Quantum Mechanics – Why Two-Electron Systems Use Singlet-Triplet Basis

Question

Why are the two-electron system usually described in singlet-triplet basis, but not computational basis
$\uparrow\uparrow$,$\uparrow\downarrow$,$\downarrow\uparrow$,$\downarrow\downarrow$? What is the advantage of that?

Answer 1

Typical spin-spin coupling has form: $$H_J=J\mathbf{S}_1\cdot \mathbf{S}_2.$$ Thus, if we take two-spin Hamiltonian $$H=\Delta_1 S_1^z+\Delta_2 S_2^z +J\mathbf{S}_1\cdot \mathbf{S}_2,$$ it will be diagonal in the singlet-triplet basis, but not in the computational basis. And working in the diagonal basis is nearly always an advantage.

This is also true for more general coupling of angular momenta, such as, e.g., the spin-orbit coupling $$H_{LS}\propto \mathbf{L}\cdot\mathbf{S},$$ or even for system containing many particles with various spin and orbital momenta. The reason for this that the Hamiltonian of the system as a whole often possesses symmetry in respect to rotations in 3D (or at least around the quantization axis), so the total angular momentum is a good quantum number, regardless of the interactions within the system.