Matrix Representations of Many-Particle Hamiltonians in Hyperspherical Coordinates

Abstract

Let us consider a system of N particles, whose Hamiltonian is given by

$$ H = - \frac{1}{2}\Delta + V({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_{1}},{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_{2}}, \ldots ,{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_{N}}) $$

(8-1)

where

$$ \Delta \equiv \sum\limits_{{a = 1}}^{N} {\left( {\frac{{{{\partial }^{2}}}}{{\partial x_{a}^{2}}} + \frac{{{{\partial }^{2}}}}{{\partial y_{a}^{2}}} + \frac{{{{\partial }^{2}}}}{{\partial z_{a}^{2}}}} \right)} = \sum\limits_{{a = 1}}^{N} {\frac{1}{{{{m}_{a}}}}\left( {\frac{{{{\partial }^{2}}}}{{\partial x_{a}^{{{{'}^{2}}}}}} + \frac{{{{\partial }^{2}}}}{{\partial y_{a}^{{{{'}^{2}}}}}} + \frac{{{{\partial }^{2}}}}{{\partial z_{a}^{{{{'}^{2}}}}}}} \right)} $$

(8-2)

and where x_a is a 3-dimensional vector

$$ {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_{a}} = ({{x}_{a}},{{y}_{a}},{{z}_{a}}) = m_{a}^{{ \frac{\hbox{$\scriptstyle 1$}}{\hbox{$\scriptstyle 2$}} }}(x_{a}^{'},y_{a}^{'},z_{a}^{'}) $$

(8-3)

representing the three mass-weighted Cartesian coordinates of the ath particle.