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Shifting integration variable when determing population densities

by "Don't panic!"
Tags: fermi dirac
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"Don't panic!"
#1
Mar26-14, 01:40 PM
P: 81
Hi,

I'm hoping someone can enlighten me on this as I'm a little bit fuzzy on the reasoning:

Say I have a space-time 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 3-vector components of [itex]B_{a}[/itex]), if I consider [itex]B_{a}[/itex] to be constant?

Thanks in advance!
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"Don't panic!"
#2
Mar26-14, 01:49 PM
P: 81
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!
dauto
#3
Mar26-14, 02:15 PM
Thanks
P: 1,948
What do you think would happen to d3p if B is not constant?

"Don't panic!"
#4
Mar26-14, 02:25 PM
P: 81
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?
"Don't panic!"
#5
Mar26-14, 02:29 PM
P: 81
or more explicitly, that you would also introduce an additional integral over [itex]d^{3}B[/itex]?
dauto
#6
Mar26-14, 02:32 PM
Thanks
P: 1,948
Slow down with the questions and answer my question in post #3
"Don't panic!"
#7
Mar26-14, 02:38 PM
P: 81
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.
dauto
#8
Mar26-14, 02:42 PM
Thanks
P: 1,948
Quote Quote by "Don't panic!" View Post
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.
Correct. If you have a explicit form for B then you might attempt a solution. You can't go much further with the general expression, I don't think
"Don't panic!"
#9
Mar26-14, 02:44 PM
P: 81
ok, that's cleared things up a bit. Thanks for your time.


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