### Abstract

The scaling of the Schrödinger equation with spatial dimension D is studied by an algebraic approach. For any spherically symmetric potential, the Hamiltonian is invariant under such scaling to order 1/D^{2}. 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 shift D up or down. These ladder operators form an SU(1,1) algebra. The spectrum generating algebra to order 1/D^{2} corresponds to harmonic motion. In the D → ∞ limit the ladder operators commute and yield a classical-like continuous energy spectrum. The relation of super symmetry and D scaling is also illustrated by deriving an analytic solution for the Hooke's law model of a two-electron atom, subject to a constraint linking the harmonic frequency to the nuclear charge and the dimension.

Original language | English |
---|---|

Pages (from-to) | 7791-7796 |

Number of pages | 6 |

Journal | The Journal of Chemical Physics |

Volume | 91 |

Issue number | 12 |

Publication status | Published - 1 Dec 1989 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*The Journal of Chemical Physics*,

*91*(12), 7791-7796.