In the Hartree model of interacting electrons one assumes that the wave function is in the form(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \Psi(x_1, x_2, ..., x_N,t) = \psi_1(x_1,t) \psi_2(x_2, t) ... \psi_N(x_N,t) [/tex]

which of course is a quite crude approximation since it for example does not take into account Pauli principle.

I have studied some recent articles in plasma physics which starts from this expression and then derives a fluid equation (for example, G. Manfredi, arxiv:quant-ph/0505004).

From the form above it is then claimed that this can be represented by a one-particle density matrix

[tex] \rho(x,y) = \sum_{i=1}^N p_i \psi_i^*(x) \psi_i(y) [/tex]

where [tex] p_i [/tex] are the "occupation probabilities".

Does any one know how to obtain this form of the density matrix?

If one would have started with a completely anti-symmetric wave function and calculated the density matrix (by tracing over [tex] N-1 [/tex] particles) then I can guess that the result would be similar to this, but then I think that the probabilites would be [tex] 1/N [/tex].

Does the form of the density matrix above possible to derive from that the particles are intdistinguishable or is it just some plausible arguments which gives the form above?

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# Statistical Mixture of N States in the Hartree-method.

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