Wave packet

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A wave packet without dispersion (real or imaginary part) Wave packet (no dispersion).gif
A wave packet without dispersion (real or imaginary part)
A wave packet with dispersion Wave packet (dispersion).gif
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. [1] 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.

Wavenumber spatial frequency of a wave

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

Wave function mathematical description of the quantum state of a system; complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it

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 made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ or Ψ.

Wave equation Second-order linear differential equation important in physics

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

Contents

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.

Quantum mechanics Branch of physics that acts as an abstract framework formulating all the laws of nature

Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

Probability amplitude

In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.

Schrödinger equation Linear partial differential equation whose solution describes the quantum-mechanical system.

The Schrödinger equation is a linear partial differential equation that describes the wave function or state 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 the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

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.

Cartesian coordinate system Coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Momentum conserved physical quantity related to the motion of a body

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction in three-dimensional space. If m is an object's mass and v is the velocity, then the momentum is

Werner Heisenberg German theoretical physicist

Werner Karl Heisenberg was a German theoretical physicist and one of the key pioneers of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series of papers with Max Born and Pascual Jordan, during the same year, this matrix formulation of quantum mechanics was substantially elaborated. He is known for the Heisenberg uncertainty principle, which he published in 1927. Heisenberg was awarded the 1932 Nobel Prize in Physics "for the creation of quantum mechanics".

Historical 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 a photon is a function of its frequency,

Isaac Newton Influential British physicist and mathematician

Sir Isaac Newton was an English mathematician, physicist, astronomer, theologian, and author who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

Electromagnetism Branch of science concerned with the phenomena of electricity and magnetism

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields, is responsible for electromagnetic radiation such as light, and is one of the four fundamental interactions in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force.

[2]

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

Ultraviolet catastrophe the problem that the Rayleigh–Jeans law for blackbody radiation is accurate at low frequencies but not at high frequencies, solved by quantum physics

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 will emit radiation in all frequency ranges, emitting more energy as the frequency increases. By calculating the total amount of radiated energy, it can be shown that a blackbody is likely to release an arbitrarily high amount of energy. This would cause all matter to instantaneously radiate all of its energy until it is near absolute zero - indicating that a new model for the behaviour of blackbodies was needed.

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.

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 presented at a Harvard seminar in March 2004. The experiment gives information about which of two paths a photon takes through the apparatus while simultaneously allowing interference between the two paths to be observed, by showing that a grid of wires, placed at the nodes of the interference pattern, does not alter the beams. Afshar claimed that the experiment violates the principle of complementarity of quantum mechanics, which states roughly that the particle and wave aspects of quantum objects cannot be observed at the same time, and specifically the Englert–Greenberger duality relation. The experiment has been repeated by a number of investigators and its results have been confirmed, but its interpretation is controversial, and some disagree that it violates complementarity, while also disagreeing amongst themselves as to why.

Basic behaviors

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

Non-dispersive

As an example of propagation without dispersion, consider wave solutions to the following wave equation from classical physics

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

where

, and

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 12 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

we obtain

and finally

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

Dispersive

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

By contrast, as an example of propagation now with dispersion, consider instead solutions to the Schrödinger equation (Pauli 2000, 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 , representing a wave packet localized in space at the origin, 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 1 + 4t² → 2t, so eventually it diffuses to an unlimited region of space. [nb 1]

The momentum profile A(k) remains invariant. The probability current is

Gaussian wave packets in quantum mechanics

Superposition of 1D plane waves (blue) that sum to form a quantum Gaussian wave packet (red) that propagates to the right while spreading. Blue dots follow each plane wave's phase velocity while the red line follows the central group velocity. Wavepacket1.gif
Superposition of 1D plane waves (blue) that sum to form a quantum Gaussian wave packet (red) that propagates to the right while spreading. Blue dots follow each plane wave's phase velocity while the red line follows the central group velocity.
Position space probability density of an initially Gaussian state trapped in an infinite potential well experiencing periodic Quantum Tunneling in a centered potential wall. Gaussian wavepacket tunneling in potential well.gif
Position space probability density of an initially Gaussian state trapped in an infinite potential well experiencing periodic Quantum Tunneling in a centered potential wall.
1D Gaussian wave packet, shown in the complex plane, for a=2 and k=4 Wavepacket-a2k4-en.gif
1D Gaussian wave packet, shown in the complex plane, for a=2 and k=4

The above dispersive Gaussian wave packet, unnormalized and just centered at the origin, instead, at t=0, can now be written in 3D, now in standard units: [3] [4]

where a is a positive real number, the square of the width of the wave packet,

The Fourier transform is also a Gaussian in terms of the wavenumber, t=0, the k-vector, (with inverse width,

so that

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 inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor. [5]

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 √a 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): [6] For instance, after 1 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 Δx=a/2, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount ħ/2a, a spread in velocity of ħ/m2a, and thus in the future position by ħt /m2a. 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 (for large t).

The Airy wave train

In contrast to the above Gaussian wave packet, it has been observed [7] 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(xB³t ²)) exp(iB³t(x−2B³t²/3)). (For simplicity, ħ=1, m=1/2, and B is a constant, cf. nondimensionalization.)

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

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) x for all times. (To the extent that it could 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(xB³t ²) + (p/BtB²)² = 0, [8]

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

Free propagator

The narrow-width limit of the Gaussian wave packet solution discussed is the free propagator kernel K. For other differential equations, this is usually called the Green's function, [9] 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, with m and ħ set equal to one, when a is the infinitesimal quantity ε, the Gaussian initial condition, rescaled so that its integral is one,

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 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,

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

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,

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 y,

it becomes the oscillatory wave,

Now, since every function can be written as a weighted sum of such narrow spikes,

the time evolution of every functionψ0 is determined by this propagation kernel K,

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 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,

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

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. [10]

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 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

for any smooth test function f.

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 H,

which is the infinitesimal diffusion operator,

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:

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,

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 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,

See also

Remarks

  1. 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.

Notes

  1. Manners 2000
  2. Einstein 1905
  3. Pauli 2000
  4. Abers & Pearson 2004
  5. Schiff 1968
  6. Fitzpatrick
  7. Berry & Balazs 1979
  8. From a general pedagogy web-site by Curtright.
  9. Jackson 1975
  10. Feynman & Hibbs 1965

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References