Systems of Interacting Particles: Method of Second Quantization

Ph. Droz-Vincent

Michele Cini

Vladimı́r Kvasnička

Radi A. Jishi

Nothing can be made out of nothing. William Shakespeare , King Lear Historically, quantization of the motion of particles was developed first. The state was described by a wave function and observables by operators. When dealing with interactions between particles and fields, such as the electromagnetic field, the fields were treated classically. Classical field equations look like the quantum mechanical equations for the wave function of the field quanta. For example, the Klein–Gordon classical field equation is similar to the quantum mechanical wave equation for a relativistic spinless particle. Quantizing the fields, leading to quantum field theory, appears to be quantizing a theory that has already been quantized; hence the name “second quantization.” In reality, there is only one quantization and one quantum theory. The method of second quantization is important in the study of many-particle systems. It enables us to express many-body operators in terms of creation and annihilation operators, thus rendering calculations less cumbersome. Moreover, the method makes it possible to treat systems with a variable number of particles; that is why the method initially emerged in the context of quantum field theory. In Chapter 1 we indicated that any one-particle wave function may be expanded in a complete set of states. In this chapter, we show that products of single-particle states, when properly symmetrized, form an orthonormal basis for the expansion of the wave function of an N -particle system.

Zhou Duan-Lu, Yu Si-Xia, Sun Chang-Pu

A practical method is developed to deal with the second quantization of the many-body system containing the composite particles.In our treatment,the modes associated with composite particles are regarded approximately as independent ones compared with those of unbound particles.The field operators of the composite particles thus arise naturally in the second quantization Hamiltonian.To be emphasized,the second quantization Hamiltonian has the regular structures which correspond clearly to different physical processes.

Hanna Makaruk

Quantum computers by their nature are many particle quantum systems. Both the many-particle arrangement and being quantum are necessary for the existence of the entangled states, which are responsible for the parallelism of the quantum computers. Second quantization is a very important approximate method of describing such systems. This lecture will present the general idea of the second quantization, and discuss shortly some of the most important formulations of second quantization.