Physics Letters A 366 (2007) 30–35
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Relationship between quantum-mechanical systems
with and without monopoles
Levon Mardoyan
a
, Armen Nersessian
a,b
, Armen Yeranyan
a,∗
a
Yerevan State University, 1 Alex Manoogian street, Yerevan 375025, Armenia
b
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
Abstract
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
oscillator.
©
2007 Elsevier B.V. All rights reserved.
1. Introduction
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 [1] and McIntosh and Cisneros
[2], 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 [3]. 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 [4]. 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 [5]). 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 [6] (see also [7]). 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
(1.1)ds
2
= g(r)dr dr,r=|r|, r = (x
1
,...,x
d
).
We will show that incorporation of the monopole with the addition to the potential of the specific “centrifugal term”,
(1.2)U(r)→ U(r)+
˜s
2
2g(r)r
2
,
*
Corresponding author.
E-mail addresses: mardoyan@ysu.am (L. Mardoyan), arnerses@yerphi.am (A. Nersessian), ayeran@ysu.am (A. Yeranyan).
0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2007.01.049