Algebraic development of many-body perturbation theory in ...

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D Symbolic computations with NCoperators 901. Second Quantisation Representation (SQR)2. Angular Momentum Theory (AMT)3. Rayleigh–Schrödinger Perturbation Theory (RSPT)4. Unstructured External Programming (UEP)The packages store Mathematica codes that are loaded into a Mathematica session by usingthe function «“NCoperators.m” or Get[“NCoperators.m”]. The NCoperatorsis designed so that each block can be brought into action separately. This is easily done withGet[“PackageName.m”]. All these functions (Get[]) are located in a single packageNCoperators.m.D.1 SQR and AMT blocksThe SQR and AMT blocks are closely related and therefore they will be considered together.The present blocks contain a huge number of functions, each of them being useful for a specialcase of interest. In the present overview, only a few of them will be discussed.In theoretical atomic physics, a typical task is to find the angular coefficients that relate twodistinct ( momenta coupling schemes. Consider, for example, the momenta recoupling coefficientj1 j 2 (j 12 )j 3 j ∣ ∣j 2 j 3 (j 23 )j 1 j ) . The expression is known, and it is{ }(−1) 2j+j 2+j 3 −j 23[j 12 , j 23 ] 1/2 j1 j 2 j 12.j 3 j j 23Making use of NCoperators, the algorithm to obtain the latter formula is displayed in Fig. 1.Fig. 1: A computation of recoupling coefficient ( j 1 j 2 (j 12 )j 3 j ∣ ∣ j2 j 3 (j 23 )j 1 j ) with NCoperatorsOut[25] contains two Clebsch–Gordan coefficients labelled by CGcoeff[] – the standardMathematica function ClebschGordan[] is insufficient for symbolic manipulations.The function Recoupling[] initiates the momenta recoupling followed by Refs. [10–12].SymbolSum[] plays a role of a standard Mathematica function Sum[] adapted to symboliccomputations. The function Make[] initiates the momenta recoupling within a given Clebsch–Gordan coefficient. The simplification of considered sum is performed by SumSimplify.Out[31] is printed in a standard Unix output form specified by the font family «cmmi10». Obtainedformula is easy to convert to a L A TEX language by making use of Mathematica function
D Symbolic computations with NCoperators 91Export[“DirectoryName/FileName.tex”,%31]. A brief description of all functionsis lited by using Definition[function] or simply ?function. The example ispresented in Fig. 2.Fig. 2: The usage of Definition[] in NCoperatorsIn Fig. 2, the function TAntiCommutator1[] along with TAntiCommutator2[],TAntiCommutator3[] and TAntiCommutator4[] initiates the (four) anticommutationrules for irreducible tensor operators a λα im αi , ã λα im αi (RSPT block). To compare with, the anticommutationproperties of the Fock space operators a αi , a † α iare realised through the NCoperatorsfunctions AntiCommutator1[], AntiCommutator2[], AntiCommutator3[] andAntiCommutator4[]. The examples are listed in Fig. 3. Particularly, In[61] computesthe one-body terms {Ô2(αβ)Ô1(µν)} 2 which are found in the Wick’s series for ĥ(3) mn;ξwith theindices m = 2, n = 1, ξ = 2 (refer to Eqs. (2.42), (4.3)); NonCommutOpMulti[n,α,β] isnothing else but Ôn(αβ). In addition, the function NonCommutOpMulti[n,α,β,m,µ,ν]corresponds to Ôn(αβ)Ôm(µν).Fig. 3: Manipulations with the antisymmetric Fock space operators in NCoperatorsFig. 4: Wigner–Eckart theorem in NCoperatorsThere are many more functions in NCoperators.Many tasks in atomic physics arerelated to the Wigner–Eckart theorem. Thistheorem is also found in the present package.The example of usage is demonstratedin Fig. 4, where the projection-independentquantity [ ]j α ‖v (k) ‖j β denotes a usual reducedmatrix element of the irreducible tensor operatorv (k) . Note, throughout the present text,the notation v k is preferred.
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INSTITUTE OF THEORETICAL PHYSICS AN
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VILNIAUS UNIVERSITETOTEORINĖS FIZI
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5Contents1 Introduction . . . . . .
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75 The generation of ĥ(3) 11;1 ter
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1 Introduction 9lation, the multi-c
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1 Introduction 111.3 The scientific
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2 Partitioning of function space an
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3 Irreducible tensor operator techn
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Page 73 and 74: A Basis coefficients 73(α1 α 2 (
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