This calculator gives a simplified preview of time-dependent wave-function propagation using a form that is mathematically similar to both free-particle quantum mechanics and paraxial wave optics.
The main idea is that a localized Gaussian packet does not stay perfectly localized as time passes.
Even in free propagation, different spatial-frequency components accumulate different phase shifts, so the packet broadens.
In optics language, this resembles diffraction and beam spreading.
In quantum language, it resembles free-particle wave-packet dispersion.
The simulator starts from an initial Gaussian profile
\(\psi(x,0)\)
centered at
\(x_0\)
with width
\(\sigma_0\)
and carrier wavenumber
\(k_0\).
The user also chooses an effective ratio
\(\hbar/m\),
which controls how quickly the packet spreads and how fast its center drifts.
This is not intended to represent a full microscopic light field or a complete quantum-optics treatment.
It is an educational analogy that highlights the mathematics of propagation.
For free evolution, the packet center moves according to
\[
x_c(t)=x_0+\left(\frac{\hbar}{m}\right)k_0 t.
\]
This means that if
\(k_0\neq 0\),
the whole packet drifts in space as time increases.
If
\(k_0=0\),
the center remains fixed and only spreading occurs.
The spreading itself is controlled by the parameter
\[
\tau=\frac{(\hbar/m)t}{2\sigma_0^2}.
\]
From this, the packet width at time
\(t\)
becomes
\[
\sigma_x(t)=\sigma_0\sqrt{1+\tau^2}
=
\sigma_0\sqrt{1+\left(\frac{(\hbar/m)t}{2\sigma_0^2}\right)^2}.
\]
This formula captures the essential physical trend:
narrow initial packets spread faster, while broader initial packets spread more slowly.
In that sense, localization in space comes with a price in momentum-space width, and the packet disperses as it propagates.
The probability density shown by the calculator is modeled as
\[
|\psi(x,t)|^2=
\frac{1}{\sqrt{2\pi}\,\sigma_x(t)}
\exp\!\left[
-\frac{(x-x_c(t))^2}{2\sigma_x(t)^2}
\right].
\]
This is the quantity most clearly related to observable position probability in quantum mechanics and to intensity-envelope style intuition in an optics analogy.
As time increases, the peak gets lower and wider because the total area under the probability density remains normalized while the width grows.
The phrase “Fourier phase evolution” refers to the fact that the wave packet can be decomposed into spatial-frequency components.
Each component acquires a time-dependent phase, and because those phases do not all change in the same way, the reconstructed packet broadens.
This is closely related to how optical beams diffract and how pulses disperse in wave systems more generally.
For the example case with a Gaussian packet, final time
\(t=1\),
and effective
\(\hbar/m=1\),
the output plot shows the probability density at the chosen time and compares it with the initial profile.
If the initial width is
\(\sigma_0=0.5\),
then the spreading parameter is
\(\tau=1\),
and the width becomes
\[
\sigma_x(1)=0.5\sqrt{1+1}=0.5\sqrt{2}\approx 0.707.
\]
So the packet clearly broadens, even though no external potential is applied.
That is the key point of free evolution in this model.
At a more advanced university level, one studies exact complex phase structure, momentum-space evolution, potentials, waveguide modes, stationary states, and full quantum-optics formalisms.
One can also connect the same mathematics to paraxial beam propagation in guided or free-space optical systems.
This calculator deliberately avoids those full complications and focuses on the simplest and most intuitive case:
a free Gaussian packet that drifts and spreads.
\[
x_c(t)=x_0+\left(\frac{\hbar}{m}\right)k_0 t,
\qquad
\sigma_x(t)=\sigma_0\sqrt{1+\left(\frac{(\hbar/m)t}{2\sigma_0^2}\right)^2},
\qquad
|\psi(x,t)|^2=
\frac{1}{\sqrt{2\pi}\,\sigma_x(t)}
\exp\!\left[-\frac{(x-x_c(t))^2}{2\sigma_x(t)^2}\right].
\]
These relations explain why the wave packet spreads, how its center moves, and why the probability density becomes wider and lower as time increases.
