Matrix Elements Of Two-body Operators Research Articles

Moshinsky brackets for a wide range of quantum numbers using generating functions

We used a new Python code to reproduce the brackets for the Moshinsky harmonic oscillator, which was based on the generating function. We made these brackets by transforming the wave functions of two groups of coupled particle harmonic oscillators, Φn1l1,n2l2,Λm1,m2,λ(r→1,r→2) and Φnala,nblb,Λma,mb,λ(r→a,r→b). To convert between the supplied position and momentum coordinates in both frames, we performed orthogonal transformations on nuclei with both low and high angular momentum.In our derivation, we have used the expansion of the generating functions e2p→.r→−p2 and e2cpi.pj in spherical coordinates in terms of harmonic oscillator wave functions. When we modified the Moshinsky brackets for two-coupled oscillator states, we used generating functions with two variables. The number of indices has significantly decreased compared to the oscillator brackets in previous references; this reduction in the program code's iterative process has yielded influential results. Compared to the previous version of the Moshinsky brackets code, the new Python code is easier to use. Our approach utilizes this code to assess Moshinsky brackets across a broad spectrum of quantum numbers. According to the revelation, adding more variables to the generating function makes the number of Moshinsky brackets that work for the higher body interactions increase. Program summaryProgram Title: GFBRACKETSCPC Library link to program files:https://doi.org/10.17632/jvbnwp35rm.1Licensing provisions: MITProgramming language: PythonSupplementary material: AppendixNature of problem: The generating functions were used to obtain a concise representation of the oscillator states for single particles. This was done to formulate and compute the generalized version of the Moshinsky brackets and matrix elements of two-body operators. By enlarging sets of basis oscillator states, it becomes possible to cover interactions involving more than two bodies. This can be easily achieved using the expanded generating function to compute the Moshinsky brackets for high-quantum numbers.Solution method: The Python code requires fewer iterations than similar codes generated by the method of the generating functions.Additional comments including restrictions and unusual features: When calculating Moshinsky brackets for large quantum numbers, restrictions arise. There is no data for quantum numbers greater than 10 in the formation of reaction potential. Additionally, a transformation matrix can be used to switch from single-particle to centre-of-mass coordinates on a two-particle harmonic oscillator basis. This transformation is applicable even when the masses of the two particles are unequal.

Oscillator brackets at non equal particle masses

Abstract. Efficient programs to calculate two-particle oscillator brackets (or Talmi-Moshinsky-Smirnov coefficients) have been constructed recently, Comput. Phys. Comm. 265 (2021) 108005. The corresponding Talmi-Smirnov transformation is mostly performed in case of particles of equal masses and the constructed programs have been tested in detail as to this case. However, it occurred that those programs included a bug revealing itself in case when particle masses are different. The bug is eliminated in the present version of the programs. All the various tests employed previously at equal masses of particles now reproduce the required results for the case of different masses as well. Program summaryProgram Title: OSBRACKETSCPC Library link to program files:https://doi.org/10.17632/4m594wzv94.2Licensing provisions: GPLv3Programming language: Fortran-90Journal Reference of previous version: Comput. Phys. Comm. 265 (2021) 108005Does the new version supersede the previous version?: YesReasons for the new version: Necessity to fix a bug in the programs which affected the brackets in case of the Talmi-Smirnov transformation at non equal particle masses. The author is thankful to M.A. Sharaf who pointed out the incorrect results pertaining to the previous version of the programs.Summary of revisions: The subroutine to calculate the quantity (23) from Ref. [1] is corrected. This quantity includes powers of tan⁡φ where φ is the angle specifying the kinematic rotation. In case of equal mass particles one has |tan⁡φ|=1, see [1]. Most applications of the brackets in the literature refer to this case and the previous version of the programs has been tested at |tan⁡φ|=1. However, the mentioned subroutine included a bug that reveals itself in the |tan⁡φ|≠1 case. In the present version of the programs, the bug is eliminated. All the various tests described in [1] now reproduce the required results also at |tan⁡φ|≠1, i.e., at arbitrary mass ratio of particles.Nature of problem: Single-particle basis oscillator states are widely used for studying the structure of various many-body systems. To compute matrix elements of two-body operators, the Talmi-Smirnov transformation of oscillator states is performed. Coefficients of this transformation are called oscillator brackets. Often it is necessary to retain large sets of basis oscillator states in calculations. Therefore, a fast program to compute the brackets is needed. The program should provide accurate results up to high oscillator excitations.Solution method: At zero radial quantum numbers, oscillator brackets are calculated using an explicit expression that includes few summations. Starting from such brackets, recurrence relations are employed to calculate the brackets of the general type. These relations prove to work perfectly up to very high oscillator excitations.

Calculation of oscillator (Talmi–Moshinsky–Smirnov) brackets

A program to calculate the oscillator brackets, or Talmi–Moshinsky–Smirnov coefficients, is presented. The recursion with respect to radial quantum numbers is employed. The listed runs show that the program is very fast and it produces accurate results up to very high oscillator excitations. The amount of computations per bracket does not increase with the increase of quantum numbers. In one of the presented versions, the program provides the sets of all existing brackets of a given parity pertaining to oscillator excitations and total angular momenta which lie within given ranges. In the other version, the subsets of such brackets having given angular momenta are produced. Such type arrays of the brackets are quite convenient for majority of applications. These arrays are made compact due to use of suitable combinations of partial angular momenta as array arguments. The program is easy to implement and follow. Comparisons are made with results of programs based on the explicit expression for the brackets. Program summaryProgram Title: OSBRACKETSCPC Library link to program files:https://doi.org/10.17632/4m594wzv94.1Licensing provisions: GPLv3Programming language: Fortran-90Nature of problem: Single-particle basis oscillator states are widely used for studying the structure of various many-body systems. To compute matrix elements of two-body operators, the Talmi-Smirnov transformation of oscillator states is performed. Coefficients of this transformation are called oscillator brackets. Often it is necessary to retain large sets of basis oscillator states in calculations. Therefore, a fast program to compute the brackets is needed. The program should provide accurate results up to high oscillator excitations.Solution method: At zero radial quantum numbers oscillator brackets are calculated using an explicit expression that includes only few summations. Starting from such brackets, recurrence relations are employed to calculate the brackets of the general type. These relations prove to work perfectly up to very high oscillator excitations.

Matrix elements of two-body operators between many-body symmetrized hyperspherical states.

Matrix elements of two-body operators between many-particle permutational symmetry-adapted functions in hyperspherical coordinates are constructed. The matrix elements are evaluated using the appropriate hyperspherical coefficients of fractional parentage, the Raynal-Revai coefficients, and the hyperspherical recoupling coefficients. We use the power expansion of the two-body operators and obtain an analytic expression for the matrix element of each term. These expressions are studied numerically. The results allow precise evaluations of two-body matrix elements for few-body calculations in nuclear, atomic, and molecular physics.

Generator coordinate kernels between zero- and two-quasiparticle BCS states

We give a method with which the matrix elements of operators between BCS states can be evaluated with the generalized Wick's theorem even when the excitation of a pair of quasiparticles leads to a vanishing of the overlap. We analyze the behavior of the mean field and the pairing contributions to the matrix elements of two-body operators versus the gauge angle of the orbital and find that each of them diverges as the overlap vanishes while their sum remains finite. A new definition of these quantities is proposed which removes the divergence occurring at zero overlap. A numerical application to a deformed heavy nucleus is done, which tests the feasibility of our prescription.

Matrix elements of two-body operators between three-body hyperspherical basis states

Abstract Matrix elements of two-body operators between three-body spherically symmetric hyperspherical basis states are expressed through the Clebsch-Gordon coefficients and explicitly written functions of a radial variable.

Vector-coupling approach to orbital and spin-dependent tableau matrix elements in the theory of complex spectra

The power of the Young tableau scheme for labeling a complete spin-adapted basis set in the theory of complex spectra lies in one's ability to evaluate matrix elements of irreducible tensor operators directly in terms of the tableau labels and shapes. We show that the matrix-element rules stated by Harter for one-body operators can be easily derived from simple vector-coupling considerations. The graphical method of angular momentum analysis is used to derive closed-form expressions for the matrix elements of two-body operators. This study yields several interesting new relationships between spin-dependent operators and purely orbital operators.

On the Wigner Supermultiplet Scheme

Calculation of Wigner and Racah coefficients for the group SU(4)⊃[SU(2)×SU(2)] make it possible to perform the spin—isospin sums in the cfp (fractional parentage coefficients) expansion of the matrix elements of one- and two-body operators in the Wigner supermultiplet scheme. The SU(4) coefficients needed to evaluate one- and two-particle cfp's, the matrix elements of one-body operators, and the diagonal matrix elements of two-body operators are calculated in general algebraic form for many-particle states characterized by the SU(4) irreducible representations [yy0], [y y − 1 0], [yy1], [y11], [y y − 1 y − 1], [y10], [yy y − 1], [y00], and [yyy], whose states are specified completely by the spin and isospin quantum numbers (y = arbitrary integer). Applications are made to the calculation of the matrix elements of the complete space-scalar part of the Coulomb interaction and the space-scalar part of the particle-hole interaction for nucleons in different major oscillator shells.