Second quantized form of a two-particle operator

Eugene Feenberg

Donald H. Kobe

Second quantization in nonrelativistic quantum mechanics is given in a simplified way using eigenstates of the number operator. The field creation operator is shown to be the operator which adds a particle with a delta-function wave function to the system, while the field annihilation operator removes a particle with a delta-function wave function from the system. The field creation operator is proved to be the Hermitian conjugate of the annihilation operator. The commutation (anticommutation) relations for boson (fermion) field operators are proved from the symmetry (antisymmetry) of the many-particle wave function. The second quantized form of one- and two-particle operators is obtained from their form in wave mechanics using the definition of the field creation and annihilation operators.

Péter R. Śurján

Travis S. Taylor

In this chapter, we explore the advanced formalism of second quantization, which provides a powerful framework for describing quantum systems with variable particle numbers. We begin with an overview of how operators are second quantized, introducing the essential tools of creation and annihilation operators. These operators form the foundation of Fock space, a versatile representation that accommodates the quantum states of systems with differing particle counts.

N. I. Greenberg, S. Raboy

The form of one-particle and two-particle operators is obtained in the notation of second quantization. These forms are applied to many-particle states of identical particles where the appropriate symmetry with respect to exchange of particles is invoked from the start. The form in the wave-mechanical formalism of operators suggested by classical analog is derived in a matter suitable for inclusion in courses in quantum mechanics.

Ph. Droz-Vincent