The quantum free particle on the sphere
$S_\kappa ^2$
(κ > 0) and on the hyperbolic plane $H_\kappa ^2$
(κ < 0) is studied using a formalism that considers the curvature κ as a parameter. The first part is mainly concerned with the analysis of some geometric formalisms appropriate for the description of the dynamics on the spaces ($S_\kappa ^2$
, ${\rm I\hspace{-1.79993pt}R}^2$
, $H_\kappa ^2$
) and with the transition from the classical κ-dependent system to the quantum one using the quantization of the Noether momenta. The Schrödinger separability and the quantum superintegrability are also discussed. The second part is devoted to the resolution of the κ-dependent Schrödinger equation. First the characterization of the κ-dependent “curved” plane waves is analyzed and then the specific properties of the spherical case are studied with great detail. It is proved that if κ > 0 then a discrete spectrum is obtained. The wavefunctions, that are related with a κ-dependent family of orthogonal polynomials, are explicitly obtained.REFERENCES
1.
N. D.
Birrell
and P. C.
Davies
, Quantum Fields in Curved Space
(Cambridge University Press
, Cambridge, England
, 1994
).2.
L. E.
Parker
and D. J.
Toms
, Quantum Field Theory in Curved Spacetime
(Cambridge University Press
, Cambridge, England
, 2009
).3.
A.
Comtet
, Ann. Phys.
173
, 185
(1987
).4.
G. V.
Dunne
, Ann. Phys.
215
, 233
(1992
).5.
S.
Kim
and C.
Lee
, Ann. Phys.
296
, 390
(2002
).6.
H.
Fakhri
and M.
Shariati
, J. Phys. A
37
, L539
(2004
).7.
V. V.
Gritsev
and Y. A.
Kurochkin
, Phys. Rev. B
64
, 035308
(2001
).8.
A. A.
Bogush
, V. V.
Gritsev
, Y. A.
Kurochkin
, and V. S.
Otchik
, Phys. At. Nucl.
65
, 1052
(2002
).9.
D. V.
Bulaev
, V. A.
Geyler
, and V. A.
Margulis
, Phys. Rev. B
69
, 195313
(2004
).10.
C.
Furtado
, A.
Rosas
, and S.
Azevedo
, EPL
79
, 57001
(2007
).11.
P.
Stovicek
and M.
Tusek
, Recent Advances in Operator Theory in Hilbert and Krein Spaces, Operator Theory: Advances and Applications
Vol. 198 (Birkhauser-Verlag, Bassel, 2010
),e-print arXiv:0811.3825.
12.
13.
L.
Infeld
, Phys. Rev.
59
, 737
(1941
).14.
A. F.
Stevenson
, Phys. Rev.
59
, 842
(1941
).15.
L.
Infeld
and A.
Schild
, Phys. Rev.
67
, 121
(1945
).16.
A. O.
Barut
, A.
Inomata
, and G.
Junker
, J. Phys. A
20
, 6271
(1987
).17.
A. O.
Barut
, A.
Inomata
, and G.
Junker
, J. Phys. A
23
, 6271
(1990
).18.
L. M.
Nieto
, M.
Santander
, and H. C.
Rosu
, Mod. Phys. Lett. A
14
, 2463
(1999
).19.
S.
Bellucci
and A.
Nersessian
, Phys. Rev. D
67
, 065013
(2003
).20.
S.
Bellucci
, A.
Nersessian
, and A.
Yeranyan
, Phys. Rev. D
70
, 085013
(2004
).21.
L.
Mardoyan
and A.
Nersessian
, Phys. Rev. B
72
, 233303
(2005
).22.
P. R.
Giri
, J. Phys. A
40
, 3539
(2007
).23.
C.
Grosche
, Fortschr. Phys.
42
, 509
(1994
).24.
A.
Lecheheb
, M.
Merad
, and T.
Boudjedaa
, Ann. Phys.
322
, 1233
(2007
).25.
P. W.
Higgs
, J. Phys. A
12
, 309
(1979
).26.
H. I.
Leemon
, J. Phys. A
12
, 489
(1979
).27.
P. L.
Ferreira
, J. Phys. A
16
, 2087
(1983
).28.
A. O.
Barut
and R.
Wilson
, Phys. Lett. A
110
, 351
(1985
).29.
N.
Katayama
, Nuovo Cimento B
107
, 763
(1992
).30.
D.
Bonatsos
, C.
Daskaloyannis
, and K.
Kokkotas
, Phys. Rev. A
48
, R3407
(1993
).31.
D.
Bonatsos
, C.
Daskaloyannis
, and K.
Kokkotas
, Phys. Rev. A
50
, 3700
(1994
).32.
E. G.
Kalnins
, W.
Miller
, and G. S.
Pogosyan
, J. Math. Phys.
41
, 2629
(2000
).33.
J. J.
Slawianowski
, Rep. Math. Phys.
46
, 429
(2000
).34.
V. V.
Gritsev
, Y. A.
Kurochkin
, and V. S.
Otchik
, J. Phys. A
33
, 4903
(2000
).35.
A.
Nersessian
and G.
Pogosyan
, Phys. Rev. A
63
, 020103
(2001
).36.
M. F.
Rañada
and M.
Santander
, J. Math. Phys.
43
, 431
(2002
).37.
M. F.
Rañada
and M.
Santander
, J. Math. Phys.
44
, 2149
(2003
).38.
T. G.
Vozmischeva
, Integrable Problems of Celestial Mechanics in Spaces of Constant Curvature, Astrophysics and Space Science Library
, Vol. 295 (Kluwer Academic Publishers, Dordrecht, 2003
).39.
D.
Voyatzi
and S.
Ichtiaroglou
, Celest. Mech. Dyn. Astron.
93
, 331
(2005
).40.
A. V.
Borisov
and I. S.
Mamaev
, Celest. Mech. Dyn. Astron.
96
, 1
(2006
).41.
J. F.
Cariñena
, M. F.
Rañada
, and M.
Santander
, J. Phys. A
40
, 13645
(2007
).42.
P.
Stovicek
and M.
Tusek
, Russian J. Math. Phys.
14
, 493
(2007
).43.
J. F.
Cariñena
, M. F.
Rañada
, and M.
Santander
, J. Math. Phys.
49
, 032703
(2008
).44.
M.
Gadella
, J.
Negro
, G. P.
Pronko
, and M.
Santander
, J. Phys. A
41
, 304030
(2008
).45.
V. V.
Gritsev
and Y. A.
Kurochkin
, J. Phys. A
33
, 4073
(2000
).46.
V. V.
Kudryashov
, Y. A.
Kurochkin
, E. M.
Ovsiyuk
, and V. M.
Red'kov
, SIGMA Symmetry, Integrability and Geometry. Methods and Applications
6
, 004
(2010
).47.
C.
Grosche
, G. S.
Pogosyan
, and A. N.
Sissakian
, Fortschr. Phys.
43
, 523
(1995
).48.
M. F.
Rañada
and M.
Santander
, J. Math. Phys.
40
, 5026
(1999
).49.
E. G.
Kalnins
, J. M.
Kress
, G.
Pogosyan
, and W.
Miller
, J. Phys. A
34
, 4705
(2001
).50.
E. G.
Kalnins
, J. M.
Kress
, and P.
Winternitz
, J. Math. Phys.
43
, 970
(2002
).51.
A.
Ballesteros
, F. J.
Herranz
, M.
Santander
, and T.
Sanz-Gil
, J. Phys. A
36
, L93
(2003
).52.
F. J.
Herranz
and A.
Ballesteros
, SIGMA Symmetry, Integrability and Geometry. Methods and Applications
2
, 010
(2006
).53.
A.
Ballesteros
, A.
Enciso
, F. J.
Herranz
, and O.
Ragnisco
, J. Nonlinear Math. Phys.
15
(3
), 43
(2008
). 54.
A.
Ballesteros
and F. J.
Herranz
, J. Phys. A
42
, 245203
(2009
).55.
N. L.
Balazs
and A.
Voros
, Phys. Rep.
143
, 109
(1986
).56.
M. E.
Spina
and M.
Saraceno
, J. Phys. A
32
, 7803
(1999
).57.
J. F. Cariñena,
M. F.
Rañada
, M.
Santander
, and M.
Senthilvelan
, Nonlinearity
17
, 1941
(2004
).58.
59.
J. F.
Cariñena
, M. F.
Rañada
, and M.
Santander
, Ann. Phys.
322
, 434
(2007
).60.
J. F.
Cariñena
, M. F.
Rañada
, and M.
Santander
, Ann. Phys.
322
, 2249
(2007
).61.
J. F.
Cariñena
, M. F.
Rañada
, and M.
Santander
, J. Math. Phys.
48
, 102106
(2007
).62.
63.
M.
Lakshmanan
and S.
Rajasekar
, Nonlinear Dynamics
. Integrability, Chaos and Patterns, Advanced Texts in Physics (Springer-Verlag, Berlin, 2003
).64.
P.
Létourneau
and L.
Vinet
, Ann. Phys.
243
, 144
(1995
).65.
J.
Hietarinta
, Phys. Lett. A
246
, 97
(1998
).66.
M. B.
Sheftel
, P.
Tempesta
, and P.
Winternitz
, J. Math. Phys.
42
, 659
(2001
).67.
C.
Daskaloyannis
, J. Math. Phys.
42
, 1100
(2001
).68.
S.
Gravel
and P.
Winternitz
, J. Math. Phys.
43
, 5902
(2002
).69.
M. A.
Rodriguez
and P.
Winternitz
, J. Math. Phys.
43
, 1309
(2002
).70.
J.
Bérubé
and P.
Winternitz
, J. Math. Phys.
45
, 1959
(2004
).71.
A.
Ballesteros
, F.
Herranz
, and O.
Ragnisco
, J. Phys. A
38
, 7129
(2005
).72.
E. G.
Kalnins
, W.
Miller
, and G. S.
Pogosyan
, J. Math. Phys.
47
, 033502
(2006
).73.
J. A.
Calzada
, J.
Negro
, and M. A.
del Olmo
, J. Math. Phys.
47
, 043511
(2006
).74.
M.
Gadella
, J.
Negro
, L. M.
Nieto
, G. P.
Pronko
, and M.
Santander
, J. Math. Phys.
52
, 063509
(2011
),e-print arXiv:1007.1423.
75.
W.
Klinberger
, A Course in Differential Geometry, Graduate Texts in Mathematics
(Springer-Verlag, New York, 1978
).© 2011 American Institute of Physics.
2011
American Institute of Physics
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