Some problems related to a completely antisymmetrized R-matrix theory

The complete solution is presented for the $N$ -completeness problem, i.e., the problem of determining when a set of $p$ -particle functions generates symmetric or antisymmetric functions without applying a symmetrizing or antisymmetrizing operator. Then it is shown that a symmetric or antisymmetric function has finite 2 rank if, and only if, it has finite 1 rank. Finally, the relation between $N$ representability of $N$ completeness is discussed.The complete solution is presented for the $N$-completeness problem, i.e., the problem of determining when a set of $p$-particle functions generates symmetric or antisymmetric functions without applying a symmetrizing or antisymmetrizing operator. Then it is shown that a symmetric or antisymmetric function has finite 2 rank if, and only if, it has finite 1 rank. Finally, the relation between $N$ representability of $N$ completeness is discussed.

Physical Review AMar 1, 197214 citations

Application of the Infinite Product Representation of the R-Matrix

Jinchu Huang

Jan 1, 19890 citations

N -Representability Problem: Conditions on Geminals

Mary Beth Ruskai

The $N$ -representability problem is approached by considering geminal product expansions of symmetric and antisymmetric functions. The $N$ -completeness problem, i.e., the problem of determining when a set of geminals is a suitable basis for expanding symmetric or antisymmetric $N$ -particle functions, is considered. New necessary conditions are given for both $N$ -completeness and $N$ -representability. In some cases, one can also obtain sufficient conditions; examples of such cases are discussed. The circumstances under which a density matrix can be derived from two or more different functions are also treated. Finally, extensions to higher order are also mentioned.The $N$-representability problem is approached by considering geminal product expansions of symmetric and antisymmetric functions. The $N$-completeness problem, i.e., the problem of determining when a set of geminals is a suitable basis for expanding symmetric or antisymmetric $N$-particle functions, is considered. New necessary conditions are given for both $N$-completeness and $N$-representability. In some cases, one can also obtain sufficient conditions; examples of such cases are discussed. The circumstances under which a density matrix can be derived from two or more different functions are also treated. Finally, extensions to higher order are also mentioned.

Physical ReviewJul 5, 196934 citations

A construction for R-matrices without difference property in the spectral parameter

Jon Links

arXiv:solv-int/9907010Physics Letters AJan 1, 20001 citation

INFINITE-DIMENSIONAL R MATRIX WITH COMPLETE ℤ SYMMETRY

Youichi Shibukawa, K. UENO

Jun 1, 19933 citations

THE EIGENVALUE PROBLEM OF SYMMETRIC AND ANTISYMMETRIC MATRIX OPERATORS IN PHYSICS

Badriyah Badriyah, Bambang Supriadi, Chilwatun Nasiroh +1 more

Classical physics began to be superseded by quantum physics because it could not explain microscopic phenomena. Quantum physics employs mathematical approaches such as wave mechanics and matrix mechanics, one of which is the eigenvalue problem involving matrix operators. There are many types of matrix operators; however, not all are suitable for measuring physical quantities in quantum physics. Operators in quantum physics must be Hermitian to ensure that their eigenvalues are real. Eigenvalue problems for matrix operators can be solved either analytically or numerically. Therefore, the objective of this research is to analyze symmetric and antisymmetric matrix operators in the measurement of physical quantities through eigenvalue problems. The research method utilized is a theoretical computational approach. The results demonstrate that symmetric matrix operators are Hermitian and yield real eigenvalues. In contrast, antisymmetric matrix operators are non Hermitian, yielding imaginary or complex eigenvalues. The results of solving eigen problems using Python obtained relative errors between 0 and orders., while the absolute error remained below 10%, thus confirming the validity of the results. Based on the findings, symmetric matrix operators satisfy the requirements for use in physical problems through eigenvalue analysis. The eigenvalues of antisymmetric matrices cannot represent the measurement of physical quantities directly and serve only as rotation operators. However, antisymmetric matrices can be utilized in quantum physics for physical measurements if they are transformed into Hermitian matrices. Furthermore, solving eigenvalue problems with Python is effective due to its high accuracy and small error margins.Classical physics began to be superseded by quantum physics because it could not explain microscopic phenomena. Quantum physics employs mathematical approaches such as wave mechanics and matrix mechanics, one of which is the eigenvalue problem involving matrix operators. There are many types of matrix operators; however, not all are suitable for measuring physical quantities in quantum physics. Operators in quantum physics must be Hermitian to ensure that their eigenvalues are real. Eigenvalue problems for matrix operators can be solved either analytically or numerically. Therefore, the objective of this research is to analyze symmetric and antisymmetric matrix operators in the measurement of physical quantities through eigenvalue problems. The research method utilized is a theoretical computational approach. The results demonstrate that symmetric matrix operators are Hermitian and yield real eigenvalues. In contrast, antisymmetric matrix operators are non Hermitian, yielding imaginary or complex eigenvalues. The results of solving eigen problems using Python obtained relative errors between 0 and orders., while the absolute error remained below 10%, thus confirming the validity of the results. Based on the findings, symmetric matrix operators satisfy the requirements for use in physical problems through eigenvalue analysis. The eigenvalues of antisymmetric matrices cannot represent the measurement of physical quantities directly and serve only as rotation operators. However, antisymmetric matrices can be utilized in quantum physics for physical measurements if they are transformed into Hermitian matrices. Furthermore, solving eigenvalue problems with Python is effective due to its high accuracy and small error margins.

Jurnal Ilmu Fisika dan Pembelajarannya (JIFP)Jun 26, 20260 citations

The complete solution is presented for the

N

-completeness problem, i.e., the problem of determining when a set of

p

-particle functions generates symmetric or antisymmetric functions without applying a symmetrizing or antisymmetrizing operator. Then it is shown that a symmetric or antisymmetric function has finite 2 rank if, and only if, it has finite 1 rank. Finally, the relation between

N

representability of

N

completeness is discussed.

The

N

-representability problem is approached by considering geminal product expansions of symmetric and antisymmetric functions. The

N

-completeness problem, i.e., the problem of determining when a set of geminals is a suitable basis for expanding symmetric or antisymmetric

N

-particle functions, is considered. New necessary conditions are given for both

N

-completeness and

N

-representability. In some cases, one can also obtain sufficient conditions; examples of such cases are discussed. The circumstances under which a density matrix can be derived from two or more different functions are also treated. Finally, extensions to higher order are also mentioned.