Bloch wavepackets and the Pauli exclusion principle

[pedda]

Hello,

I have a question concerning the use of wavepackets to justify the semiclassical approach in solid state physics. In Ashcroft/Mermin, the authors briefly mention that we can construct wave packets and then use them to describe the motion of the center which can be interpreted as what one usually calls the point particle electron. Now, the problem that I have is that for each state there is one Bloch vector k. If I was to form a wave packet spreading over several k, how can there be a second electron occupying the state k' that is right next to k? The packet centered around k will definitely have components of wave vector k' and vice versa. Doesn't this violate the pauli exclusion principle?

- Peter

 

[squigglywolf]

bump!
I have the same question :(. Did you manage to resolve it Peter??

 

[pedda]

Hey,

yes, I did resolve it for me, but I don't know if it is correct. The Pauli exclusion principle states that the wave function has to be antisymmetric with respect to the exchange of particles. The fact that you have two wavepackets centered around two different ks doesn't violate this principle, even if they are centered at the same place. You can write down the wavefunction for two gaussian wavepackets in the position representation. You will see that you will get something like

[itex]e^{-(x_1-ik_1)^2}e^{-(x_2-ik_2)^2}-e^{-(x_1-ik_2)^2}e^{-(x_2-ik_1)^2}[/itex]

+ some prefactors and other stuff. As you see, no problem here!

Hope this helps, Peter

 

[pedda]

I don't see any way to edit my last post, but an important part that I left out is the actual time development that appears in the denominator of the exponentials, so don't take what I've written too seriously. The most important part is that symmetrization works even for wavepackets as long as they are not centered around the same eigenvalue k.