Dimensional scaling as a symmetry operation

S. Kais, D. R. Herschbach, R. D. Levine

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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.

Original languageEnglish
Pages (from-to)7791-7796
Number of pages6
JournalThe Journal of Chemical Physics
Issue number12
Publication statusPublished - 1 Dec 1989
Externally publishedYes


ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics

Cite this

Kais, S., Herschbach, D. R., & Levine, R. D. (1989). Dimensional scaling as a symmetry operation. The Journal of Chemical Physics, 91(12), 7791-7796.