Wave Packet Expansion: Physical Reason & Conservation Laws

[Black Integra]

Why do wave packets expand when they evolve in time?
Is there any physical reason behind this?
Is it a conservation law of some quantities that makes it expanded?

 

[jtbell]

A wave packet is a sum of waves with different frequencies (and wavelengths). Waves with different frequencies travel at different speeds, and this leads to dispersion. If the speed didn't depend on frequency, there would be no dispersion.

The same thing happens classically with light waves in a medium. In general, the speed of light in a medium depends on frequency, so a "white light" pulse disperses as it travels through glass, water, optical fiber, etc.

 

[Black Integra]

Thanks for that answer :) but let me ask another question.

If we consider an electron as a wave packet moving through space hitting at the step potential, theoretically, electron with low energy could sink into the potential and then jump back to the free space, or, also, be reflected back at first place.

Can we write the wave function as these wave packets(reflect + transmit)?
I mean if I can study the evolution of probability density by time of this electron and a step?