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Propagators in Quantum Chemistry, 2nd Edition by Yngve Öhrn, Jan Linderberg

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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,…ξN ; t) of an N-particle system, where Image is a compound configuration space and spin variable for a particle. Operators acting on this Hilbert space are obtained by making the common identifications Image for the momentum and Image 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) Image and their adjoints Image 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 ...

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