Dimensional scaling as a symmetry operation

摘要

The scaling of the Schrouml;dinger equation with spatial dimensionDis studied by an algebraic approach. For any spherically symmetric potential, the Hamiltonian is invariant under such scaling to order 1/D2. For the special family of potentials that are homogeneous functions of the radial coordinate, the scaling invariance is exact to all orders in 1/D. Explicit algebraic expressions are derived for the operators which shiftDup or down. These ladder operators form an SU(1,1) algebra. The spectrum generating algebra to order 1/D2corresponds to harmonic motion. In theDrarr;infin; limit the ladder operators commute and yield a classicalhyphen;like continuous energy spectrum. The relation of supersymmetry andDscaling is also illustrated by deriving an analytic solution for the Hookersquo;s law model of a twohyphen;electron atom, subject to a constraint linking the harmonic frequency to the nuclear charge and the dimension.