• Show, using the formalism of the second quantization, that, for both bosons and fermions the total-number operator and the Hamiltonian operator commute: [N,H]=0 .
• Degenerate electron gas in 1D :
Repeat the exercise on the 3D degenerate electron gas in 1D (replace 3D integrals with 1D integrals !); this could be a model for electrons in carbon nanotubes (graphite sheets rolled up into cylinders); these long and thin carbon molecules have extraordinary material characteristics (they are believed to be the strongest material in the world). Depending on the specific way the cylinder is rolled up the nanotubes are either metallic, semiconducting or insulating; a metallic nanotube is a nearly ideal 1D metal wire.
In particular:
(a) evaluate the E0 (kinetic-energy) contribution;
(b) show that the E1 (first-order potential-energy) diverges !
Hint : !"#
𝑑𝑥=−2 𝛾+ ln 𝑞
so 4π/q2 (in 3D) must be replaced by −𝟐 𝜸+ 𝐥𝐧 𝒒 in 1D, (𝛾=0.577216… , Euler constant).
