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1) The Thomas-Fermi method is a useful way to estimate the "screening cloud" surrounding electrons in a metal. A quantification of the screening is the inverse dielectric function of the material: $$\frac{\phi_{ext}(\textbf{q})}{\epsilon(\textbf{q})} = \phi_{total}(\textbf{q})$$ Slowly-varying in this context means that the ...

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By definition, in the tensor product of Hilbert spaces $\mathscr{H}_1$ and $\mathscr{H}_2$, the two spaces are different: it is not possible to identify the creation/annihilation operators of the first space with the ones of the second. As presented by the OP, both $\psi_1$ and $\psi_2$ belong to (different subspaces of) the same full symmetric Fock space ...

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The potential has to vary slowly enough so that the electrons have time to respond as if the field were static. Risking a semi-educated guess: I suppose that the potential would have to change less than a few percent in a time on the order of the transit time of an electron across a unit cell. That is, the Fermi velocity divided by the lattice constant. ...

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I believe it has to do with the Inclusion-Exclusion Principle from combinatorics. (link: http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle) In doing so, one is making the assumption that the interaction energies between distinct, separate bodies decreases as the number of bodies being taken into account increases. Of course, this is not ...

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A more precise mathematical way of asking your question is: given a (usually unbounded) self-adjoint operator $H$ on a Hilbert space $\mathscr{H}$, am I able to characterize its spectrum? Finding "closed" solutions to the equation you are writing, means finding eigenfunctions of your operator $H$, possibly belonging to the Hilbert space $\mathscr{H}$ (since ...

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Hints: Define difference $\delta:=\Delta-\Delta_0$. Deduce from $|\delta|\ll |\Delta_0|$ that the lhs. of eq. (1) is $$\tag{A}\text{lhs}~\approx~ -\frac{\delta}{\Delta_0}.$$ Substitute $\xi=x\Delta$ in the integral on the rhs. of eq. (1). Deduce using $\hbar \omega_D \gg \Delta$ that the rhs. is \tag{B} \text{rhs}~\approx~ \int_{\mathbb{R}} \! ...

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