Why Physicists Love Gaussian Wave Packets

Last Updated on May 25, 2025 by Max

Open almost any quantum-mechanics textbook, and the first example of a moving particle you meet is not a sharp spike or a jagged pulse but a smooth, bell-shaped curve. That curve is a Gaussian wave packet. A plot of the bell-shaped curve is shown in Figure 1.

Far from being a pedagogical accident, the Gaussian has become the industry standard whenever we need a quantum state that behaves like a localized particle yet keeps the mathematics and numerics under control.

Here we outline the reasons for its popularity and point out the few situations where it isn’t the right choice.

What is a Gaussian Wave Packet?

A one-dimensional Gaussian packet centered at position \(x_0\) with mean momentum \(p_0\) is
\[\psi(x,0)=\left(\frac{1}{2\pi\sigma_0^{2}}\right)^{1/4}\exp\bigg[-\frac{(x-x_0)^2}{4\sigma_0^{2}}+\frac{i p_0 x}{\hbar}\bigg],\]

where \(\psi(x,0)\) is the wave function at \(t=0\). Here, \(x\) is the position, \(\sigma_0\) the initial spatial width of the packet, and \(\hbar\) the reduced Planck constant.

Its probability density \(|\psi(x,0)|^{2}\) is a normal (bell-curve) distribution of width \(\sigma_0\) and its momentum-space profile is another Gaussian of width \(\hbar/(2\sigma_0)\). Thus, the packet is simultaneously localised in both position and momentum.

Minimum-Uncertainty Property

The Heisenberg uncertainty principle demands \(\Delta x\,\Delta p\ge\hbar/2\). For a Gaussian packet, the product equals that lower bound, making it the minimum-uncertainty state. That optimal balance is essential when we want to model a particle as sharply as quantum mechanics allows.

Fourier-Transform Symmetry

The Fourier transform of a Gaussian is another Gaussian. This self-similarity keeps algebra simple when switching between position and momentum space and enables highly efficient algorithms based on the fast Fourier transform (FFT).

Simple Time Evolution

Under free evolution, the packet remains Gaussian; only its width changes:

\[\sigma(t)=\sigma_0\sqrt{1+\bigl(\frac{\hbar t}{m\sigma_0^{2}}\bigr)^{2}}.\]

No Bessel or Airy functions—just one square-root factor. This closed-form behaviour is invaluable for teaching, benchmarking numerical solvers, and interpreting ultrafast-electron and cold-atom experiments.

Calculational Convenience

Gaussian integrals have analytic solutions, which tame otherwise messy expressions in path integrals, Feynman propagators, semiclassical approximations, and variational methods.

In practice, a modest basis of Gaussians can approximate almost any smooth wave function to high accuracy.

Physical Realism

• Many laboratory sources naturally emit or are engineered to emit nearly Gaussian beams.
• The ground state of a harmonic trap (ultracold atoms, ions) is exactly Gaussian.
• Transmission Electron Microscopy (TEM) electron beams and femtosecond laser pulses are designed to approach a Gaussian envelope so that their shape stays neat while propagating.

Consequently, Gaussian packets map well onto real experimental initial conditions.

When Gaussians are Not Enough

Use other shapes when physics demands sharp edges or multiple peaks—for example, slit diffraction, square-well bound states, lattice Bloch waves, or any system with hard boundaries. Always match the packet to the potential and the preparation procedure.

Key Takeaways

• Gaussian packets saturate the uncertainty limit.
• Their form survives free-particle evolution exactly.
• Algebra and numerics remain tractable because of Fourier self-similarity.
• They model many experimental beams, but they are not universal—choose the packet that reflects your system.

Further Reading

[1] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed. (Cambridge University Press, Cambridge, 2020).

[2] MIT OpenCourseWare 8.04 Lecture 6 Notes, “Wave Packets and the Uncertainty Principle,” (accessed on May 25, 2025).