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Physical Review

Method for Simplifying Calculations Involving Products of Operators in Many-Particle Systems

Harry J. Lipkin

Published Oct 15, 1957• 1 citation
Publisher

Abstract

In calculations for quantum mechanical systems containing many particles, expressions involving matrix elements of products of operators are frequently encountered. This is of particular interest in the calculation of nuclear moments of inertia. The evaluation of such expressions by conventional methods is laborious when antisymmetric wave functions are used. A method is proposed which greatly simplifies the calculation of these product expressions.

DOI
10.1103/physrev.108.191

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In calculations for quantum mechanical systems containing many particles, expressions involving matrix elements of products of operators are frequently encountered. This is of particular interest in the calculation of nuclear moments of inertia. The evaluation of such expressions by conventional methods is laborious when antisymmetric wave functions are used. A method is proposed which greatly simplifies the calculation of these product expressions.
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation TTT of a real algebraic Lie group GGG. This requires defining an inner product on the Hilbert space mathbbH\\mathbb{H}mathbbH that carries the representation TTT. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of TTT, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (mathfrakFmathbbHD\\mathfrak{F}_{\\mathbb{H}_D}mathfrakFmathbbHD​​, mathbbHD\\mathbb{H}_DmathbbHD​, mathfrakFmathfrakHD\\mathfrak{F}^{\\mathfrak{H}_D}mathfrakFmathfrakHD​) of Bargmann spaces. Coherent state wave functions -- the elements of mathfrakFmathbbHD\\mathfrak{F}_{\\mathbb{H}_D}mathfrakFmathbbHD​​ and of mathfrakFmathbbHD\\mathfrak{F}^{\\mathbb{H}_D}mathfrakFmathbbHD​ -- are used to define the inner product on mathbbHD\\mathbb{H}_DmathbbHD​ in a way that simplifies the calculation of the overlaps. This inner product and the group action Gamma\\GammaGamma of GGG on mathfrakFmathbbHD\\mathfrak{F}^{\\mathbb{H}_D}mathfrakFmathbbHD​ are used to formulate expressions for the matrix elements of TTT with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for Gamma\\GammaGamma relies on matrix factorizations in the complexification of GGG even though the representation TTT does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and Gamma\\GammaGamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in mathbbH\\mathbb{H}mathbbH. The scalar and vector coherent state methods will both be applied to Sp(nnn) and Sp(n,mathbbRn,\\mathbb{R}n,mathbbR).
In the context of many-body quantum mechanics, one often encounters the need to evaluate expectation values of operators or their matrix elements. For instance, given a Hamiltonian https://www.w3.org/1998/Math/MathML" display="inline"> H ^ , we typically want to evaluate its expectation value, which is the energy of the system, https://www.w3.org/1998/Math/MathML" display="inline"> E = ⟨ H ^ ⟩ . In second quantization, the matrix element of an operator can always be expressed as a vacuum expectation value. This generally yields an expression which is hard to evaluate. Fortunately, a powerful tool to simplify these calculations exists: Wick's theorem. This theorem provides a systematic way to break down expressions involving products of creation and annihilation operators into sums of simpler terms.
A diagrammatical method, based on the second quantization formalism, time-independent Wick's theorem and Feynman-like diagrams, is developed for the calculation of arbitrary matrix elements of n-particle spin-independent operators between the states represented by antisymmetrized products of geminals of an arbitrary kind (i.e., strongly or weakly orthogonal, interacting, nonorthogonal or identical geminals). The method greatly facilitates any algebraic manipulation with geminal type functions, particularly when strong orthogonality is not assumed, and enables one to write down directly the algebraic expressions for the matrix elements of an arbitrary spin-independent operator between the single antisymmetrized products of arbitrary geminals, solely on the basis of simple graphs, which are easily constructed.