Wave Packets

In quantum mechanics, particles are described not by a single position but by a wave function $\psi(x,t)$. A wave packet is a localized disturbance built from the superposition of many plane waves with slightly different wave numbers $k$ (and thus different momenta $p = \hbar k$).

Why Wave Packets?

A single plane wave $e^{ikx}$ extends infinitely in both directions — it describes a particle with perfectly known momentum but completely unknown position. By combining many such waves, we create a localized bump in $|\psi|^2$ (the probability density), at the cost of introducing a spread in momentum.

This trade-off is codified in Heisenberg's uncertainty principle:

$\Delta x \, \Delta p \;\geq\; \frac{\hbar}{2}$

The Gaussian Wave Packet

The simplest and most commonly used wave packet has a Gaussian envelope:

$\psi(x,t) = e^{ik_0 x - i\omega t} \cdot e^{-x^2 / 4\sigma_k^2}$

Parameter	Symbol	Physical Meaning
Central wave number	$k_0$	Average momentum of the particle ($p_0 = \hbar k_0$)
Wave number width	$\Delta k$ ($\sigma_k$)	Spread in momentum — wider $\Delta k$ means better position localization
Time	$t$	Evolution of the packet — in free space, it spreads over time (dispersion)

The probability density $|\psi|^2$ tells you where the particle is most likely to be found.

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Things to Try

1. Narrow the wave-number width ($\Delta k \to 0.5$) — the wave packet becomes very wide in position space. This is the uncertainty principle in action: less spread in momentum forces more spread in position.
2. Widen the wave-number width ($\Delta k \to 5$) — the packet becomes tightly localized. At the extreme, the real and imaginary parts oscillate rapidly inside a narrow envelope.
3. Increase $k_0$ — the carrier wave oscillates faster (higher momentum particle), but the envelope width stays the same.
4. Advance time — watch the packet spread out (disperse). In this simplified model the spreading is slow, but in a full quantum treatment the width grows as $\sigma(t) = \sigma_0 \sqrt{1 + (\hbar t / 2m\sigma_0^2)^2}$.
5. Compare with the ghost reference — the faint dotted curve shows the default $|\psi|^2$ at $k_0=10$, $\Delta k=2$, $t=0$.

Key Concepts

• Group velocity vs. phase velocity: The envelope (the bump) travels at the group velocity $v_g = d\omega/dk$, while the internal oscillations travel at the phase velocity $v_p = \omega/k$. These can differ!
• Dispersion: If $\omega(k)$ is not a linear function of $k$, different components travel at different speeds, causing the packet to spread over time. This is why quantum particles "delocalize" as they evolve.
• Born interpretation: $|\psi(x,t)|^2\, dx$ is the probability of finding the particle between $x$ and $x + dx$. The teal-filled curve in the plot is this probability density.

A note on this simulation — The time evolution shown here is simplified. A full treatment requires solving the free-particle Schrodinger equation, which produces a complex-valued Gaussian that spreads in a specific way depending on the particle mass. The qualitative behaviour (spreading, oscillation) is faithfully captured.