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I'm having some trouble, that I was hoping someone here could assist me with. I do hope that I have started the topic in an appropriate subforum - please redirect me otherwise.

Specifically, I'm having a hard time understanding the matrix elements of the density matrix, [itex] \varrho[/itex]. For instance, I would like to determine the density matrix element [itex]\langle \boldsymbol{\mathrm{k}} | \varrho | \boldsymbol{\mathrm{k}}+\boldsymbol{\mathrm{q}} \rangle [/itex], i.e. matrix elements of [itex] \varrho[/itex] in a momentum basis.

The reason for me wanting to do this, is that I am trying to understand an old paper by N.D. Mermin [1]. In this paper, the particle density, [itex]\rho(\boldsymbol{\mathrm{q}})[/itex], and current density, [itex]\boldsymbol{\mathrm{J}}(\boldsymbol{\mathrm{q}})[/itex], expectation values are introduced as (slightly rewritten - the essence remains the same):

[tex] \langle \rho(\boldsymbol{\mathrm{q}})\rangle = \sum_{\boldsymbol{\mathrm{p}}} \langle \boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}} | \varrho | \boldsymbol{\mathrm{p}} \rangle \\

\langle \boldsymbol{\mathrm{J}}(\boldsymbol{\mathrm{q}}) \rangle = \sum_{\boldsymbol{\mathrm{p}}} (\boldsymbol{\mathrm{p}}+\frac{1}{2} \boldsymbol{\mathrm{q}} )\langle \boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}} | \varrho | \boldsymbol{\mathrm{p}} \rangle [/tex]

I'm used to the following second quantized forms of the particle density operator and particle currents:

[tex] \rho(\boldsymbol{\mathrm{q}}) = \sum_{\boldsymbol{\mathrm{p}}} c_{\boldsymbol{\mathrm{p}}}^\dagger c_{\boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}}} \\

\boldsymbol{\mathrm{J}}(\boldsymbol{\mathrm{q}}) = \sum_{\boldsymbol{\mathrm{p}}} (\boldsymbol{\mathrm{p}}+\frac{1}{2} \boldsymbol{\mathrm{q}} )c_{\boldsymbol{\mathrm{p}}}^\dagger c_{\boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}}} [/tex]

My question is, how do I derive (preferably starting from the second-quantized form of the operators) the expectation values [itex]\langle \rho(\boldsymbol{\mathrm{q}})\rangle[/itex] and [itex]\langle \boldsymbol{\mathrm{J}}\boldsymbol{\mathrm{q}}) \rangle[/itex], expressed as sums over density matrix elements?

I feel that, essentially, this should be a simple problem to do in a stringent manner - but I just can't seem to make the necessary connections.

[1] Lindhard Dielectric Function in the Relaxation-Time Approximation

EDIT: Edit since I had apparently not understood the use of [ tex ] [ /itex ].

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# Density matrix elements, momentum basis, second quantization

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