Bloch parameter k:-

For free particles k = wave number =

expected value of momentum

For a particle bound to a periodic potential or crystal ,    = crystal momentum.

   is not the actual momentum but the momentum related to the constant of motion which incorporates the crystal interaction.

This crystal momentum parameter k is also periodic with a period of . The E-K diagram is therefore the Energy versus crystal momentum characteristics of an electron in the crystal.

Energy Band Solution

Energy band solution indicates only the allowed energy and momentum states but not about time evolution of electron's position etc.given E, k gives possible values of position of finding the electron with a certain probability i.e, position is uncertain.

Heisenberg Uncertainty Principle.

If E is known exactly, uncertainty in t is infinity, we can not find anything about the electron's position.

Therefore for a particle motion we need wave packets constant E wave function grouped about a peak energy.

Probability of finding the represented particle in a given region of space = 1 for some specified time.

Center of mass of a particle moving with a velocity - classical idea.

wave packet also a mass - QM idea

Packet of traveling wave with center frequency and center wave number k then describes the particle motion represented by this group.

group velocity,

E, k gives the center values of energy and crystal momentum.