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I have a quite mysterious and cumbersome question concerning with the expectation values for a system of identical particles. For example, suppose I have a system of N identical bosons given by the wavefunction ψ(x1,x2,...xN), which is of course symmetrized. My concern is:

1. What is <x1> really means? I know mathematically it is given by:

∫dx1dx2...dxN x1 ψ*(x1,x2,...xN)ψ(x1,x2,...xN)

but I think this is unphysical? We cannot measure this value, right? since physical observable for identical particles should be symmetric, like x1+x2+x3, right? But can I say <x1>=<x2>=<x3> = 1/3<x1+x2+x3> due to symmetric argument?

2. Suppose I really want to compute the above quantity, how to write it in second quantized form, that is using field operators? Is it given by:

<x1> = ∫ x1 <ψdagger(x1)ψ(x1)> dx1 ? where ψ is now a field annihilation operator that annihilate a particle at the point x1. If yes, can someone give me some ideas on how to show it starting from the wavefunction description?

3. Continue the same idea, then what is <(x2-x1)^2> means? Also, how to write it in second quantized form?

Thank you for all

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# Expectation Value for system of identical particles

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