wave-packet spreading
Wave-packet spreading is the inevitable widening, over time, of a localized free particle's wavefunction. Squeeze a quantum particle into a small region and let it go, and the bundle of probability does not stay put — it gradually broadens, so that as time passes the particle becomes more and more uncertain in position. A tightly localized packet spreads fastest of all.
The cause is the very uncertainty relation that gave the packet its initial form. A narrow packet contains a wide range of momenta, and different momentum components travel at different speeds. Like runners who start together but move at different paces, the faster and slower parts of the packet pull apart, stretching the bundle out. The smaller you make the starting region, the broader its momentum spread and the quicker it disperses.
This spreading is a genuine physical phenomenon for free particles, not merely a mathematical artifact. It is why a single electron, if not confined or continually measured, becomes spread over a large region surprisingly quickly. Confining potentials, like the walls of an atom or a quantum dot, fight against this tendency, which is why bound states hold a stable shape while truly free particles drift ever wider.
A free packet's width grows over time; the tighter its start, the faster it disperses.
Spreading applies to free particles. A particle held in a stable bound state, such as an electron in an atom, does not endlessly spread, because the binding potential reshapes its evolution into a steady stationary pattern.