Chapter

The Coulomb problem in <i>n</i> space dimensions

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662920.003.0021
The Coulomb problem in n space dimensions

More Like This

Show all results sharing this subject:

  • Mathematical and Statistical Physics

GO

Show Summary Details

Preview

The Hamiltonian for the Coulomb problem in n space dimensions is given. It involves the two n-vectors R and P, which form a Heisenberg algebra. It follows that the n(n–1)/2 components of orbital angular momentum Lij formed out of R and P generate the algebra 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 Lij, 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

Full text: subscription required

How to subscribe Recommend to my Librarian

Buy this work at Oxford University Press »

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.