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. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile 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 amplitude, its norm squared describing the probability density that a particle or particles in a particular state will be measured to have a given position or momentum. The wave equation is in this case the Schrödinger equation. It is possible to deduce the time evolution of a quantum mechanical system, similar to the process of the Hamiltonian formalism in classical mechanics. 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 position of the physical object's localized probability is specified by the position of the packet solution. Moreover, the narrower the spatial wave packet, and therefore the better localized 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 a characteristic feature of the Heisenberg uncertainty principle, and will be illustrated below.
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 function of its frequency,
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 amplitudes. The particle-like nature of the world has been confirmed by experiment over a century, while the wave-like phenomena could be characterized as consequences of the wave packet aspect of quantum particles, see wave-particle duality. According to the principle of complementarity, the wave-like and particle-like characteristics never manifest themselves at the same time, i.e. in the same experiment — see however the Afshar experiment and the lively discussion around it.
Basic behaviors of wave packets
As an example of propagation without dispersion, consider wave solutions to the following wave equation,
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
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
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
As in the plane-wave case the wave packet travels to the right for ω(k) = kc, since u(x, t)= F(x − ct), and to the left for ω(k) = −kc, since u(x,t) = F(x + ct).
The factor Template:Frac 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:
For instance, choosing
The imaginary part is a sine wave with perpendicular polarisation to the cosine wave. The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.
By contrast, as an example of propagation now with dispersion, consider instead solutions to the Schrödinger equation (with m and ħ set equal to one),
yielding the dispersion relation
Once again, restricting attention to one dimension, the solution to the Schrödinger equation satisfying the initial condition u(x,0) = exp(−x²+ikox) is seen to be
An impression of the dispersive behavior of this wave packet is obtained by looking at the probability density,
It is evident that this dispersive wave packet, while moving with constant group velocity ko, is delocalizing rapidly: it has a width increasing with time as Template:Radical → 2t, so eventually it diffuses to an unlimited region of space.[nb 1]
Gaussian wave packets in quantum mechanics
where Template:Mvar is a positive real number, the square of the width of the wave packet, a = 2⟨r·r⟩/3⟨1⟩ = 2 (Δx)2.
The Fourier transform is also a Gaussian in terms of the wavenumber, Template:Mvar=0, the k-vector, (with inverse width, 1/a = 2⟨k·k⟩/3⟨1⟩ = 2 (Δpx/ħ)2, so that Δx Δpx=ħ/2, i.e. it saturates the uncertainty relation),
Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is
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,
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,
in which √Template:Mvar is the width of P(r) at t = 0; r is the distance from the origin; the speed of the particle is zero; and the time origin t = 0 can be chosen arbitrarily.
The width of the Gaussian is the interesting quantity which can be read off from the probability density, |Ψ|2,
This width eventually grows linearly in time, as ħt/(m√a), indicating wave-packet spreading.
For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., 10−10 m) then the width of the packet doubles in about 10−16 s. Clearly, particle wave packets spread out very rapidly indeed (in free space): For instance, after 1 ms, the width will have grown to about a kilometer.
This linear growth is a reflection of the momentum uncertainty: the wave packet is confined to a narrow Δx=Template:Radical, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount ħ/Template:Radical, a spread in velocity of ħ/mTemplate:Radical, and thus in the future position by ħt /mTemplate:Radical. The uncertainty relation is then a strict inequality, very far from saturation, indeed! The initial uncertainty ΔxΔp = ħ/2 has now increased by a factor of ħt/ma.
The Airy wave train
In contrast to the above Gaussian wave packet, it has been observed that a particular wave function based on Airy functions, propagates freely without envelope dispersion, maintaining its shape. It accelerates undistorted in the absence of a force field: ψ=Ai(B(x−B³t ²)) exp(iB³t(x−2B³t²/3)). (For simplicity, Template:Mvar=1, m=1/2, and B is a constant, cf. nondimensionalization.)
Nevertheless, Ehrenfest's theorem is still valid in this force-free situation, because the state is both non-normalizable and has an undefined (infinite) ⟨x⟩ for all times. (To the extent that it can be defined, ⟨p⟩ = 0 for all times, despite the apparent acceleration of the front.)
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 B(x−B³t ²) + (p/B − tB²)² = 0,
Note the momentum distribution obtained by integrating over all Template:Mvar is constant. Since this is the probability density in momentum space, it is evident that the wave function itself is not normalizable.
The narrow-width limit of the Gaussian wave packet solution discussed is the free propagator kernel Template:Mvar. 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 Template:Mvar.
becomes a delta function, δ(x), so that its time evolution,
yields the propagator.
Note that a very narrow initial wave packet instantly becomes infinitely wide, but with a phase which is more rapidly oscillatory at large values of x. This might seem strange—the solution goes from being localized at one point to being "everywhere" at all later times, but it is a reflection of the enormous momentum uncertainty of a localized particle, as explained above.
Further note that the norm of the wave function is infinite, which is also correct, since the square of a delta function is divergent in the same way.
The factor involving Template:Mvar is an infinitesimal quantity which is there to make sure that integrals over Template:Mvar are well defined. In the limit that ε→0, Template:Mvar becomes purely oscillatory, and integrals of Template:Mvar 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,
In the limit when t is small, the propagator, of course, goes to a delta function,
To see this, note that the integral over all space of Template:Mvar equals 1 at all times,
since this integral is the inner-product of K with the uniform wave function. But the phase factor in the exponent has a nonzero spatial derivative everywhere except at the origin, and so when the time is small there are fast phase cancellations at all but one point. This is rigorously true when the limit ε→0 is taken at the very end.
So the propagation kernel is the (future) time evolution of a delta function, and it is continuous, in a sense: it goes to the initial delta function at small times. If the initial wave function is an infinitely narrow spike at position Template:Mvar,
it becomes the oscillatory wave,
Now, since every function can be written as a weighted sum of such narrow spikes,
Thus, this is a formal way to express the fundamental solution or general solution. The interpretation of this expression is that the amplitude for a particle to be found at point Template:Mvar at time Template:Mvar is the amplitude that it started at Template:Mvar, times the amplitude that it went from Template:Mvar to Template:Mvar, summed over all the possible starting points. In other words, it is a convolution of the kernel Template:Mvar with the arbitrary initial condition Template:Mvar0,
which can be interpreted as follows: the amplitude to travel from Template:Mvar to Template:Mvar in time Template:Mvar+Template:Mvar' is the sum of the amplitude to travel from Template:Mvar to Template:Mvar in time Template:Mvar, multiplied by the amplitude to travel from Template:Mvar to Template:Mvar in time Template:Mvar', 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 wave packets 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 (also see the heat equation),
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,
and, since the integral of ρt is constant while the width is becoming narrow at small times, this function approaches a delta function at t=0,
again only in the sense of distributions, so that
The spreading Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity,
which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator Template:Mvar,
which is the infinitesimal diffusion operator,
A matrix has two indices, which in continuous space makes it a function of Template:Mvar and Template:Mvar'. In this case, because of translation invariance, the matrix element Template:Mvar 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:
Translation invariance means that continuous matrix multiplication,
is essentially convolution,
The exponential can be defined over a range of ts which include complex values, so long as integrals over the propagation kernel stay convergent,
The limit of this expression for Template:Mvar approaching the pure imaginary axis is the above Schrödinger propagator encountered,
which illustrates the above time evolution of Gaussians.
From the fundamental identity of exponentiation, or path integration,
holds for all complex z values, where the integrals are absolutely convergent so that the operators are well defined.
Thus, quantum evolution of a Gaussian, which is the complex diffusion kernel K,
amounts to the time-evolved state,
This illustrates the above diffusive form of the complex Gaussian solutions,
- By contrast, the introduction of interaction terms in dispersive equations, such as for the quantum harmonic oscillator, may result in the emergence of envelope-non-dispersive, classical-looking solutions—see coherent states: Such "minimum uncertainty states" do saturate the uncertainty principle permanently.
|CitationClass=citation }} This annus mirabilis paper on the photoelectric effect was received by Annalen der Physik 18 March 1905.
|CitationClass=citation }} (Dover, 2010, ISBN 0-486-47722-3.)
- A simulation of a wave package in 2D (According to FOURIER-Synthesis in 2D)