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A wave packet without dispersion

A wave packet with dispersion

In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Depending on the evolution equation, the wave packet's envelope may remain constant (no dispersion, see figure) or it may change (dispersion) while propagating.

Quantum mechanics ascribes a special significance to the wave packet: it is interpreted as a "probability wave", describing the probability that a particle or particles in a particular state will be measured to have a given position and momentum. It is in this way related to the wave function. Through application of the Schrödinger equation in quantum mechanics, it is possible to deduce the time evolution of a system, similar to the process of the Hamiltonian formalism in classical mechanics. The wave packet is thus a mathematical solution to the Schrödinger equation. The area under the absolute square of the wave packet solution is interpreted as the probability density of finding the particle in a region. The dispersive character of solutions of the Schrödinger equation has played an important role in rejecting Schrödinger's original interpretation, and accepting the Born rule.

In the coordinate representation of the wave (such as the Cartesian coordinate system), the localized position of the physical object's probability is given by the position of the packet. Moreover, the narrower the spatial wave packet, and therefore the better defined the position of the wave packet, the larger the spread in the momentum of the wave. This trade-off between spread in position and spread in momentum is one example of the Heisenberg uncertainty principle.

Background

In the early 1900s, it became apparent that classical mechanics had some major failings. Isaac Newton originally proposed the idea that light came in discrete packets, which he called "corpuscles", but the wave-like behavior of many light phenomena quickly led scientists to favor a wave description of electromagnetism. It wasn't until the 1930s that the particle nature of light really began to be widely accepted in physics. The development of quantum mechanics — and its success at explaining confusing experimental results — was at the root of this acceptance. Thus, one of the basic concepts in the formulation of quantum mechanics is that of light coming in discrete bundles called photons. The energy of light photon is a discrete function of its frequency,

$E=h\nu ~.$

The photon's energy is equal to Planck's constant, h, multiplied by its frequency, ν. (This resolved a significant problem in classical physics, called the ultraviolet catastrophe.)

The ideas of quantum mechanics continued to be developed throughout the 20th century. The picture that was developed was of a particulate world, with all phenomena and matter made of and interacting with discrete particles; however, these particles were described by a probability wave. The interactions, locations, and all of physics would be reduced to the calculations of these probability amplitude waves. The particle-like nature of the world was significantly confirmed by experiment, while, at the very same time, the wave-like phenomena could be characterized as consequences of the wave-packet nature of particles.

Basic behaviors of wave packets

As an example of propagation without dispersion, consider wave solutions to the following wave equation:

${\partial ^{2}u \over \partial t^{2}}=c^{2}{\nabla ^{2}u}~,$

where c is the speed of the wave's propagation in a given medium. Using the physics time convention, exp(−iωt), the wave equation has plane-wave solutions

$u({\mathbf {x}},t)=e^{i{({\mathbf {k\cdot x}}}-\omega t)}~,$

where

$\omega ^{2}=|{\mathbf {k}}|^{2}c^{2}$, and $|{\mathbf {k}}|^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}.$

This relation between ω and k should be valid so that the plane wave is a solution to the wave equation. It is called a dispersion relation.

To simplify, consider only waves propagating in one dimension (extension to three dimensions is straightforward). Then the general solution is

$u(x,t)=Ae^{i(kx-\omega t)}+Be^{-i(kx+\omega t)},\,$

in which we may take ω= kc . The first term represents a wave propagating in the positive x-direction since it is a function of x−ct only; the second term, being a function of x+ct, represents a wave propagating in the negative x-direction.

A wave packet is a localized disturbance that results from the sum of many different wave 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. From the basic solutions in one dimension, a general form of a wave packet can be expressed as

$u(x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }A(k)~e^{i(kx-\omega (k)t)}\,dk$.

As in the plane-wave case the wave packet travels to the right for ω(k)=kc (since then u(x,t)=F(x−ct)) and to the left for ω(k)=−kc (since then u(x,t) = F(x+ct)).

The factor $1/{\sqrt {2\pi }}$ comes from Fourier transform conventions. The amplitude A(k) contains the coefficients of the linear superposition of the plane-wave solutions. These coefficients can in turn be expressed as a function of u(x,t) evaluated at t=0 by inverting the Fourier transform relation above:

$A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }u(x,0)~e^{-ikx}\,dx$.

For instance, 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

$u(x,t)=e^{-(x-ct)^{2}+ik_{0}(x-ct)}.$

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

As an example of propagation with dispersion, consider solutions to the Schrödinger equation (with m and ħ set equal to one)

$i{\partial u \over \partial t}=-{\frac {1}{2}}{\nabla ^{2}u},$

yielding as dispersion relation

$\omega ={\frac {1}{2}}|{\mathbf {k}}|^{2}.$

Once again, restricting ourselves to one dimension the solution to the Schrödinger equation satisfying the initial condition $\scriptstyle u(x,0)=e^{-x^{2}+ik_{0}x}$ is found according to

$u(x,t)={\frac {e^{-{\frac {k_{0}^{2}}{4}}}}{\sqrt {1+2it}}}e^{-{\frac {(x-{\frac {ik_{0}}{2}})^{2}}{1+2it}}}.$

An impression of the dispersive behaviour of this wave packet is obtained by looking at

$|u(x,t)|={\frac {1}{(1+4t^{2})^{1/4}}}e^{-{\frac {(x-k_{0}t)^{2}}{1+4t^{2}}}}$

(note that |u(x,t)| itself is not a solution of the Schrödinger equation).

It is evident that this dispersive wave packet, while moving with constant group velocity k₀, has a width increasing with time as $(1+4t^{2})^{1/2}$.

Gaussian wavepackets in quantum mechanics

File:Wave function of a Gaussian state moving at constant momentum.gif

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

Position space probability density of an initially Gaussian state trapped in an infinite potential well experimenting periodic Quantum Tunneling in a centered potential wall.

The above Gaussian wavepacket, unnormalized and just centered at the origin, instead, can now be written in 3D:

$\psi (x)=e^{-{\mathbf {r}}\cdot {\mathbf {r}}/2a}~,$

where a is a positive real number, the square of the width of the wavepacket, a = 2⟨r·r⟩/3⟨1⟩ = 2 (Δx)².

The Fourier transform is also a Gaussian in terms of the wavenumber, the k-vector, (with inverse width, 1/a = 2⟨k·k⟩/3⟨1⟩ = 2 (Δp_x/ħ)², so that Δx Δp_x=ħ/2, i.e. it saturates the uncertainty relation),

$\psi ({\mathbf {k}})=(2\pi a)^{3/2}e^{-a{\mathbf {k}}\cdot {\mathbf {k}}/2}~.$

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

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

The inverse Fourier transform is still a Gaussian, but the parameter a 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 of Ψ over all space is invariant, because it is the inner product of Ψ with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any energy eigenstate η(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 of η. When η has zero energy, like the infinite wavelength wave, it doesn't change at all. The integral ∫|Ψ|²d³r is also invariant, which is a statement of the conservation of probability. Explicitly,

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

The width of the Gaussian is the interesting quantity which can be read off from |Ψ|²:

${\sqrt {a^{2}+(\hbar t/m)^{2} \over a}}\,$.

The width eventually grows linearly in time, as ħt /m√a, indicating wave-packet spreading.

This linear growth is a reflection of the momentum uncertainty: the wavepacket is confined to a narrow width √a, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount ħ/2√a, a spread in velocity of ħ/2m√a, and thus in the future position by ħt /m√a. (The uncertainty relation is then a strict inequality, far from saturation.)

The Airy wave train

In contrast to the above Gaussian wavepacket, it has been observed that a particular wavefunction based on Airy functions, propagates freely without dispersion, maintaining its shape. It accelerates undistorted in the absence of a force field: Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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Truncated view of time development for the Airy front in phase space. (Click to animate.)

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In phase space, this is evident in the pure state Wigner quasiprobability distribution of this wavetrain, whose shape in x and p is invariant as time progresses, but whose features accelerate to the right, in accelerating parabolas Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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$W(x,p;t)=W(x-B^{3}t^{2},p-B^{3}t;0)={1 \over 2^{1/3}\pi B}~\mathrm {Ai} \left(2^{2/3}\left(Bx+{p^{2} \over B^{2}}-2Bpt\right)\right).$

Note the momentum distribution obtained by integrating over all Template:Mvar is constant.

Free propagator

The narrow-width limit of the Gaussian wavepacket solution discussed is the free propagator kernel K. For other differential equations, this is sometimes 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 K.

Returning to one dimension for simplicity, when a is the infinitesimal quantity ε, the Gaussian initial condition, rescaled so that its integral is one:

$\psi _{0}(x)={1 \over {\sqrt {2\pi \epsilon }}}e^{-{x^{2} \over 2\epsilon }}\,$

becomes a delta function, δ(x), so that its time evolution,

$K_{t}(x)={1 \over {\sqrt {2\pi (it+\epsilon )}}}e^{-x^{2} \over 2it+\epsilon }\,$

gives the propagator.

Note that a very narrow initial wavepacket instantly becomes infinitely wide, but with a phase which is more rapidly oscillatory at large values of x. This might seem strange—the solution goes from being localized at one point to being "everywhere" at all later times, but it is a reflection of the enormous momentum uncertainty of a localized particle. Also note that the norm of the wavefunction is infinite, but this is also correct, since the square of a delta function is divergent in the same way.

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

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

$K_{t}(x,y)=K_{t}(x-y)={1 \over {\sqrt {2\pi it}}}e^{-i(x-y)^{2} \over 2t}\,.$

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

$\lim _{t\rightarrow 0}K_{t}(x-y)=\delta (x-y)~,$

but only in the sense of distributions: The integral of this quantity multiplied by an arbitrary differentiable test function gives the value of the test function at zero. To see this, note that the integral over all space of K equals 1 at all times:

$\int _{x}K_{t}(x)=1\,,$

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

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

$\psi _{0}(x)=\delta (x-y)\,,$

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,

$\psi _{0}(x)=\int _{y}\psi _{0}(y)\delta (x-y)\,,$

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

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

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

$\psi _{t}=K*\psi _{0}\,.$

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

$\int _{y}K(x-y;t)K(y-z;t')=K(x-z;t+t')\,,$

which can be interpreted as follows: the amplitude to travel from x to z in time t+t ' is the sum of the amplitude to travel from x to y in time t, multiplied by the amplitude to travel from y to z in time t ', summed over all possible intermediate states y. This is a property of an arbitrary quantum system, and by subdividing the time into many segments, it allows the time evolution to be expressed as a path integral.

Analytic continuation to diffusion

The spreading of wavepackets in quantum mechanics is directly related to the spreading of probability densities in diffusion. For a particle which is randomly walking, the probability density function at any point satisfies the diffusion equation:

${\partial \over \partial t}\rho ={1 \over 2}{\partial ^{2} \over \partial x^{2}}\rho ~,$

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

A solution of this equation is the spreading Gaussian,

$\rho _{t}(x)={1 \over {\sqrt {2\pi t}}}e^{-x^{2} \over 2t}~,$

and since the integral of ρ_t is constant, while the width is becoming narrow at small times, this function approaches a delta function at t=0:

$\lim _{t\rightarrow 0}\rho _{t}(x)=\delta (x)\,$

again only in the sense of distributions, so that

$\lim _{t\rightarrow 0}\int _{x}f(x)\rho _{t}(x)=f(0)\,$

for any smooth test function f.

The spreading Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution 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 operator H:

$K_{t}(x)=e^{-tH}\,,$

which is the infinitesimal diffusion operator,

$H=-{\nabla ^{2} \over 2}\,.$

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

$K_{t}(x,x')=K_{t}(x-x')\,.$

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 of ts 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 of z is positive, for large values of x, K is exponentially decreasing, and integrals over K are indeed absolutely convergent.

The limit of this expression for z coming close to the pure imaginary axis is the Schrödinger propagator:

$K_{t}^{\rm {Schr}}=K_{it+\epsilon }=e^{-(it+\epsilon )H}\,,$

and this gives a more conceptual explanation for the time evolution of Gaussians.

From the fundamental identity of exponentiation, or path integration:

$K_{z}*K_{z'}=K_{z+z'}\,$

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

Thus, quantum evolution starting from a Gaussian, which is the diffusion kernel K,

$\psi _{0}(x)=K_{a}(x)=K_{a}*\delta (x)\,$

gives the time evolved state,

$\psi _{t}=K_{it}*K_{a}=K_{a+it}\,.$

This illustrates the above diffusive form of the Gaussian solutions,

$\psi _{t}(x)={1 \over {\sqrt {2\pi (a+it)}}}e^{-{x^{2} \over 2(a+it)}}\,.$

See also

• Wave
• Wave propagation
• Fourier analysis
• Group velocity
• Phase velocity
• Wavelet

External links

• A simulation of a wave package in 2D (According to FOURIER-Synthesis in 2D)