Quantum mechanics of the 1∕x2 potential

Essin, Andrew M.; Griffiths, David J.

doi:doi:10.1119/1.2165248

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American Journal of Physics -- February 2006 -- Volume 74, Issue 2, pp. 109

Quantum mechanics of the 1∕x² potential

Andrew M. Essin¹ and David J. Griffiths²

¹Department of Physics, University of California, Berkeley, California 94720
²Department of Physics, Reed College, Portland, Oregon 97202

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In quantum mechanics a localized attractive potential typically supports a (possibly infinite) set of bound states, characterized by a discrete spectrum of allowed energies, together with a continuum of scattering states, characterized (in one dimension) by an energy-dependent phase shift. The 1∕x² potential on 0<x<∞ confounds all of our intuitions and expectations. Resolving its paradoxes requires sophisticated theoretical machinery: regularization, renormalization, anomalous symmetry-breaking, and self-adjoint extensions. Our goal is to introduce the essential ideas at a level accessible to advanced undergraduates.