What is the spreading for rectangular wave packets? I know the spatial width for rectangular wave packets, but what is the spreading for a rectangular wave packet?
 A: In a word (well, two): not pretty. A rectangular wavepacket has a sharp discontinuity, which means that its momentum-space representation (i.e. its Fourier transform) has a significant support at very large momentum, and as soon as you give the time-dependent Schrödinger equation any time to act, those short-wavelength, high-momentum components will whizz away fast.
The wavepacket itself is exactly solvable: you start with the free-particle propagator, which is expliclty known as a Fresnel oscillating exponential,
$$
K(x,x';t)
={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ik(x-x')}e^{-{\frac {i}{2}\hbar k^{2}t}}\mathrm dk
=\sqrt{\frac {1}{2\pi it}}e^{-{\frac {(x-x')^{2}}{2i t}}}
$$
and then you just integrate to get the wavepacket at later times,
\begin{align}
\psi(x,t) 
& = \int_{-\infty}^\infty K(x,x';t)\psi(x',0)\mathrm dt
\\ & =
\sqrt{\frac {1}{2\pi it}}\int_{-L/2}^{L/2} e^{-{\frac {(x-x')^{2}}{2i t}}}\mathrm dx'
\\ & =
\frac{i}{2}
   \left[
 \text{erfi}\left(\sqrt{\frac{i}{2t}}\left(x+\frac{L}{2}\right)\right)
-\text{erfi}\left(\sqrt{\frac{i}{2t}}\left(x-\frac{L}{2}\right)\right)
\right],
\end{align}
in terms of imaginary error functions evaluated along the complex-plane diagonal at which $|\mathrm{erfi}(\sqrt{i}x)|$ is oscillatory around unit modulus.
At short times, this wavepacket has some extreme ringing caused by the discontinuity, but at longer times this gives way to a behaviour that becomes very similar to that of the propagator itself (i.e. to a wavepacket that starts off as a delta function). More graphically:

Mathematica source: Import["http://halirutan.github.io/Mathematica-SE-Tools/decode.m"]["http://i.stack.imgur.com/cnnL8.png"]
(Here the dark and light blue lines are the real and imaginary parts of $\psi(x,t)$, phase-shifted so $\psi(0,t)$ is always real and positive for visual clarity. Mind the logarithmic timescale, though: it makes it easier to see the details of the dynamics, but the animation doesn't accurately represent how the evolution would look like on linear time.)
