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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, a wave packet (or wave train) is a short "burst" or "
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
" 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
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
s, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
, and hence the wave packet, are solutions of a
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
. Depending on the wave equation, the wave packet's profile may remain constant (no
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...
, see figure) or it may change (dispersion) while propagating.
Quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
ascribes a special significance to the wave packet; it is interpreted as a
probability amplitude In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the qu ...
, its
norm squared In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
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 The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
, and through its application it is possible to deduce the
time evolution Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
of a quantum mechanical system, similar to the process of the Hamiltonian formalism in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
. 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 The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of findi ...
. In the coordinate representation of the wave (such as the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
), 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 In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the Heisenberg
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
, and will be illustrated below.


Historical background

In the early 1900s, it became apparent that classical mechanics had some major failings.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
. It wasn't until the 1930s that the particle nature of light really began to be widely accepted in physics. The development of quantum mechanicsand its success at explaining confusing experimental resultswas 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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
. The energy of a photon is a function of its frequency, E = h\nu. The photon's energy is equal to Planck's constant, , multiplied by its frequency, . This resolved a problem in classical physics, called the
ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
. 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 The Afshar experiment is a variation of the double-slit experiment in quantum mechanics, devised and carried out by Shahriar Afshar while at the private, Boston-based Institute for Radiation-Induced Mass Studies (IRIMS). The results were presente ...
and the lively discussion around it.


Basic behaviors


Non-dispersive

As an example of propagation ''without dispersion'', consider wave solutions to the following
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
from
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
= c^2 \, \nabla^2 u, where is the speed of the wave's propagation in a given medium. Using the physics time convention, , the wave equation has
plane-wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...
solutions u(\mathbf,t) = e^, where \omega^2 =, \mathbf, ^2 c^2, and , \mathbf, ^2 = k_x^2 + k_y^2+ k_z^2. This relation between and should be valid so that the plane wave is a solution to the wave equation. It is called a
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
. To simplify, consider only waves propagating in one dimension (extension to three dimensions is straightforward). Then the general solution is u(x,t)= A e^ + B e^, in which we may take . The first term represents a wave propagating in the positive since it is a function of only; the second term, being a function of , represents a wave propagating in the negative . 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 \int^_ A(k) ~ e^ \, dk. As in the plane-wave case the wave packet travels to the right for , since , and to the left for , since . The factor comes from
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
conventions. The amplitude contains the coefficients of the linear superposition of the plane-wave solutions. These coefficients can in turn be expressed as a function of evaluated at by inverting the Fourier transform relation above: A(k) = \frac \int^_ u(x,0) ~ e^ \, dx. For instance, choosing u(x,0) = e^, we obtain A(k) = \frac e^, and finally \begin u(x,t) &= e^\\ &= e^ \left cos\left(2\pi \frac\right)+ i\sin\left(2\pi\frac\right)\right \end The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.


Dispersive

By contrast, as an example of propagation now ''with dispersion'', consider instead solutions to the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
(nondimensionalized with , , and ''ħ'' set equal to one), i = -\frac , yielding the dispersion relation \omega = \frac, \mathbf, ^2. Once again, restricting attention to one dimension, the solution to the Schrödinger equation satisfying the initial condition \psi(x,0)= \sqrt \exp\left(\right), representing a wave packet localized in space at the origin, is seen to be \begin \psi(x,t) &= \frac e^ ~ e^\\ &= \frac e^~ e^ ~. \end An impression of the dispersive behavior of this wave packet is obtained by looking at the probability density: , \psi(x,t), ^2 = \frac~e^~. It is evident that this dispersive wave packet, while moving with constant
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
, is delocalizing rapidly: it has a
width Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
increasing with time as , so eventually it diffuses to an unlimited region of space.By contrast, the introduction of ''interaction terms'' in dispersive equations, such as for the
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最� ...
, may result in the emergence of envelope-non-dispersive, classical-looking solutions—see
coherent states In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
: Such "minimum uncertainty states" do saturate the uncertainty principle permanently.
The momentum profile remains invariant. The
probability current In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is th ...
is j=\rho v = \frac (\psi^* \nabla \psi - \psi \nabla \psi^*)= \rho \left (k_0+\frac\right ) .


Gaussian wave packets in quantum mechanics

The above dispersive Gaussian wave packet, unnormalized and just centered at the origin, instead, at =0, can now be written in 3D, now in standard units: \psi(\mathbf,0) = e^, where is a positive real number, the ''square of the width of the wave packet'', a = 2\langle \mathbf r \cdot \mathbf r\rangle/3\langle 1\rangle = 2 (\Delta x)^2. The Fourier transform is also a Gaussian in terms of the wavenumber, =0, the k-vector, (with inverse width, 1/a = 2\langle\mathbf k\cdot \mathbf k\rangle/3\langle 1\rangle = 2 (\Delta p_x/\hbar)^2, so that \Delta x \Delta p_x = \hbar/2, i.e., it saturates the uncertainty relation), \psi(\mathbf,0) = (2\pi a)^ e^. Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is The inverse Fourier transform is still a Gaussian, but now the parameter has become complex, and there is an overall normalization factor. The integral of over all space is invariant, because it is the inner product of with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any
energy eigenstate A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ...
, the inner product, \langle \eta , \psi \rangle = \int \eta(\mathbf) \psi(\mathbf)d^3\mathbf, only changes in time in a simple way: its phase rotates with a frequency determined by the energy of . When has zero energy, like the infinite wavelength wave, it doesn't change at all. The integral is also invariant, which is a statement of the conservation of probability. Explicitly, in which is the width of at ; is the distance from the origin; the speed of the particle is zero; and the time origin can be chosen arbitrarily. The width of the Gaussian is the interesting quantity which can be read off from the probability density, , \sqrt. This width eventually grows linearly in time, as , indicating wave-packet spreading. For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., m) then the width of the packet doubles in about s. Clearly, particle wave packets spread out very rapidly indeed (in free space): For instance, after ms, the width will have grown to about a kilometer. This linear growth is a reflection of the (time-invariant) momentum uncertainty: the wave packet is confined to a narrow , and so has a momentum which is uncertain (according to the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
) by the amount , a spread in velocity of , and thus in the future position by . The uncertainty relation is then a strict inequality, very far from saturation, indeed! The initial uncertainty has now increased by a factor of (for large ).


The Airy wave train

In contrast to the above Gaussian wave packet, it has been observed that a particular wave function based on
Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solut ...
s, propagates freely without envelope dispersion, maintaining its shape. It accelerates undistorted in the absence of a force field: . (For simplicity, , , and ''B'' is a constant, cf.
nondimensionalization Nondimensionalization is the partial or full removal of dimensional analysis, physical dimensions from an mathematical equation, equation involving physical quantity, physical quantities by a suitable substitution of variables. This technique can ...
.) Nevertheless, there is no dissonance with Ehrenfest's theorem in this force-free situation, because the state is both non-normalizable and has an undefined (infinite) for all times. (To the extent that it could be defined, for all times, despite the apparent acceleration of the front.) In
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
, this is evident in the
pure state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
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 , W(x,p;t) = W(x-B^3 t^2, p-B^3 t ;0) = ~ \mathrm \left(2^ \left(Bx + - 2Bpt\right)\right). Note the momentum distribution obtained by integrating over all is constant. Since this is the probability density in momentum space, it is evident that the wave function itself is not normalizable. In 2018, the first experimental observation of the cubic phase of accelerating Airy wave packets was achieved by a collaboration of researchers from Israeli, German, and American universities.


Free propagator

The narrow-width limit of the Gaussian wave packet solution discussed is the free ''propagator kernel'' . 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 . Returning to one dimension for simplicity, with ''m'' and ħ set equal to one, when is the infinitesimal quantity , the Gaussian initial condition, rescaled so that its integral is one, \psi_0(x) = e^ \, becomes a
delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, , so that its time evolution, K_t(x) = e^\, 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 In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
is divergent in the same way. The factor involving is an infinitesimal quantity which is there to make sure that integrals over are well defined. In the limit that , becomes purely oscillatory, and integrals of 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) = e^ \, . In the limit when ''t'' is small, the propagator goes to a delta function \lim_ 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 Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
gives the value of the test function at zero. To see this, note that the integral over all space of equals 1 at all times, \int K_t(x) dx = 1 \, , 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 , \psi_0(x) = \delta(x - y) \, , it becomes the oscillatory wave, \psi_t(x) = e^ \, . Now, since every function can be written as a weighted sum of such narrow spikes, \psi_0(x) = \int \psi_0(y) \delta(x-y) dy \, , the time evolution of ''every function'' 0 is determined by this propagation kernel , Thus, this is a formal way to express the
fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
or ''general solution''. The interpretation of this expression is that the amplitude for a particle to be found at point at time is the amplitude that it started at , times the amplitude that it went from to , ''summed over all the possible starting points''. In other words, it is a
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the kernel with the arbitrary initial condition , \psi_t = K * \psi_0 \, . Since the amplitude to travel from to after a time +' can be considered in two steps, the propagator obeys the composition identity, \int K(x-y;t)K(y-z;t')dy = K(x-z;t+t')~ , which can be interpreted as follows: the amplitude to travel from to in time +' is the sum of the amplitude to travel from to in time , multiplied by the amplitude to travel from to in time ', 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 Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...
. For a particle which is randomly walking, the probability density function at any point satisfies the
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's la ...
(also see the
heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
), \rho = \rho ~, where the factor of 2, which can be removed by rescaling either time or space, is only for convenience. A solution of this equation is the spreading Gaussian, \rho_t(x) = e^ ~, 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, \lim_ \rho_t(x) = \delta(x) again only in the sense of distributions, so that \lim_ \int_x f(x) \rho_t(x) = f(0) for any smooth
test function Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
. The spreading Gaussian is the propagation kernel for the diffusion equation and it obeys the
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
identity, K_ = K_*K_ \, , which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator , K_t(x) = e^ \, , which is the infinitesimal diffusion operator, H= - \, . A matrix has two indices, which in continuous space makes it a function of and '. In this case, because of translation invariance, the matrix element 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_ A(x,x')B(x',x'') \, , is essentially convolution, C(\Delta) = C(x-x'') = \int_ A(x-x') B(x'-x'') = \int_ A(\Delta-y)B(y) \, . The exponential can be defined over a range of ''t''s which include complex values, so long as integrals over the propagation kernel stay convergent, K_z(x) = e^ \, . As long as the real part of is positive, for large values of , is exponentially decreasing, and integrals over are indeed absolutely convergent. The limit of this expression for approaching the pure imaginary axis is the above Schrödinger propagator encountered, K_t^ = K_ = e^ \, , which illustrates the above time evolution of Gaussians. From the fundamental identity of exponentiation, or path integration, K_z * K_ = K_ \, 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'', \psi_0(x) = K_a(x) = K_a * \delta(x) \, amounts to the time-evolved state, \psi_t = K_ * K_a = K_ \, . This illustrates the above diffusive form of the complex Gaussian solutions, \psi_t(x) = e^ \, .


See also

*
Wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
*
Wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative ...
*
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
*
Group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
*
Phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
*
Free particle In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
*
Coherent states In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
*
Waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electro ...
*
Wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
*
Matter wave Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wav ...
*
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
*
Introduction to quantum mechanics Quantum mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the be ...
*
Soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...


Remarks


Notes


References

* This
annus mirabilis ''Annus mirabilis'' (pl. ''anni mirabiles'') is a Latin phrase that means "marvelous year", "wonderful year", "miraculous year", or "amazing year". This term has been used to refer to several years during which events of major importance are re ...
paper on the photoelectric effect was received by ''Annalen der Physik'' 18 March 1905. * * * * * * * * (
Dover Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maids ...
, 2010, .) *


External links

* *
1d Wave packet plot in Google

1d Wave train and probability density plot in Google

2d Wave packet plot in Google

2d Wave train plot in Google

2d probability density plot in Google

Quantum physics online : Interactive simulation of a free wavepacket


Interactive 2D wave packet dynamics simulation
A simulation of a wave package in 2D (According to FOURIER-Synthesis in 2D)
* {{Citation, last=Curtfirst=T.L., url=http://www.physics.miami.edu/~curtright/TimeDependentWignerFunctions.html, title=Time-dependent Wigner Functions Wave mechanics Quantum mechanics