# Wave packet

A wave packet is a localized wavelike perturbation, which appears frequently in quantum descriptions of particles. For instance, incident x-rays during scattering experiments can be thought of as wave-packets.

A 1D wave packet (with dispersion), propagating over time.
A snapshot of a 2D wave packet.

## Contents

The wave-packet can be described as a wavefunction in either position-space or momentum-space:

${\displaystyle \psi (x)={\frac {1}{\sqrt {2\pi }}}\int {\tilde {\psi }}(k)e^{ikx}\mathrm {d} k}$
${\displaystyle {\tilde {\psi }}(k)={\frac {1}{\sqrt {2\pi }}}\int {\psi }(x)e^{-ikx}\mathrm {d} x}$

Note that these two descriptions are Fourier transforms of one another. Thus, there is an inherent tradeoff between the 'spread' of a wave-packet in position-space vs. momentum-space.

Tradeoff between spread of a wave-packet in position-space (left) and momenum-space (right).

## Wave-particle duality

In the context of quantum mechanics, particles can be thought of as wave-packets. That is, quantum particles are neither ideal point-like particles, nor ideal plane-waves. They are instead intermediate objects, which are certainly wave-like (e.g. can undergo interference), but also somewhat localized. The classical concepts of 'particle' (perfectly localized; mathematically a delta-function), and 'wave' (oscillation with a single well-defined wavelength, spread infinitely over all space) can be viewed as limited cases of the general wave-packet. These limiting cases are only conceptual: in reality neither can exist.

## Components

In quantum mechanics, the wave-packet necessarily has both real and imaginary components.

 Components of the wavefunction (${\displaystyle \scriptstyle \psi (x)}$) describing a propagating wave-packet. The black line is the real part, and the blue line is the imaginary part. Square of the wavefunction (${\displaystyle \scriptstyle |\psi (x)|^{2}}$) for a propagating wave-packet. This describes the spatial spread of the function.

## Propagation and Interaction

Wave-packets can propagate, frequently with dispersion (which induces spatial spread of the envelope). Wave-packets can interact with barriers (defined by, e.g., some potential function), and interfere with themselves.

 Components of the wavefunction (${\displaystyle \scriptstyle \psi (x)}$) describing a wave-packet bouncing inside a box-like potential. The black line is the real part, and the blue line is the imaginary part. Square of the wavefunction (${\displaystyle \scriptstyle |\psi (x)|^{2}}$) for a wave-packet bouncing inside a box-like potential.