Importance of Second Quantization

In the many-electron problem, one is focused on solving the Schrodinger equation which for stationary states T reads  

1) E is the electronic energy, while F is the many-electron wave function. In Eq. (1.2) atomic units have been used (h = e = m = 1) with usual notations for the kinetic energy operator of the i-th electron (— Aj), atomic number Z, the coordinates rj and r, and the electronic coordinate difference r = rj — rj.  

In the general case, approximate solutions to Eq. (1.1) can be obtained only. The main goal of quantum chemistry, and some fields of solid state physics as well, is to develop approximate theories suitable to solve the Schrodinger equation [Eq. (1.1)] for many-electron systems. Many approaches have been developed, which are equivalent in the sense that they yield the same numerical result at the same level of approximation. For a particular problem, however, they are not equivalent in the sense that one can be significantly simpler than the other one. 

Among possible approaches, the so-called second quantization plays an important role. The ultimate goal of the second quantized approach to the many-electron problem is to offer a formalism which is substantially simpler than the traditional one in many cases. As a matter of fact, most difficulties of the traditional or first quantized approach arises from the Pauli principle which requires the wave function W of Eq. (1.1) to be antisymmetric in the electronic variables. This is an additional requirement which does not result from the Schrodinger equation and requires a special formalism the using of Slater determinants for constructing appropriate solutions to Eq. (1.1). The Slater determinant is not a very pictorial mathematical entity, and the evaluation of matrix elements over determinantal wave functions makes the first quantized quantum chemistry somewhat complicated for beginners. In the second quantized 

Another useful feature of the second quantized formalism is that the second quantized representant of the Hamiltonian (and any other physical observable) is independent of the number of electrons, N, in contrast to the first quantized form of the Hamiltonian, Eq. (1.2). Thus, chemical systems containing different numbers of electrons, for example, can be described by one and the same (or very similar) Hamiltonian. 

---

A third, but not negligible importance of second quantization is that this approach inherently deals with one-electron functions, although the theory is not restricted to one-electron methods. The occupancies of one-electron orbitals are the basic quantities one is dealing with in second quantization. This is a very descriptive picture which is in the mind of every chemist who tries to understand something about the electronic structure of molecules. 

---