Legendre Polynomials and Related Functions

Valeriya Akhmedova, Emil Tofik ogly Akhmedov

Vladimir I. Man’ko, Liubov A. Markovich

Using entropic inequalities for Shannon entropies new inequalities for some classical polynomials are obtained. To this end, photon distribution functions for one-, two- and multi-mode squeezed states in terms of Hermite, Laguerre, Legendre polynomials and Gauss' hypergeometric functions are used. The dependence between the violation of the quadrature uncertainty relation, the sign and the existence of the distribution function of such states is considered.

Ali Akbari, Norman Henry March, Ángel Rubio

The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction $u({r}_{12})$ is studied. One representation with contracted information is first explored by constructing the momentum density $\ensuremath{\rho}(\mathbf{p})$ in terms of the wave function of the relative motion, say ${\ensuremath{\Psi}}_{R}({r}_{12})$, which naturally depends on the choice of $u({r}_{12})$. For $u({r}_{12})={e}^{2}∕{r}_{12}$, the so-called Hookean atom, and for the inverse square law $u({r}_{12})=\ensuremath{\lambda}∕{r}_{12}^{2}$, plots are presented of the above density $\ensuremath{\rho}(\mathbf{p})$ in momentum space. The correlated kinetic energy is recovered from averaging ${p}^{2}∕2m$, $m$ denoting the electron mass, with respect to $\ensuremath{\rho}(\mathbf{p})$. The second method developed is in coordinate space and expands the density matrix $\ensuremath{\gamma}({\mathbf{r}}_{1},{\mathbf{r}}_{2})$ in Legendre polynomials, using relative coordinate ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$, center-of-mass coordinate $({\mathbf{r}}_{1}+{\mathbf{r}}_{2})∕2$ and the angle, $\ensuremath{\theta}$ say, between these two vectors. For the Moshinsky atom in which $u({r}_{12})=\frac{1}{2}k{r}_{12}^{2}$ only the $s$ term $(l=0)$ contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms $(s+d)$ of the Legendre expansion. The Wigner function is finally calculated analytically for both Moshinsky and inverse square law models.

H. C. Fan, Liyun Hu

Shi‐Hai Dong

Mark W. Coffey

Supersymmetric quantum mechanics has many applications, and typically uses a raising and lowering operator formalism. For one dimensional problems, we show how such raising and lowering operators may be generalized to include an arbitrary function. As a result, the usual Rodrigues' formula of the theory of orthogonal polynomials may be recovered in special cases, and it may otherwise be generalized to incorporate an arbitrary function. We provide example generalized operators for several important classical orthogonal polynomials, including Chebyshev, Gegenbauer, and other polynomials. In particular, as concerns Legendre polynomials and associated Legendre functions, we supplement and generalize results of Bazeia and Das.