Connection to Bra and Ket Formalism

This section is devoted to point out some formal similarities between the second quantized representation and the so-called hra and ket formalism introduced into quantum theory originally by Dirac (1958). A short summary can be found in Lowdin (1985). Although we will not refer to the details of bra-ket formalism in the rest of the book, we feel that the brief comparison below may be useful, offering a deeper understanding of second quantization. Bra and ket functions have already been introduced in Sect. 1 and they have been utilized throughout, but a more detailed discussion is justified by its relevance to the second quantized formalism. 

For the sake of simplicity, let us discuss a one-electron system. Consider a set of orthonormalized functions Bra- and ket-functions, respectively, are 

We can consider Q as an operator which is defined by its action on a ket-function  

It is easy to see that Q is a projector. If an arbitrary one-electron function [f is expanded in terms of the basis functions (p as  

Q projected out the i-th component of /. The idempotency of Q is also obvious  

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Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

In conclusion, a few words should be said about the equivalence between the ket-bra formalism frequently used in this article and the particle-hole formalism based on the ideas of second quantization T commonly used in the special propagator theories and the EOM method. Both formalisms are used to construct a basis for the operator space, and the essential difference is that the latter treats particles having specific symmetry properties—i.e., fermions or bosons—whereas the former is not yet adapted to any particular symmetry. In order to get a connection between the two schemes, it may be convenient in the ket-bra formalism to introduce a so-called Fock space for different numbers of particles... 

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