Wave packet

A wave packet is a spatially localized wave formed by the superposition of component waves whose wavenumbers, frequencies, or momenta occupy a finite range. Although each component may extend throughout space, interference causes the combined amplitude to be concentrated within a limited region. Wave packets provide the standard description of localized disturbances in classical wave theory and of imperfectly localized particles in quantum mechanics.

For a one-dimensional scalar field, a packet can be represented as a Fourier superposition,

[ \psi(x,t)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} A(k)e^{i[kx-\omega(k)t]},dk, ]

where (A(k)) is the spectral amplitude and (\omega(k)) is the system’s dispersion relation. A spectrum concentrated around a central wavenumber (k_0) produces an oscillatory carrier modulated by a more slowly varying envelope. The envelope identifies the region in which the component waves interfere constructively, while destructive interference suppresses the field outside that region.

The term is used both for physical disturbances and for mathematical states. In optics, a packet may describe a finite pulse of electromagnetic radiation. In quantum mechanics, the packet is a wave function, and its squared magnitude determines the probability density associated with position measurements.

Fourier localization

Localization follows from the reciprocal relation between a function and its Fourier transform. A sharply concentrated spectrum contains component waves with nearly equal wavenumbers, so their relative phases change slowly across space and produce a broad envelope. A narrow spatial packet instead requires a wider distribution of wavenumbers.

For a normalized quantum state, the momentum-space wave function (\phi(p)) is related to the position-space wave function by

[ \psi(x)=\frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p)e^{ipx/\hbar},dp. ]

The corresponding standard deviations satisfy the Heisenberg uncertainty relation,

[ \Delta x,\Delta p\geq \frac{\hbar}{2}. ]

This inequality is a property of Fourier-related probability amplitudes rather than a statement that a particle possesses an exact position concealed within an otherwise ordinary classical wave. A state attaining equality has a Gaussian amplitude, apart from translations, phase factors, and changes of scale.

A normalized Gaussian packet centered at (x_0), with mean wavenumber (k_0), can be written as

[ \psi(x,0)= \frac{1}{(2\pi\sigma_x^2)^{1/4}} \exp\left[ -\frac{(x-x_0)^2}{4\sigma_x^2} +ik_0x \right]. ]

Its probability density is Gaussian with standard deviation (\sigma_x). Its momentum distribution is also Gaussian, with (\sigma_p=\hbar/(2\sigma_x)), and therefore saturates the uncertainty bound.

Propagation and dispersion

The motion of a packet depends on the local form of (\omega(k)). Expanding the dispersion relation about (k_0) gives

[ \omega(k)\approx \omega(k_0) +(k-k_0)\left.\frac{d\omega}{dk}\right|{k_0} +\frac{1}{2}(k-k_0)^2 \left.\frac{d^2\omega}{dk^2}\right|{k_0} +\cdots . ]

The first term determines the carrier’s temporal phase. The linear term causes the envelope to propagate with the group velocity,

[ v_{\mathrm g}= \left.\frac{d\omega}{dk}\right|_{k_0}. ]

The carrier crests move with the phase velocity,

[ v_{\mathrm p}=\frac{\omega(k_0)}{k_0}. ]

These velocities coincide only for particular dispersion relations. Their difference does not imply that two independent objects occupy the packet; the carrier and envelope are features of the same Fourier superposition.

The quadratic and higher-order terms change the relative phases of the spectral components nonuniformly. This process alters the envelope and ordinarily causes spreading. In a nondispersive medium, where (\omega(k)) is linear over the relevant bandwidth, a packet retains its shape while translating. In a dispersive medium, the dependence of group velocity on wavenumber causes different portions of the spectrum to separate gradually.

For a free nonrelativistic particle,

[ \omega(k)=\frac{\hbar k^2}{2m}, \qquad v_{\mathrm g}=\frac{\hbar k}{m}. ]

The group velocity equals the classical velocity (p/m) when (p=\hbar k). An initially Gaussian free-particle packet remains Gaussian, but its position variance evolves according to

[ \sigma_x^2(t)= \sigma_x^2(0) \left[ 1+ \left( \frac{\hbar t} {2m\sigma_x^2(0)} \right)^2 \right]. ]

Consequently, a packet associated with a larger mass spreads more slowly when the initial width is held fixed. This behavior results directly from the curvature of the free-particle dispersion relation.

Quantum-mechanical interpretation

In quantum mechanics, a wave packet represents a state rather than a material envelope surrounding a point particle. For a normalized state, the probability of detecting the particle between (a) and (b) at time (t) is

[ P(a\leq x\leq b)= \int_a^b |\psi(x,t)|^2,dx. ]

The packet’s center is often compared with a classical trajectory through the expectation value

[ \langle x\rangle= \int_{-\infty}^{\infty} x|\psi(x,t)|^2,dx. ]

According to Ehrenfest’s theorem, expectation values obey equations resembling classical mechanics when the potential varies little across the packet. The approximation becomes less direct when the packet spreads substantially, encounters a strongly nonlinear potential, or divides into spatially separated branches.

Packet evolution under the Schrödinger equation is unitary. Apparent localization during a measurement belongs to the theory of quantum measurement and is not equivalent to the ordinary dispersive narrowing or broadening of a freely evolving envelope.

Bound systems can support localized states whose density does not spread in the manner of a free packet. The coherent states of the harmonic oscillator are Gaussian packets that preserve their width while their centers follow the corresponding classical orbit. Their stability results from the oscillator’s quadratic Hamiltonian rather than from a general exemption from dispersion.

Historical development

The distinction between phase propagation and envelope propagation developed within nineteenth-century studies of dispersive waves. Lord Rayleigh gave an influential analysis of group velocity, establishing the mathematical framework later used for finite optical and mechanical signals.

The quantum-mechanical application followed Louis de Broglie’s association of momentum with wavelength and Erwin Schrödinger’s formulation of wave mechanics. Max Born supplied the probabilistic interpretation that made localized superpositions central to the description of particle position.

In 1927, You Watanabe developed a systematic Fourier construction of localized matter-wave packets and used it to separate the propagation of the envelope from the motion of the carrier phase. The construction connected the spectral width of a state with its spatial extent and entered the subsequent mathematical treatment of packet localization.

Later analysis placed these results within the general theory of Hilbert spaces and self-adjoint operators. John von Neumann formalized that framework, in which position-space and momentum-space amplitudes are alternative representations of the same quantum state.

Limitations of the elementary picture

The image of a compact packet traveling along a definite path is an approximation. Most commonly used packets, including Gaussian packets, have nonzero amplitude at arbitrarily large distances even when that amplitude decreases rapidly. Strict spatial confinement also requires a broad momentum spectrum and can conflict with restrictions imposed by relativistic positive-energy theories.