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At first the Hamiltonian of the system is written as ## H=H_{el}+H_b+H_{el-b} ## where the first part is the Hamiltonian for electrons and their interactions among themselves, the second part is the Hamiltonian for the interaction among the fixed background ions (no kinetic part because they're assumed to be fixed) and the last part is the Hamiltonian for the interaction between electrons and the background ions.

My question is about the last two parts. In the books I mentioned, they're written as:

## H_b=\frac 1 2 e^2 \int d^3 x' \int d^3 x'' \rho(\mathbf x')\rho(\mathbf x'') \frac{e^{-\mu|\mathbf x'-\mathbf x''|}}{|\mathbf x'-\mathbf x''|} ##

## H_{el-b}=-e^2 \sum_i \int d^3x \rho(\mathbf x) \frac{e^{-\mu|\mathbf x-\mathbf x_i|}}{|\mathbf x-\mathbf x_i|} ##

Where ## \rho(\mathbf x ) ## is describes the distribution of the background ions which is assume to be uniform(## \rho(\mathbf x )=\frac N V##). Until now everything is fine. But then there are these two lines of calculation which are obviously wrong!

## \left. \begin{array}{l} z \equiv |\mathbf x'-\mathbf x''| \\ \rho (\mathbf x)=\frac N V \end{array} \right\} \Rightarrow H_b=\frac 1 2 e^2 (\frac N V)^2 \int d^3 x \int d^3 z \frac{e^{-\mu z}}{z}= \frac 1 2 e^2 \frac{N^2}{V} \frac{4\pi}{\mu^2}##

This is wrong because it ignores that ## |\mathbf x'-\mathbf x''| ## depends on the angular coordinates too. The most you can do is choosing z axis to be along one of those vectors and then ## |\mathbf x'-\mathbf x''| ## will depend only on ## \theta ## but you can't completely ignore the dependence on angular coordinates. I get it, the term ## H_b ## is a constant term in the Hamiltonian and can be ignored but it just annoys me when such well respected books simply don't care about such issues.

But then comes the part about ## H_{el-b} ## which is more serious!

## H_{el-b}=-e^2 \sum_i \frac N V \int d^3 x \frac{e^{-\mu|\mathbf x-\mathbf x_i|}}{|\mathbf x-\mathbf x_i|} =-e^2 \sum_i \frac N V \int d^3 z \frac{e^{-\mu z}}{z}=-e^2 \frac{N^2}{V} \frac{4\pi}{\mu^2}##

Aside from the fact that this integration has the same problem as the above one, they simply draw the conclusion that this term is a c-number! But...come on! You can easily see that the position operators for the electrons appear in ## H_{el-b} ##. Whatever they do to that integral, they should end up with ## H_{el-b}=\sum_i f(\mathbf x_i) ##. If they want to simply conclude that this operator is a c-number, they should give really good reasons, not just...I don't even know what to call it! Actually this can't be true because an operator is an operator and a c-number is a c-number. I can only accept it as an approximation that they can actually ignore this operator but such approximations really need to be discussed!

Any objections or comments on this?