A quantum wave packet is a localized superposition of many momentum components. Unlike a single plane wave, which extends over all space, a wave packet can represent a particle that is initially concentrated near some position \(x_0\). One of the most important lessons of quantum mechanics is that such packets usually do not remain rigid: they spread in time because the different momentum components evolve with different phases.
A common starting point is a Gaussian packet. In position space, the initial probability density can be written schematically as
\[
|\psi(x,0)|^2 \propto \exp\!\left[-\frac{(x-x_0)^2}{2\sigma_x^2}\right],
\]
where \(\sigma_x\) measures the initial spatial width. In momentum space, the packet is also Gaussian, with a momentum spread inversely related to the position spread. For a minimum-uncertainty Gaussian in units \(\hbar = 1\), the initial momentum uncertainty is
\[
\Delta p = \frac{1}{2\sigma_x},
\]
so the uncertainty product is
\[
\Delta x\,\Delta p = \frac12.
\]
This relation helps explain why narrow packets spread more quickly: if \(\sigma_x\) is small, then the momentum spread is large, and the different momentum components separate more strongly during evolution.
For a free particle with \(\hbar=m=1\), the standard Gaussian spreading formula is
\[
\sigma_x(t)=\sigma_x(0)\sqrt{1+\frac{t^2}{4\sigma_x(0)^4}}.
\]
This shows directly that the packet width increases with time. If the packet also has mean momentum \(k_0\), then its center moves according to
\[
\langle x \rangle_t = x_0 + k_0 t.
\]
For the sample free-particle case with \(\sigma_x=1\) and \(t=1\), the evolved width is
\[
\sigma_x(1)=1\cdot \sqrt{1+\frac{1}{4}}=\sqrt{1.25}\approx 1.118.
\]
So even after a moderate time, the packet is already broader than it was initially. The probability density \(|\psi|^2\) therefore becomes lower at the center and wider across space.
This calculator is intentionally a teaser. In the free-particle mode, it uses the standard Gaussian evolution result. In the trap mode, it uses a simplified Gaussian-style preview of motion in a quadratic potential. That is useful for visualization, but it is not a full quantum solver for arbitrary potentials, barriers, scattering, or full Fourier-integral evolution.
In more advanced treatments, the time-dependent Schrödinger equation is solved either analytically or numerically. Fourier methods evolve each momentum component with a phase factor such as \(e^{-i p^2 t / 2m\hbar}\) for a free particle, and inverse transforms reconstruct \(\psi(x,t)\). That deeper framework explains precisely why wave packets spread and why barrier or bound-state problems require more than the simple formulas used here.
Even with these simplifications, the central physical message remains accurate: localization in position implies uncertainty in momentum, and that uncertainty causes quantum wave packets to spread in time. This is one of the clearest visual signatures of quantum evolution.
