Chapter 2
Second Quantization
The concept of second quantization (SQ), originally developed in quantum field theory, has proven to be an indispensable tool in many-body theory since it allows one to
represent many-electron states and operators in an utmost flexible and compact way.
In this chapter, we shall review the SQ formalism at some length. The presentation
of the SQ operator algebra in the first three subsections is essentially based on a
concise formulation in the appendix of a textbook by Baumgärtner and Schuck [1],
who acknowledge unpublished lecture notes by W. Brenig. A combination of SQ
and the SC rules, described in Sect. 2.4, leads to a practical means for evaluating
many-electron matrix elements. The SQ field operators relate to underlying oneparticle states. The transformations of the field operators induced by changes of the
one-particle representation are considered in the final subsection.
2.1 Definition of Creation and Destruction Operators
The starting point is a complete set (basis) of orthonormal one-particle states |q,
q|q
= δ qq
q
|qq| = ˆ
1
and the corresponding basis set of normalized antisymmetric N -electron product
states,
|q 1 q 2 . . . q N , q 1 < q 2 < · · · < q N
as introduced in Chap. 1. So far, the electron number N was assumed to be arbitrary
but fixed. Now, we extend the scope to allow for variable electron numbers. For this
purpose, the concept of the Fock space is introduced,
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_2
19
Second Quantization
The concept of second quantization (SQ), originally developed in quantum field theory, has proven to be an indispensable tool in many-body theory since it allows one to
represent many-electron states and operators in an utmost flexible and compact way.
In this chapter, we shall review the SQ formalism at some length. The presentation
of the SQ operator algebra in the first three subsections is essentially based on a
concise formulation in the appendix of a textbook by Baumgärtner and Schuck [1],
who acknowledge unpublished lecture notes by W. Brenig. A combination of SQ
and the SC rules, described in Sect. 2.4, leads to a practical means for evaluating
many-electron matrix elements. The SQ field operators relate to underlying oneparticle states. The transformations of the field operators induced by changes of the
one-particle representation are considered in the final subsection.
2.1 Definition of Creation and Destruction Operators
The starting point is a complete set (basis) of orthonormal one-particle states |q,
q|q
= δ qq
q
|qq| = ˆ
1
and the corresponding basis set of normalized antisymmetric N -electron product
states,
|q 1 q 2 . . . q N , q 1 < q 2 < · · · < q N
as introduced in Chap. 1. So far, the electron number N was assumed to be arbitrary
but fixed. Now, we extend the scope to allow for variable electron numbers. For this
purpose, the concept of the Fock space is introduced,
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_2
19