Dimensional scaling as a symmetry operation

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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 supersymmetry 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.

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