Degenerate Electron Gas Unrealistic model: Uniform positive charge density n L Very large volume V=L 3 Eventually take limit: System is neutral Positive charge is a uniform background – not quantum mechanical. Wave functions of electrons: Particle in a box

Total Hamiltonian Electron piece Background piece Electron-background piece We will need a convergence factor to handle infinities. At end, need to take limit:

Background Pieces Background piece Electron-background piece Translational invariance Note: last term is infinite when we go to desired limits.

Electron Piece All of the physical effects are in H e. Evaluate in second quantization form. Kinetic Piece:

Kinetic Energy Operator Number operator

Now Start on P.E. Momentum conservation Helicity conservation Coulomb interaction does not flip spins

Potential Operator Let Conservation of momentum automatically

No Scattering Term Number Operators Anticommutation relations Cancels remaining N 2 /V piece Still an infinity as μ→0, but it’s just a number and and we can redefine zero of energy.

New Hamiltonian Define dimensionless parameters: Perturbation in high density limit (r 0 << a 0 ).

Notes on High Density Perturbation Expansion In high density limit: See problems 5 & 6 Term diverges as r s →0. Have to handle carefully at second order. We will evaluate a (kinetic energy term) and b (potential energy term) and leave the rest to more powerful machinery that we develop later (I hope). In high density limit: Kinetic energy of ground state plane waves. “Exchange” perturbation

Ground State – Fermi Level In ground state, two electron (one spin up, on spin down) fill each momentum state starting at the lowest and going up until all of the electrons are used up. The energy where they stop is call the “Fermi level.” Fermi Level

Sum over states below Fermi level Note: we are still integrating in rectangular coordinates. Fermi Level – Number Operator Spherical integral Sum over λ

Ground state energy – First Term

Ground state energy – Second Term Makes vacancy in two states Must fill the same two vacancies Thus, either: This is the non-scattering case q=0 Already subtracted.

Working on the Matrix Element Intersection of two spheres

Important Integral Volume of Integration Take zero of coordinate system at k and z along q; do k integral first in cylindrical coordinates for half of volume (then double).

Finish the Integration

Expansion – First Two Terms Kinetic Term Exchange Term Minimum at 4.82 a 0