390:B
6383:x
6380:(
6377:B
6374:[
6365:=
6352:B
6346:m
6339:ħ
6320:,
6315:]
6312:)
6307:2
6303:m
6299:6
6295:/
6289:2
6285:t
6279:3
6275:B
6271:(
6265:x
6262:[
6259:)
6253:m
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34:.
20:)
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