Undergrad Why particles have group velocity?

[arda]

I just confused about it.Why can't we discribe a particle just one wave function instead of wave packet(group of waves with different phase velocities)?

 

[BvU]

And how would you do that ?
Can you quote context, reference, example ?

 

[arda]

My referance is Quantum Mechanics Concepts and Applications
Second Education written by Nouredine Zettili
Page is 38 section 1.8 Wave Packets

 

[BvU]

Doesn't help me.

In general:
A single wave function for a particle is generally a summation (integral) with an average position and average momentum.
A single wave of the type $\psi = e^{ikx}$ is unnormalizable in $x$, has infinitely sharp momentum ($\hbar k$) but no position.

A wave packet (e.a gaussian wave packet, which google) has a position (with some variance) and a momentum (with some variance). A phase velocity and a group velocity.

 

[BvU]

And, eh, a belated $\quad$ $\quad$ !

 

[Demystifier]

I just confused about it.Why can't we discribe a particle just one wave function instead of wave packet(group of waves with different phase velocities)?

A wave packet is one wave function. But it can be written as a sum of other, simpler wave functions. This sum is nothing but the Fourier expansion (or transform) of the wave function. It is not much different from the fact that 375 is one number, but it can be written as a sum of simpler numbers as
$$375 = 3\cdot 100 + 7\cdot 10 + 5\cdot 1$$
With an abuse of language, someone could say that 375 is a "packet" of numbers.

 

[jihai]

Actually, we describe particles using wave functions. For example, with electrons in solid state physics, we basically solve Schrödinger equation to get eigenvalues and eigenfunctions. However, the picture of the wave function is something spreaded in the whole solid which is inconsistent with classical concept of electrons. So we use the wave packet to explain the transition from the quantum mechanics to classical picture.

 

[arda]

Thank you all!