Dimensional scaling as a symmetry operation

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

Research output: Contribution to journalArticle

123 Citations (Scopus)

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/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
Volume91
Issue number12
Publication statusPublished - 1 Dec 1989
Externally publishedYes

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Ladders
Algebra
scaling
Hamiltonians
symmetry
Invariance
ladders
operators
Mathematical operators
algebra
harmonic motion
Atoms
Electrons
continuous spectra
invariance
energy spectra
harmonics
shift
atoms
electrons

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.

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

In: The Journal of Chemical Physics, Vol. 91, No. 12, 01.12.1989, p. 7791-7796.

Research output: Contribution to journalArticle

Kais, S, Herschbach, DR & Levine, RD 1989, 'Dimensional scaling as a symmetry operation', The Journal of Chemical Physics, vol. 91, no. 12, pp. 7791-7796.
Kais S, Herschbach DR, Levine RD. Dimensional scaling as a symmetry operation. The Journal of Chemical Physics. 1989 Dec 1;91(12):7791-7796.
Kais, S. ; Herschbach, D. R. ; Levine, R. D. / Dimensional scaling as a symmetry operation. In: The Journal of Chemical Physics. 1989 ; Vol. 91, No. 12. pp. 7791-7796.
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