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.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.

Physical Review AApr 1, 199517 citations

Operators and Matrix Elements

Brian L. Silver

Elsevier eBooksJan 1, 19760 citations

Diagrammatic technique for calculating matrix elements of collective operators in superradiance

Ching Tsung Lee

Adopting the so-called "genealogical construction," one can express the eigenstates of collective operators corresponding to a specified mode for an $N$ -atom system in terms of those for an ($N\ensuremath{-}1$)-atom system. Using these Dicke states as bases and using the Wigner-Eckart theorem, a matrix element of a collective operator of an arbitrary mode can be written as the product of an $m$ -dependent factor and an $m$ -independent reduced matrix element (RME). A set of recursion formulas for the RME is obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups is then introduced. This gives a simple and systematic way of calculating the RME. This method is especially useful when the cooperation number $\mathcal{r}$ is close to $\frac{N}{2}$ , where almost exact asymptotic expressions can be obtained easily. The result shows explicitly the geometry dependence of superradiance and the relative importance of $\mathcal{r}$ -conserving and $\mathcal{r}$ -nonconserving processes. This clears up the chief difficulty encountered in the Dicke-Schwendimann approach to the problem of $N$ two-level atoms, spread over large regions, interacting with a multimode radiation field.Adopting the so-called "genealogical construction," one can express the eigenstates of collective operators corresponding to a specified mode for an $N$-atom system in terms of those for an ($N\ensuremath{-}1$)-atom system. Using these Dicke states as bases and using the Wigner-Eckart theorem, a matrix element of a collective operator of an arbitrary mode can be written as the product of an $m$-dependent factor and an $m$-independent reduced matrix element (RME). A set of recursion formulas for the RME is obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups is then introduced. This gives a simple and systematic way of calculating the RME. This method is especially useful when the cooperation number $\mathcal{r}$ is close to $\frac{N}{2}$, where almost exact asymptotic expressions can be obtained easily. The result shows explicitly the geometry dependence of superradiance and the relative importance of $\mathcal{r}$-conserving and $\mathcal{r}$-nonconserving processes. This clears up the chief difficulty encountered in the Dicke-Schwendimann approach to the problem of $N$ two-level atoms, spread over large regions, interacting with a multimode radiation field.

Physical Review AAug 1, 197510 citations

Matrix elements of local operators

Brian P. Anderson, David H. Sharp, Arthur S. Wightman

Jan 1, 19740 citations

Superradiant effects on pulse propagation in resonant media

C. Lee

Adopting the so-called genealogical construction, the eigenstates of collective operators can be expressed corresponding to a specified mode for an N-atom system in terms of those for an (N-1)-atom system. Matrix element of a collective operator of an arbitrary mode is presented which can be written as the product of an m-dependent factor and an m-independent reduced matrix element (RME). A set of recursion formulas for the RME was obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups was then introduced. This gave a simple and systematic way of calculating the RME. Results show explicitly the geometry dependence of superradiance and the relative importance of r-conserving and r-nonconserving processes and clears up the chief difficulty encounted in the problem of N two-level atoms, spread over large regions, interacting with a multimode radiation field.Adopting the so-called genealogical construction, the eigenstates of collective operators can be expressed corresponding to a specified mode for an N-atom system in terms of those for an (N-1)-atom system. Matrix element of a collective operator of an arbitrary mode is presented which can be written as the product of an m-dependent factor and an m-independent reduced matrix element (RME). A set of recursion formulas for the RME was obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups was then introduced. This gave a simple and systematic way of calculating the RME. Results show explicitly the geometry dependence of superradiance and the relative importance of r-conserving and r-nonconserving processes and clears up the chief difficulty encounted in the problem of N two-level atoms, spread over large regions, interacting with a multimode radiation field.

Alabama Univ., Huntsville ReportSep 1, 19750 citations

Quantum matrix elements

William Martin McClain

Springer eBooksJan 1, 20090 citations

Adopting the so-called "genealogical construction," one can express the eigenstates of collective operators corresponding to a specified mode for an

N

-atom system in terms of those for an ($N\ensuremath{-}1$)-atom system. Using these Dicke states as bases and using the Wigner-Eckart theorem, a matrix element of a collective operator of an arbitrary mode can be written as the product of an

m

-dependent factor and an

m

-independent reduced matrix element (RME). A set of recursion formulas for the RME is obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups is then introduced. This gives a simple and systematic way of calculating the RME. This method is especially useful when the cooperation number

\mathcal{r}

is close to

\frac{N}{2}

, where almost exact asymptotic expressions can be obtained easily. The result shows explicitly the geometry dependence of superradiance and the relative importance of

\mathcal{r}

-conserving and

\mathcal{r}

-nonconserving processes. This clears up the chief difficulty encountered in the Dicke-Schwendimann approach to the problem of

N

two-level atoms, spread over large regions, interacting with a multimode radiation field.