The jellium Hamiltonian in second quantisation

Units are SI and the gas is non-relativistic, so the course's metric (+, −, −, −) never enters; 1 Ry = ħ²/(2ma₀²) = 13.606 eV. In a cube of volume V = L³ with periodic boundaries the orbitals are φₖσ = V^(−1/2)e^(ik⋅r)χ_σ with k = 2πn/L for n a triple of integers, and the kinetic energy is Σεₖc†ₖσcₖσ, εₖ = ħ²k²/2m. Unit 10's ½Σ⟨ij|v|kl⟩c†ᵢc†ⱼcₗcₖ then needs ⟨k + q, k′ − q|v|k, k′⟩. Writing r₁ = r₂ + r, the r₂ integral returns V times a Kronecker δ on total momentum and leaves v_q/V, with v_q = ∫d³r e^(−iq⋅r)v(r): a pair scatters by exchanging ħq, spin rides through each vertex unchanged, and only three of the four momenta are free. For v = e²e^(−μr)/(4πε₀r) the angular integral leaves (e²/(ε₀q))∫₀^∞ sin(qr)e^(−μr)dr = e²/(ε₀(q² + μ²)), derived. Dimensions check: v_q is in J m³, so v_q/V is an energy.

A uniform charge has no Fourier component except at q = 0, so the background enters only through two constants, which join the q = 0 part of the pair sum; all three are finite while μ ≠ 0 because ∫d³r e^(−μr)/r = 4π/μ². Its energy with itself is ½(e²/4πε₀)n²V(4π/μ²) = e²N²/(2ε₀Vμ²), with n = N/V, and its attraction to the electrons is −e²N²/(ε₀Vμ²). The q = 0 part of the pair sum is (v₀/2V)Σc†ₖσc†ₖ′σ′cₖ′σ′cₖσ; anticommuting cₖσ two places to the left turns the operator into N̂² − N̂, so it equals (e²/(2ε₀Vμ²))N(N − 1). All three are derived, and they add to −e²N/(2ε₀Vμ²). The N² parts cancel exactly, which is neutrality, not an approximation. What survives is the −N̂: the background terms count all N² pairs, self-pairs included, while the normal-ordered electron sum counts only the N(N − 1) distinct ones. One-particle check: for N = 1 the pair term vanishes, and the two background terms give −e²/(2ε₀Vμ²), exactly the formula.

Per electron the leftover is −e²/(2ε₀μ²V), while each term is of order e²n/(2ε₀μ²). For sodium, n = 2.65 × 10²⁸ m⁻³, with μ⁻¹ = 1 μm in a 1 cm cube, the three are +2.40 × 10⁸, −4.80 × 10⁸ and +2.40 × 10⁸ eV per electron, and they sum to −9.05 × 10⁻¹⁵ eV. At fixed μ the leftover falls as 1/V and vanishes in the thermodynamic limit; send μ → 0 at fixed V instead and it diverges. So V → ∞ comes first and q = 0 is dropped exactly. Only then does μ → 0, in the q ≠ 0 terms, where it is harmless: as V → ∞ the sum over q becomes V/(2π)³ times ∫d³q, and because d³q/q² integrates over angles to 4πdq, which stays finite near q = 0, the cutoff alters a sliver of width about μ that vanishes with it. The result, derived, is H = Σεₖc†c + (1/2V)Σ′v_q c†c†cc with v_q = e²/(ε₀q²), the prime leaving out q = 0. Free-field check: e → 0 leaves the ideal Fermi gas.

A uniform gas has one length, the radius rₛa₀ of the sphere that holds one electron: (4π/3)(rₛa₀)³ = 1/n, with a₀ = 4πε₀ħ²/(me²) = 0.529 Å. Measured densities put the simple metals at rₛ ≈ 2 to 6: 2.07 for aluminium, 3.93 for sodium and 5.69 for caesium. Rescale k = k̃/(rₛa₀), q = q̃/(rₛa₀) and V = Ṽ(rₛa₀)³, where Ṽ = 4πN/3. Then εₖ = (Ry/rₛ²)k̃², and since e²/(ε₀a₀) = 8π Ry, v_q/2V = (Ry/rₛ²)(3rₛ/N)q̃⁻². So H = (Ry/rₛ²)[Σk̃²c†c + (3rₛ/N)Σ′q̃⁻²c†c†cc], derived, and the allowed k̃ = (2π/L̃)n, with L̃ = (4πN/3)¹⁄³, depend on N alone. Dimensions check: the bracket is a pure number and Ry/rₛ² carries the energy. Density enters only through the prefactor and the explicit rₛ before the interaction: kinetic energy per electron scales as 1/rₛ², interaction as 1/rₛ. Because rₛ ∝ me²n^(−1/3)/(ε₀ħ²), the limit rₛ → 0 is at once the dense gas and the free-field limit e → 0, a check on any expansion in rₛ.

The kinetic energy here is Pauli's, not thermal: to hold N fermions the occupied plane waves reach a wavenumber of order 1/(rₛa₀), so kinetic energy per electron grows as n²⁄³, while the Coulomb energy at the mean spacing, 2/rₛ Ry, grows only as n¹⁄³. Rescaling a 1/r^p interaction the same way gives it a relative weight rₛ^(2−p): p = 1 is weak when dense, p = 2 never changes, and interactions falling faster than 1/r², like the short-range ones of topic 15.07's Bose gas, are weak when dilute. The next two topics turn the bracket into numbers, derived there and quoted here: one determinant gives E/N = 2.21/rₛ² − 0.916/rₛ Ry, an interaction-to-kinetic ratio of 0.414rₛ. That is 0.86 for aluminium, 1.63 for sodium and 2.36 for caesium, so metals sit where the small parameter is not small, and the series is not even a power series, correlation adding a ln rₛ term (Gell-Mann and Brueckner, quoted). At the dilute end, quantum Monte Carlo puts the Wigner crystal near rₛ ≈ 100 (quoted).

The ions. Point ions on a lattice have Fourier components at every reciprocal lattice vector G, and an electron–ion term ΣU_G c†_(k+G)σcₖσ would let the lattice absorb momentum ħG and open gaps at the Brillouin-zone faces; jellium keeps only G = 0, where the background sits. For the alkali metals the loss is small: de Haas–van Alphen measurements find sodium's Fermi surface spherical to about one part in a thousand, so its electrons barely see the lattice. In aluminium the lattice already reshapes the Fermi surface, and in the transition metals, with d bands, jellium fails outright. The ions are also frozen into a rigid background, so there are no phonons, and the only pressure is the electrons' own, whose zero topic 15.04 finds. Relativity is safe: with the free Fermi gas's k_F = 1.92/(rₛa₀), quoted from statistical physics, v_F/c = 1.92α/rₛ = 0.0036 for sodium, and the non-relativistic H holds until rₛ falls to about 0.014.