The Free Gaussian Wave Packet and Its Spreading

The Idea in Plain Language

Imagine you prepare a particle in a very narrow region of space. To confine it to that small region, you must combine many different momentum components, because a single momentum state is spread out infinitely. The narrower the initial position packet, the wider the range of momenta you need. Once released, each momentum component travels at its own group velocity. Since different momenta correspond to different velocities, the components drift apart. The packet broadens. This is not diffusion due to collisions or thermal effects; it is a purely quantum-mechanical consequence of the superposition of plane waves with different wavelengths. The Gaussian shape is special because it minimises the product of position and momentum uncertainties, so it spreads in the most “efficient” way possible, and its width at any time can be calculated exactly.

Initial State and Momentum Distribution

We begin with a normalised Gaussian wave function in position space at \( t = 0 \):

Here \( \sigma_0 \) is the initial standard deviation of the position distribution. The corresponding momentum-space wave function is obtained via the Fourier transform. Because the Fourier transform of a Gaussian is another Gaussian, the momentum distribution is also Gaussian, with standard deviation \( \Delta p = \hbar / (2\sigma_0) \). This satisfies the minimum uncertainty relation \( \Delta x \, \Delta p = \hbar/2 \). The key point is that the initial localisation \( \sigma_0 \) directly determines the spread in momenta: tighter position means broader momentum, and vice versa.

Time Evolution and the Spreading Law

For a free particle, the time evolution of the momentum-space amplitude is simply a phase rotation: \( \tilde{\psi}(p, t) = \tilde{\psi}(p, 0) \exp(-i E_p t / \hbar) \), where \( E_p = p^2 / 2m \). Transforming back to position space, the wave function at time \( t \) remains Gaussian, but its width increases. The result, derived in the cited card D-012, is:

This formula tells us that at \( t = 0 \), the width is \( \sigma_0 \), and as \( t \to \infty \), the width grows linearly with time: \( \sigma(t) \approx \frac{\hbar t}{2m\sigma_0} \). The spreading is quadratic in time for small \( t \), but becomes linear for large \( t \). The characteristic spreading time is defined as \( \tau = 2m\sigma_0^2 / \hbar \). After one such time unit, the width has increased by a factor of \( \sqrt{2} \). This timescale depends only on the mass and the initial width, not on the particle’s average momentum or energy.

Physical Interpretation and Limits

The spreading can be understood as the dephasing of the momentum components. Each momentum component \( p \) travels at the group velocity \( p/m \) (its phase velocity is only half that). The spread in velocities is \( \Delta v = \Delta p / m = \hbar / (2m\sigma_0) \). Over time \( t \), the components separate by roughly \( \Delta v \, t \), which matches the asymptotic spreading rate \( \hbar t / (2m\sigma_0) \). In the classical limit, where \( \sigma_0 \) is large compared to the de Broglie wavelength, the spreading is negligible over typical observation times. For macroscopic objects, \( m \) is so large that \( \tau \) is enormous, and the packet effectively behaves like a classical particle. For microscopic particles, such as electrons, \( \tau \) can be very short, and spreading is a dominant effect in experiments like electron diffraction and atomic interferometry.