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Proceedings of the Physical Society

Second quantization of an arbitrary operator

Donald H. Kobe

Published May 1, 1966• 3 citations
Publisher

Abstract

The representation of an arbitrary operator in second quantization is obtained for both fermion and boson systems by using the isomorphism between second quantization and a graded configuration space.
DOI
10.1088/0370-1328/88/1/303

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A review of the second quantization formulation of quantum mechanics is given based on the idea of a graded Hilbert space whose elements are represented by column matrices containing zero, one, two, ···, particle functions. The creation and annihilation operators are defined in terms of intuitive ideas about adding a particle to or removing a particle from the system. The (anti)commutation relations are shown to follow from the (anti)symmetry of the wavefunctions. The field operators which describe the creation or annihilation of a particle at a point are then obtained and their properties discussed. These operators are then used to express general operators in second quantization. Finally, a basis is introduced and the equations of occupation-number space are obtained.