Systems of Particles and Second Quantization

We need to extend the quantum mechanical theory to the case of N particles; separately, the particles would be described by a Schrödinger equation or the Pauli equation, depending on their spins. As in the classical case, the Hamiltonian of the system will be the sum of those of the particles plus an interaction term (possibly). For independent particles (no interaction), the Hamiltonian is, in obvious notation \( \hat{H}(1,2,\ldots ,N)= \sum _{n}^N \hat{h}(n), \) where h(n) describes particle n. The wave function \(\varPsi (1,2,\ldots ,N)\) depends on all the orbital and spin degrees of freedom of the particles, so it is a spinor in the spin space of each particle.