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How BEC being described by the singleparticle density matrix? 
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#1
Jul2114, 06:10 AM

P: 1

Hello everybody,
this is my first time being here. I am a beginner learning some introductions on BoseEinstein Condensation (BEC) on my own. Often times in the literature (say, [1], [2] (p.409) ) it comes the onebody(singleparticle) density matrix, as [tex]<\psi\mathbf{\Psi(r)^\dagger\Psi(r')}\psi>=N\int dx_2...dx_N~\psi^*(r,x_2,...,x_N)\psi(r',x_2',...,x_N') [/tex] I am not sure how to derive the above equation... My first step is to write [itex]<\psi\mathbf{\Psi(r)^\dagger\Psi(r')}\psi>[/itex] as [tex] <\psi\mathbf{\Psi(r)^\dagger\Psi(r')}\psi>=\int dx_1...dx_N \int dx_1'...dx_N' \psi_t^*(x_1,...,x_N)<x_1,...,x_N\mathbf{\Psi(r)^\dagger\Psi(r')}x_1' ,...,x_N'>\psi_t(x_1,...,x_N) [/tex] then I am not sure how to handle [itex]<x_1,...,x_N\mathbf{\Psi(r)^\dagger\Psi(r')}x_1',...,x_N'>[/itex]. Any ideas? thanks in advance for help and comments, C.H. 


#2
Jul2114, 07:57 AM

Sci Advisor
P: 3,633

You probably know that e.g. ##\mathbf{\Psi(x_1)\Psi(x_2)}0>=x_1,x_2>## and so on for the position eigenstates of n particles in genera. Furthermore, you know the commutation properties of the Psi operators, ##\{\mathbf{\Psi^+(x_1),\Psi(x_2)}\}=\delta(x_1x_2)##. This should be sufficient to work out the matrix element.



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