The time-dependent Schrödinger equation is one of the central equations of quantum mechanics:
\[
i\hbar\,\frac{\partial \psi}{\partial t}=\hat H\psi.
\]
For a one-dimensional free particle, the potential is zero, so the Hamiltonian becomes
\[
\hat H=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}.
\]
In this case, a convenient and physically important initial state is a Gaussian wave packet.
A Gaussian packet is localized in position while still containing oscillatory phase structure associated with a central wavenumber \(k_0\). It is useful because it provides a simple bridge between wave behavior and particle-like motion. The center of the packet moves approximately like a particle, while the packet also spreads over time because different Fourier components evolve differently.
For a free-particle Gaussian packet, the group velocity is
\[
v_g=\frac{\hbar k_0}{m}.
\]
This gives the motion of the packet center:
\[
x_c(t)=x_0+v_g t.
\]
So if \(x_0=0\), \(\hbar=1\), \(m=1\), and \(k_0=3\), then
\[
v_g=3,
\]
and at time \(t=1\),
\[
x_c(1)=3.
\]
The packet width does not remain fixed. Instead, the free quantum packet spreads. A useful width formula is
\[
\sigma(t)=\sigma_0\sqrt{1+\left(\frac{\hbar t}{2m\sigma_0^2}\right)^2}.
\]
This shows directly that the width increases with time. When the initial width \(\sigma_0\) is small, spreading is more pronounced. When the mass is larger, the spreading is slower.
The central momentum scale is
\[
p_0=\hbar k_0,
\]
and the central angular-frequency scale is
\[
\omega_0=\frac{\hbar k_0^2}{2m}.
\]
These quantities set the phase oscillation of the carrier wave inside the moving envelope.
The quantity most often interpreted physically is the probability density
\[
|\psi(x,t)|^2.
\]
It tells how likely the particle is to be found near position \(x\) at time \(t\). In this preview tool, the animation emphasizes \(|\psi|^2\) and the envelope behavior, because those are often easier to interpret than the full complex-valued wave function.
This calculator does not solve the full Schrödinger equation for arbitrary potentials. Instead, it provides a clean free-particle Gaussian packet model to illustrate the essential qualitative ideas: group motion, spreading, phase oscillation, and the interpretation of probability density.
At more advanced level, one studies harmonic oscillator states, tunneling, scattering, and numerical Fourier evolution in nontrivial potentials. But the free Gaussian packet is the standard first example because it captures so much of quantum wave-packet behavior in a mathematically manageable form.
