Thomas-Fermi theory with an external magnetic field


Of concern is a rigorous Thomas-Fermi theory of ground state electron densities for quantum mechanical systems in an external magnetic field. The energy functional takes the form script c sign(ρ1, ρ2) = Σi=12∫ R′ρi(x)5/3 dx+1/2∫R′∫ R′[ρ(x)ρ(y)/|x-y|]dx dy+∫R3V(x)ρ(x) dx +∫R′(B(x)(ρ1(x)-ρ2)dx; here c, is a positive constant, ρ1 [resp. ρ2] is the density of spin-up [resp. spin-down] electrons, ρ = ρ1+ρ 2 is the total electron density, V is a given potential (typically a Coulomb potential describing electron-nuclear attraction), and B describes the effect of the external magnetic field. Let N1 = ∫R′ρ i(x)dx be the number of spin-up and spin-down electrons for i = 1,2, and let N = N1 + N2 be the total number of electrons. Specifying N and minimizing script c sign(ρ1,ρ2) generally leads to a spin polarized system. For example, if B≤0 and B≢0, then ρ1≥ρ2 and N 1>N2. This and a number of related results are proved. © 1991 American Institute of Physics.

Publication Title

Journal of Mathematical Physics