Physics Letters A 366 (2007) 30–35
Relationship between quantum-mechanical systems
with and without monopoles
, Armen Nersessian
, Armen Yeranyan
Yerevan State University, 1 Alex Manoogian street, Yerevan 375025, Armenia
Yerevan Physics Institute, 2 Alikhanian Brothers street, Yerevan 375036, Armenia
Received 10 November 2006; received in revised form 28 December 2006; accepted 16 January 2007
Available online 8 February 2007
Communicated by P.R. Holland
It is shown that the inclusion of the monopole ﬁeld in the three- and ﬁve-dimensional spherically symmetric quantum-mechanical systems, with
the addition of the special centrifugal term, leads to the lift of the range of the total and azimuth quantum numbers only. Meanwhile the functional
dependence of the energy spectra on quantum numbers does not undergo any changes. We also present a new integrable model of the spherical
2007 Elsevier B.V. All rights reserved.
During the last decades there has been much activity in the study of the integrable quantum-mechanical systems speciﬁed by the
presence of monopole-like ﬁeld conﬁgurations. It was initiated by the pioneer works by Zwanziger  and McIntosh and Cisneros
, where the analog of Coulomb problem with a Dirac monopole has been suggested, which inherits whole (nonlinear) symmetry
algebra of the Coulomb system (MICZ-Kepler system). The similarity between MICZ-Kepler and Coulomb systems has quite
transparent explanation in terms of four-dimensional space: these systems could be obtained from the four-dimensional oscillator
by the reduction by U(1) group action . In the same way, one can construct the ﬁve-dimensional analog of the MICZ-Kepler
problem, reducing the eight-dimensional oscillator by SU(2) group action . In this case, instead of Dirac monopole the SU(2)
Yang monopole appears in the system. The uniqueness of these reduction procedures insists on their close relation with the ﬁrst
and second Hopf maps (the detailed quantum-mechanical description of this correspondence could be found in ). Let us mention
that the existing analogs of these Coulomb-like systems on the curved spaces are also speciﬁed with the closed similarity of the
systems with and without monopoles  (see also ). The actual observable difference between these systems results in the lift of
the range of the total angular momentum, which in its turn leads to the degeneracy of the ground state.
In this Letter we show that the closed similarity between rotationally invariant three-/ﬁve-dimensional quantum-mechanical
systems with and without Dirac/Yang monopole is the general peculiarity of these systems. We consider the quantum mechanics
with central potential on the d = 3, 5-dimensional spaces equipped with so(d)-invariant metrics
= g(r)dr dr,r=|r|, r = (x
We will show that incorporation of the monopole with the addition to the potential of the speciﬁc “centrifugal term”,
E-mail addresses: email@example.com (L. Mardoyan), firstname.lastname@example.org (A. Nersessian), email@example.com (A. Yeranyan).
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