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The Hamiltonian for the Coulomb problem in *n* space dimensions is given. It involves the two *n*-vectors ** R** and

**, which form a Heisenberg algebra. It follows that the**

*P**n*(

*n*–1)/2 components of orbital angular momentum

*L*formed out of

_{ij}**and**

*R***generate the algebra**

*P**so*(

*n*). The Lenz–Runge

*n*-vector is introduced and shown to be a constant of the motion. A properly rescaled Lenz–Runge vector is shown to generate, together with the

*L*,

_{ij}*so*(

*n*+1). The Hamiltonian can be expressed in terms of the quadratic Casimir operator of this

*so*(

*n*+1). This yields the Balmer formula for the energy levels of the hydrogen atom. Biographical notes on Coulomb, Heisenberg, Lenz and Runge are given.

*Keywords: *
Coulomb problem;
Heisenberg algebra;
Lenz–Runge n-vector;
hydrogen atom energy levels

*Chapter.*
*2470 words.*

*Subjects: *
Mathematical and Statistical Physics

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