Compressibility of the electron gas: Analytical results for width effects within the Hartree-Fock approximation

We present analytical results for the compressibility of the interacting electron gas within the Hartree-Fock approximation. For the three-dimensional and the ideally two-dimensional electron gas, the well-known results from the literature are found. For the finite width effects in the two-dimensional (quantum well) and the one-dimensional (quantum wire) electron gas, analytical results are described. The explicit form of the interaction potential between the charged carriers enters into the expression for the compressibility: finite-width effects reduce the corrections induced by interaction effects. Our predictions could be tested by capacitance measurements.