Wave packet explained

In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an envelope. A wave packet can be analyzed into, or can be synthesized from, a potentially-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. Any signal of a limited width in time or space requires many frequency components around a center frequency within a bandwidth inversely proportional to that width; even a gaussian function is considered a wave packet because its Fourier transform is a "packet" of waves of frequencies clustered around a central frequency.^[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) or it may change (dispersion) while propagating.

Historical background

Ideas related to wave packets – modulation, carrier waves, phase velocity, and group velocity – date from the mid-1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.

Erwin Schrödinger introduced the idea of wave packets just after publishing his famous wave equation.^[2] He solved his wave equation for a quantum harmonic oscillator, introduced the superposition principle, and used it to show that a compact state could persist. While this work did result in the important concept of coherent states, the wave packet concept did not endure. The year after Schrödinger's paper, Werner Heisenberg published his paper on the uncertainty principle, showing in the process, that Schrödinger's results only applied to quantum harmonic oscillators, not for example to Coulomb potential characteristic of atoms.^[2]

The following year, 1927, Charles Galton Darwin explored Schrödinger's equation for an unbound electron in free space, assuming an initial Gaussian wave packet.^[3] Darwin showed that at time

t later the position x of the packet traveling at velocity v would be

$$x\_0\; +\; vt\; \backslash pm\; \backslash sqrt$$

where

\sigma is the uncertainty in the initial position.

Later in 1927 Paul Ehrenfest showed that the time,

T for a matter wave packet of width \Delta x and mass m to spread by a factor of 2 was $T\backslash approx\; \backslash Delta\; x\; \backslash sqrt$. Since \hbar is so small, wave packets on the scale of macroscopic objects, with large width and mass, double only at cosmic time scales.^[2]

Significance in quantum mechanics

Quantum mechanics describes the nature of atomic and subatomic systems using Schrödinger's wave equation. The classical limit of quantum mechanics and many formulations of quantum scattering use wave packets formed from various solutions to this equation.Quantum wave packet profiles change while propagating; they show dispersion. Physicists have concluded that "wave packets would not do as representations of subatomic particles".^[2]

Wave packets and the classical limit

Schrodinger developed wave packets in hopes of interpreting quantum wave solutions as locally compact wave groups.^[2] Such packets tradeoff position localization for spreading momentum. In the coordinate representation of the wave (such as the Cartesian coordinate system), the position of the particle's localized probability is specified by the position of the packet solution. The narrower the spatial wave packet, and therefore the better localized the position of the wave packet, the larger the spread in the momentum of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the Heisenberg uncertainty principle.

One kind of optimal tradeoff minimizes the product of position uncertainty

\Delta x and momentum uncertainty \Delta p _{x} .^[4] If we place such a packet at rest it stays at rest: the average value of the position and momentum match a classical particle. However it spreads out in all directions with a velocity given by the optimal momentum uncertainty \Delta p _{x} . The spread is so fast that in the distance of once around an atom the wave packet is unrecognizable.

Wave packets and quantum scattering

Particle interactions are called scattering in physics; wave packet mathematics play an important role in quantum scattering approaches. A monochromatic (single momentum) source produces convergence difficulties in the scattering models.^[5] Scattering problems also have classical limits. Whenever the scattering target (for example an atom) has a size much smaller than wave packet, the center of the wave packet follows scattering classical trajectories. In other cases, the wave packet distorts and scatters as it interacts with the target.^[6]

Basic behaviors

See also: Dispersion (water waves) and Dispersion (optics).

Non-dispersive

Without dispersion the wave packet maintains its shape as it propagates.As an example of propagation without dispersion, consider wave solutions to the following wave equation from classical physics$$=\; c^2\; \backslash ,\; \backslash nabla^2\; u,$$

where is the speed of the wave's propagation in a given medium.

Using the physics time convention,, the wave equation has plane-wave solutions$$u(\backslash mathbf,t)\; =\; e^,$$

where$$\backslash omega^2\; =|\backslash mathbf|^2\; c^2,$$ and

| k | ^{2} = k ^{2} _{x} + k ^{2+} _{y} k ^{2 .} _{z}

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.

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$$u(x,t)\; =\; \backslash frac\; \backslash int^\_\; A(k)\; ~\; e^\; \backslash ,\; dk.$$

As in the plane-wave case the wave packet travels to the right for, since, and to the left for, since .

The factor

1 / \sqrt{ 2 \pi} comes from Fourier transform 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:$$A(k)\; =\; \backslash frac\; \backslash int^\_\; u(x,0)\; ~\; e^\; \backslash ,\; dx.$$

For instance, choosing$$u(x,0)\; =\; e^,$$

we obtain$$A(k)\; =\; \backslash frac\; e^,$$

and finally

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 dispersion where a wave changes shape during propagation, consider instead solutions to the free Schrödinger equation (nondimensionalized with,, and ħ set equal to one),$$i\; =\; -\backslash frac,$$yielding the dispersion relation^[7] $$\backslash omega\; =\; \backslash frac|\backslash mathbf|^2.$$

Once again, restricting attention to one dimension, the solution to the Schrödinger equation satisfying the initial condition $\backslash psi(x,0)=\; \backslash sqrt[4]\; \backslash exp\backslash left(\backslash right)$, 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:$$|\backslash psi(x,t)|^2\; =\; \backslash frac~e^~.$$It is evident that this dispersive wave packet, while moving with constant group velocity, is delocalizing rapidly: it has a width increasing with time as, so eventually it diffuses to an unlimited region of space.^[8]

The momentum profile remains invariant. The probability current is$$j=\backslash rho\; v\; =\; \backslash frac\; (\backslash psi^*\; \backslash nabla\; \backslash psi\; -\; \backslash psi\; \backslash nabla\; \backslash psi^*)=\; \backslash rho\; \backslash left\; (k\_0+\backslash frac\backslash right)\; .$$

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:$$\backslash psi(\backslash mathbf,0)\; =\; e^,$$where is a positive real number, the square of the width of the wave packet,$$a\; =\; 2\backslash langle\; \backslash mathbf\; r\; \backslash cdot\; \backslash mathbf\; r\backslash rangle/3\backslash langle\; 1\backslash rangle\; =\; 2\; (\backslash Delta\; x)^2.$$

The Fourier transform is also a Gaussian in terms of the wavenumber,the k-vector, (with inverse width,$$1/a\; =\; 2\backslash langle\backslash mathbf\; k\backslash cdot\; \backslash mathbf\; k\backslash rangle/3\backslash langle\; 1\backslash rangle\; =\; 2\; (\backslash Delta\; p\_x/\backslash hbar)^2,$$so that$$\backslash Delta\; x\; \backslash Delta\; p\_x\; =\; \backslash hbar/2,$$i.e., it satisfies the uncertainty relation),$$\backslash psi(\backslash mathbf,0)\; =\; (2\backslash pi\; a)^\; e^.$$

Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution isThe inverse Fourier transform is still a Gaussian, but now the parameter has become complex, and there is an overall normalization factor.^[4]

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, the inner product,$$\backslash langle\; \backslash eta\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\; \backslash eta(\backslash mathbf)\; \backslash psi(\backslash mathbf)d^3\backslash mathbf,$$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,,

$$\backslash sqrt.$$This width eventually grows linearly in time, as, indicating wave-packet spreading.^[9]

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) 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).

2D

A gaussian 2D quantum wave function:

\psi(x,y,t) = \psi(x,t) \psi(y,t) \psi (x,t) = ( (2 a ^{2})/(\pi) ) ^{1/4} (e ^{i})/((a ^{4} + (4 \hbar ^{2} t ^{2})/(m ^{2}) ) ^{1/4}) e ^{i k _{0} x} \exp [ - ((x - (\hbar k _{0})/(m) t ) ^{2})/(a ^{2} + (2 i \hbar t)/(m)) ]

where

\phi = - \theta - (\hbar k ^{2} _{0})/(2 m) t tg ( 2 \theta) = (2 \hbar t)/(m a ^{2}) ^[10]

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:$$\backslash psi\; =\; \backslash operatorname(B(x-B^3t^2))\; \backslash ,\; e^\; \backslash ,\; .$$(For simplicity,,, and B is a constant, cf. nondimensionalization.)

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, 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,$$W(x,p;t)\; =\; W(x-B^3\; t^2,\; p-B^3\; t\; ;0)\; =\; ~\; \backslash mathrm\; \backslash left(2^\; \backslash left(Bx\; +\; -\; 2Bpt\backslash right)\backslash right).$$

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.

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,$$\backslash psi\_0(x)\; =\; e^\; \backslash ,$$becomes a delta function,, so that its time evolution,$$K\_t(x)\; =\; e^\backslash ,$$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 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,$$K\_t(x,y)\; =\; K\_t(x-y)\; =\; e^\; \backslash ,\; .$$

In the limit when t is small, the propagator goes to a delta function$$\backslash lim\_\; K\_t(x-y)\; =\; \backslash delta(x-y)\; ~,$$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 equals 1 at all times,$$\backslash int\; K\_t(x)\; dx\; =\; 1\; \backslash ,,$$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,$$\backslash psi\_0(x)\; =\; \backslash delta(x\; -\; y)\; \backslash ,,$$it becomes the oscillatory wave,$$\backslash psi\_t(x)\; =\; e^\; \backslash ,\; .$$

Now, since every function can be written as a weighted sum of such narrow spikes,$$\backslash psi\_0(x)\; =\; \backslash int\; \backslash psi\_0(y)\; \backslash delta(x-y)\; dy\; \backslash ,,$$the time evolution of every function ₀ is determined by this propagation kernel,

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 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 of the kernel with the arbitrary initial condition,$$\backslash psi\_t\; =\; K\; *\; \backslash psi\_0\; \backslash ,\; .$$

Since the amplitude to travel from to after a time +' can be considered in two steps, the propagator obeys the composition identity,$$\backslash int\; K(x-y;t)K(y-z;t\text{'})dy\; =\; K(x-z;t+t\text{'})~,$$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

See also: Heat kernel. 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),$$\backslash rho\; =\; \backslash rho\; ~,$$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,$$\backslash rho\_t(x)\; =\; e^\; ~,$$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,$$\backslash lim\_\; \backslash rho\_t(x)\; =\; \backslash delta(x)$$again only in the sense of distributions, so that$$\backslash lim\_\; \backslash int\_x\; f(x)\; \backslash rho\_t(x)\; =\; f(0)$$for any smooth test function .

The spreading Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity,$$K\_\; =\; K\_*K\_\; \backslash ,,$$which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator,$$K\_t(x)\; =\; e^\; \backslash ,,$$which is the infinitesimal diffusion operator,$$H=\; -\; \backslash ,\; .$$

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:$$K\_t(x,x\text{'})\; =\; K\_t(x-x\text{'})\; \backslash ,\; .$$

Translation invariance means that continuous matrix multiplication,$$C(x,x$$) = \int_ A(x,x')B(x',x) \,,is essentially convolution,$$C(\backslash Delta)\; =\; C(x-x$$) = \int_ A(x-x') B(x'-x) = \int_ A(\Delta-y)B(y) \, .

The exponential can be defined over a range of ts which include complex values, so long as integrals over the propagation kernel stay convergent,$$K\_z(x)\; =\; e^\; \backslash ,\; .$$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,$$K\_t^\; =\; K\_\; =\; e^\; \backslash ,,$$which illustrates the above time evolution of Gaussians.

From the fundamental identity of exponentiation, or path integration,$$K\_z\; *\; K\_\; =\; K\_\; \backslash ,$$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,$$\backslash psi\_0(x)\; =\; K\_a(x)\; =\; K\_a\; *\; \backslash delta(x)\; \backslash ,$$amounts to the time-evolved state,$$\backslash psi\_t\; =\; K\_\; *\; K\_a\; =\; K\_\; \backslash ,\; .$$

This illustrates the above diffusive form of the complex Gaussian solutions,$$\backslash psi\_t(x)\; =\; e^\; \backslash ,\; .$$

See also

• Wave
• Wave propagation
• Fourier analysis
• Group velocity
• Phase velocity
• Free particle
• Coherent states
• Waveform
• Wavelet
• Matter wave
• Pulse (signal processing)
• Pulse (physics)
• Schrödinger equation
• Introduction to quantum mechanics
• Soliton

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
• Web-Schrödinger: Interactive 2D wave packet dynamics simulation
• A simulation of a wave package in 2D (According to FOURIER-Synthesis in 2D)

Notes and References

1. Web site: Schwartz . Matthew . Lecture 11: Wavepackets and dispersion . live . https://web.archive.org/web/20230318213306/https://scholar.harvard.edu/files/schwartz/files/lecture11-wavepackets.pdf . 2023-03-18 . 2023-06-22 . scholar.harvard.edu.
2. Book: Kragh, Helge . Compendium of Quantum Physics . https://link.springer.com/10.1007/978-3-540-70626-7_232 . Wave Packet . 2009 . Springer Berlin Heidelberg . 978-3-540-70622-9 . Greenberger . Daniel . Berlin, Heidelberg . 828–830 . en . 10.1007/978-3-540-70626-7_232 . Hentschel . Klaus . Weinert . Friedel.
3. Darwin, Charles Galton. "Free motion in the wave mechanics." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117.776 (1927): 258-293.
4. Book: Schiff, Leonard I. . Quantum mechanics . 1995 . McGraw-Hill . 978-0-07-055287-6 . 3. ed., 29. print . International series in pure and applied physics . New York.
5. Book: Newton, Roger G. . Scattering theory of waves and particles . 1982 . Springer . 978-0-387-10950-3 . 2. Texts and monographs in physics . New York, Heidelberg, Berlin.
6. Book: Susskind . Leonard . Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics] . Friedman . Art . Susskind . Leonard . 2014 . Basic Books . 978-0-465-08061-8 . The theoretical minimum / Leonard Susskind and George Hrabovsky . New York, NY.
7. Book: Hall, Brian C. . Quantum Theory for Mathematicians . Springer . New York Heidelberg Dordrecht London . 2013 . 978-1-4614-7115-8 . 91-92.
8. 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.
9. Darwin, C. G. (1927). "Free motion in the wave mechanics", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117 (776), 258-293.
10. Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, complement G_I, §3-a