Wave Packets

A single plane wave $\psi_k(x,t) = e^{i(kx - \omega t)}$ has a perfectly definite momentum $p = \hbar k$ but is completely delocalized — it has the same amplitude everywhere in space. Real particles are not like that: an electron fired through a slit passes through a finite region, and a neutron detected on a screen lands at a definite spot. To model a localized particle we need a different kind of wave function.

Superposing many wavelengths

The key idea is Fourier's theorem: any reasonably well-behaved function can be built by adding together sinusoids of different frequencies. Starting from plane waves, we write

$\psi(x,0) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \phi(k)\, e^{ikx}\, dk,$

where $\phi(k)$ is the momentum-space amplitude — it tells us how much of each wave vector $k$ (each momentum $p = \hbar k$ ) is present. If $\phi(k)$ is sharply peaked around some central value $k_0$ , then waves with nearby wavelengths interfere constructively near $x = 0$ and destructively everywhere else. The result is a wave packet: a lump that is localized in space while still being a valid quantum state.

The Gaussian wave packet

The cleanest example is the Gaussian packet. At $t = 0$ in position space,

$\psi(x,0) = \left(\frac{1}{2\pi\,\sigma_x^2}\right)^{1/4} \exp\!\left(-\frac{x^2}{4\sigma_x^2}\right) e^{ik_0 x},$

where $\sigma_x$ is the spatial width (standard deviation) and $e^{ik_0 x}$ gives it a central momentum $p_0 = \hbar k_0$ . Taking the Fourier transform shows that the corresponding momentum-space profile is also Gaussian,

$\phi(k) = \left(\frac{2\sigma_x^2}{\pi}\right)^{1/4} \exp\!\left(-\sigma_x^2 (k - k_0)^2\right),$

with momentum-space width $\sigma_k = 1/(2\sigma_x)$ .

Converting to physical momentum $p = \hbar k$ gives $\sigma_p = \hbar\sigma_k = \hbar/(2\sigma_x)$ . Therefore

$\sigma_x\,\sigma_p = \frac{\hbar}{2}.$

The Gaussian packet is the minimum-uncertainty state: it saturates Heisenberg's inequality $\sigma_x\,\sigma_p \geq \hbar/2$ . Every other wave packet shape spreads the product further. This is not an accident of the Gaussian choice — it is a direct consequence of the Fourier relationship between position and momentum representations.

The width–bandwidth trade-off

The product $\sigma_x \sigma_k = 1/2$ that emerges from the Fourier transform is a purely mathematical identity, independent of quantum mechanics: any signal localized in space of width $\sigma_x$ requires a spread of wave vectors of at least $\sigma_k \sim 1/(2\sigma_x)$ . Quantum mechanics enters when we set $p = \hbar k$ , converting the bandwidth into a momentum spread. A narrower packet in position automatically means a broader spread in momentum, and vice versa.

This trade-off has a direct physical consequence: a free particle's wave packet spreads over time. Each component $e^{i(kx-\omega t)}$ travels at its own phase velocity $v_\phi = \omega/k$ . For a free particle, $E = p^2/(2m)$ gives $\omega = \hbar k^2/(2m)$ , so components with larger $|k|$ travel faster. After a time $t$ , the packet broadens: a Gaussian of initial width $\sigma_x$ spreads to

$\sigma_x(t) = \sigma_x\sqrt{1 + \left(\frac{\hbar t}{2m\sigma_x^2}\right)^2}.$

At short times the growth is negligible; at long times $\sigma_x(t)$ grows linearly in $t$ , driven by the momentum spread of the packet.

What the wave packet means for measurement

Before measurement, a wave packet represents a particle that is probably near the center of the packet but could be found anywhere within a few standard deviations $\sigma_x$ . The probability density at position $x$ is $|\psi(x,0)|^2$ , which for the Gaussian packet is a bell curve of width $\sigma_x$ centered at the origin. After a position measurement that locates the particle within a small region, the wave function "collapses" to a much narrower state — which, by the trade-off above, immediately carries a large spread in momentum.

This is not a limitation of our instruments; it is a feature of nature. The wave packet picture makes the origin of position–momentum uncertainty concrete: uncertainty is the price of localization in a Fourier-based description of matter waves.