The Schrödinger equation written in the second quantization formalism: derivation from first principles

Abstract

Courses about Quantum Mechanics are generally developed at the level of the mathematical formalism, its syntactic level, seldom treating the interpretation of the theory, its semantic level. In the majority of cases, this first formal approach is not followed by any other course addressing the numerous interpretations of the theory. It has become apparent that this strategy leads to a deficit in the professional formation of the students, exceedingly driving them to technical areas of application of Quantum Mechanics. However, applications such as those on quantum computing and entanglement, as they become more and more encrusted in the deepest foundations of the theory, are asking more and more for the development of this kind of ability. In this paper we develop an approach to deal with second quantization in the realm of Quantum Mechanics. We show that the second quantization Schrödinger equation can be mathematically derived from a classical Hamiltonian written in the phase space $(Q,P)$, obtained by a canonical transformation upon the original phase space $(q,p)$, using an axiomatic quantization procedure developed elsewhere. We apply this quantization process to a bosonic system, modelled by the harmonic oscillator problem. We then make reverse engineering to show that Schwinger's second quantization approach to fermionic systems furnishes the path to derive a Schrödinger equation explicitly written in terms of the usual momenta and positions operators $(\hat{q},\hat{p})$ for such systems. Finally, we use these mathematical developments to address the commonly accepted statement that 'the spin has no classical analog'.