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Shifting integration variable when determing population densitiesby "Don't panic!"
Tags: fermi dirac 
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#1
Mar2614, 01:40 PM

P: 49

Hi,
I'm hoping someone can enlighten me on this as I'm a little bit fuzzy on the reasoning: Say I have a spacetime dependent field [itex]B_{a}[/itex] that interacts with fermions such that it affects their energy dispersion. It appears in the energies in the form [tex]E\sim\sqrt{\left(\vec{p}+\vec{B}\right)m^{2}}+B_{0}[/tex] Why is it, that when I then calculate the number density of fermions in such a scenario, i.e. [tex]n\sim\int^{+\infty}_{\infty}\frac{d^{3}p}{\left(2\pi\right)^{3}}\frac{1}{\exp{\left(E/k_{_{B}}T\right)}+1}[/tex] (where in this case the chemical potential is negligible) that I can only shift the integration variable, such that [itex]\vec{p}\rightarrow \vec{p}+\vec{B}[/itex] (thus "absorbing" the 3vector components of [itex]B_{a}[/itex]), if I consider [itex]B_{a}[/itex] to be constant? Thanks in advance! 


#2
Mar2614, 01:49 PM

P: 49

Apologies for the spelling mistake in the title of the thread by the way, should be "determining" , but don't know how to retroactively edit it!



#3
Mar2614, 02:15 PM

Thanks
P: 1,948

What do you think would happen to d^{3}p if B is not constant?



#4
Mar2614, 02:25 PM

P: 49

Shifting integration variable when determing population densities
Would it be that it becomes time dependent and thus coupled to the fluctuations in B over time?



#5
Mar2614, 02:29 PM

P: 49

or more explicitly, that you would also introduce an additional integral over [itex]d^{3}B[/itex]?



#6
Mar2614, 02:32 PM

Thanks
P: 1,948

Slow down with the questions and answer my question in post #3



#7
Mar2614, 02:38 PM

P: 49

sorry, they were my attempts at a possible answer (shouldn't have included the question marks)!
I assume that you would have [itex]d^{3}p\rightarrow d^{3}\left(p+B\right)=d^{3}p'[/itex] and so, as B is not constant, one could not talk of set momentum states for the fermions as they would fluctuate in time depending on the fluctuations in B. 


#8
Mar2614, 02:42 PM

Thanks
P: 1,948




#9
Mar2614, 02:44 PM

P: 49

ok, that's cleared things up a bit. Thanks for your time.



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