Hello everyone,(adsbygoogle = window.adsbygoogle || []).push({});

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 ].

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Density matrix elements, momentum basis, second quantization

Loading...

Similar Threads - Density matrix elements | Date |
---|---|

I Indices of a Density Matrix | Monday at 8:04 PM |

Decoherence in the long time limit of density matrix element | Dec 15, 2015 |

Density matrix off-diagonal elements | Dec 21, 2012 |

Negative off-diagonal elements in density matrix? | Jan 12, 2012 |

Off diagonal element of density matrix | May 14, 2009 |

**Physics Forums - The Fusion of Science and Community**