Propagators in Quantum Chemistry, 2nd Edition

Appendix B

First and Second Quantization

Quantum mechanics takes many different guises. For instance, one can use a Hilbert space realization in terms of time-dependent wavefunctions Φ (ξ₁,ξ₂,…ξ_N ; t) of an N-particle system, where is a compound configuration space and spin variable for a particle. Operators acting on this Hilbert space are obtained by making the common identifications for the momentum and for position vector of a particle, which can be referred to as first quantization, producing quantum mechanical operators out of classical expressions. One can equally well use time-dependent field operators (so-called second quantization) and their adjoints to build Hilbert spaces (or rather Fock spaces) and corresponding quantum mechanical operators. The connection between different formulations must be that they give the same expectation values of operators.

Thus, we have, say, an N-electron operator in first quantization, which is a sum of single-particle operators

An example ...