### 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 |
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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 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*The Journal of Chemical Physics*,

*91*(12), 7791-7796.

**Dimensional scaling as a symmetry operation.** / Kais, S.; Herschbach, D. R.; Levine, R. D.

Research output: Contribution to journal › Article

*The Journal of Chemical Physics*, vol. 91, no. 12, pp. 7791-7796.

}

TY - JOUR

T1 - Dimensional scaling as a symmetry operation

AU - Kais, S.

AU - Herschbach, D. R.

AU - Levine, R. D.

PY - 1989/12/1

Y1 - 1989/12/1

N2 - 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/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 shift D up or down. These ladder operators form an SU(1,1) algebra. The spectrum generating algebra to order 1/D2 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.

AB - 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/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 shift D up or down. These ladder operators form an SU(1,1) algebra. The spectrum generating algebra to order 1/D2 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.

UR - http://www.scopus.com/inward/record.url?scp=0000290618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000290618&partnerID=8YFLogxK

M3 - Article

VL - 91

SP - 7791

EP - 7796

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 12

ER -