Explicit Phase Space Transformations and Their Application in Noncommutative Quantum Mechanics
Main Article Content
Abstract
We study a problem of transformations mapping noncommutative phase spaces into commutative ones. We find a simple way to obtain explicit formulas of such transformations in 3D and indicate matrix equations for numerical computation in higher dimensions. Then we use these formulas to calculate the energy levels of the hydrogen-like atom with six noncommutative parameters. We also find and prove new relations between the hydrogen eigenfunctions corresponding to the n-th energy level.
Keywords: Noncommutativity, noncommutative quantum mechanics, Hydrogen-like atom. PACS numbers: 11.10.Nx, 02.40.Gh, 31.15.-p, 03.65.-w, 03.65.Fd.
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[3] M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001), and references therein.
[4] R. J. Szabo, Phys. Rep. 378, 207 (2003), and references therein.
[5] A. Connes , Noncommutative Geometry (Academic Press. Inc, 1994); A. Connes, M. R. Douglas and A. Schwarz, J. High Energy Phys. 9802, 003 (1998).
[6] V. P. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267 (2001);
J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D 64, 067901 (2001);
J. Gamboa, F. Mendez, M. Loewe and J. C. Rojas, Int. J. Mod. Phys. A 17, 2555 (2002);
J. Gamboa, F. Mendez, M. Loewe and J. C. Rojas, Mod. Phys. Lett. A 16, 2075 (2001);
S. Bellucci, A. Nersessian and C. Sochichiu, Phys. Lett. B 522, 345 (2001);
P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602 (2002); A. Deriglazov, Phys. Lett. B 530, 235 (2002);
P. A. Horvathy, L. Martina and P. Stichel, Phys. Lett. B 564, 149 (2003);
[7] H. Falomir, J. Gamboa, J. Lopez-Sarrion, F. Mendez and P. A. G. Pisani, Phys. Lett. B 680, 384 (2009);
M. Gomes, V. G. Kupriyanov and A. J. da Silva, Phys. Rev. D 81, 085024 (2010);
A. Das, H. Falomir, M. Nieto, J. Gamboa and F. Mendez, Phys. Rev. D 84, 045002 (2011);
H. Falomir, J. Gamboa, M. Loewe and M. Nieto, J. Phys. A 45, 135308 (2012);
H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D 85, 025009 (2012).
[8] T. C. Adorno, M. C. Baldiotti, M. Chaichian, D. M. Gitman and A. Tureanu, Phys. Lett. B 682, 235 (2009);
T. C. Adorno, M. C. Baldiotti and D. M. Gitman, Phys. Rev. D 82, 123516 (2010);
T. C. Adorno, D. M. Gitman, A. E. Shabad and D. V. Vassilevich, Phys. Rev. D 84, 085031 (2011);
T. C. Adorno, D. M. Gitman and A. E. Shabad, Phys. Rev. D 86, 027702 (2012);
A. F. Ferrari, M. Gomes and C. A. Stechhahn, Phys. Rev. D 82, 045009 (2010);
K. Ma and S. Dulat, Phys. Rev. A 84, 012104 (2011); X. Yu and K. Li, Phys. Rev. A 84, 033617 (2011);
N. Chair, A. Al Jamel, M. Sarhan, M. Abu Sini and E. M. Rabie, J. Phys. A 44, 095306 (2011);
R. V. Maluf, Int. J. Mod. Phys. A 26, 4991 (2011); L. Khodja and S. Zaim, Int. J. Mod. Phys. A 27, 1250100 (2012);
A. Jafari, Int. J. Mod. Phys. A 29, 1450143 (2014).
[9] X. Calmet and A. Kobakhidze, Phys. Rev. D 72, 045010 (2005); M. Dobrski, Phys. Rev. D 84, 065005 (2011).
[10] J. W. Moffat, Phys. Lett. B 491, 345 (2000); A. H. Chamseddine, Int. J. Mod. Phys. A 16, 759 (2001);
X. Calmet and A. Kobakhidze, Phys. Rev. D 74, 047702 (2006);
E. Harikumar and V. O Rivelles, Class. Quantum Grav. 23, 7551 (2006);
H. S. Yang, Int. J. Mod. Phys. A 24, 4473 (2009).
[11] P. Nicolini, A. Smailagic and E. Spallucci, Phys. Lett. B 632, 547 (2006);
J. C. Lopez-Dominguez, O. Obregon, C. Ramirez and M. Sabido, Phys. Rev. D 74, 084024 (2006);
Y. S. Myung, Y.-W. Kim and Y.-J. Park, J. High Energy Phys. 0702, 012 (2007);
R. Banerjee, B. R. Majhi and S. Samanta, Phys. Rev. D 77, 124035 (2008);
R. Banerjee, B. R. Majhi and S. K. Modak, Class. Quantum Grav. 26, 085010 (2008);
M. Chaichian, A. Tureanu, M. R. Setare and G. Zet, J. High Energy Phys. 0804, 064 (2008);
A. Kobakhidze, Phys. Rev. D 79, 047701 (2009); P. Nicolini, Int. J. Mod. Phys. A 24, 1229 (2009);
L. Modesto and P. Nicolini, Phys. Rev. D 82, 104035 (2010);
J. R. Mureika and P. Nicolini, Phys. Rev. D 84, 044020 (2011);
R. B. Mann and P. Nicolini, Phys. Rev. D 84, 064014 (2011).
[12] [12] F. Lizzi, G. Mangano, G. Miele and G. Sparano, Int. J. Mod. Phys. A 11, 2907 (1996);
S. Alexander, R. Brandenberger and J. Magueijo, Phys. Rev. D 67, 081301(R) (2003);
W. Nelson and M. Sakellariadou, Phys. Lett. B 680, 263 (2009);
W. Nelson and M. Sakellariadou, Phys. Rev. D 81, 085038 (2010);
M. Buck, M. Fairbairn and M. Sakellariadou, Phys. Rev. D 82, 043509 (2010);
M. Rinaldi, Class. Quantum Grav. 28, 105022 (2011);
T. S. Koivisto and D. F. Mota, J. High Energy Phys. 1102, 061 (2011);
U. D. Machado and R. Opher, Class. Quantum Grav. 29, 065003 (2012);
M. Marcolli, E. Pierpaoli and K. Teh, Commun. Math. Phys. 309, 341 (2012).
[13] X. Calmet, Europhys. Lett. 77, 19002 (2007);
M. Chaichian, A. Tureanu, M. R. Setare and G. Zet, J. High Energy Phys. 0804, 064 (2008);
R. Garattini and P. Nicolini, Phys. Rev. D 83, 064021 (2011).
[14] L. Alvarez-Gaume and S. R. Wadia, Phys. Lett. B 501, 319 (2001);
L. Alvarez-Gaume and J. L. F. Barbon, Int. J. Mod. Phys. A 16, 1123 (2001).
[15] J.-z. Zhang, Phys. Lett. B 584, 204 (2004); Phys. Rev. Lett. 93, 043002 (2004).
[16] A. Smailagic and E. Spallucci, Phys. Rev. D 65, 107701 (2002); J. Phys. A 35, 363 (2002).
[17] L. Jonke and S. Meljanac, Eur. Phys. J. C29, 433 (2003);
[18] K. Li, J. Wang and C. Chen, Mod. Phys. Lett. A 20, 2165 (2005).
[19] A. E. F. Djemai and H. Smail, Commun. Theor. Phys. 41, 837 (2004);
O. Bertolami, J. G. Rosa, C. M. L. de Aragao, P. Castorina and D. Zappala, Phys. Rev. D 72, 025010;
Mod. Phys. Lett. A 21, 795 (2006);
C. Bastos, O. Bertolami, N. Dias and J. Prata, Int. J. Mod. Phys. A 28, 1350064 (2013).
[20] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); Eur. Phys. J. C 36, 251 (2004).
[21] T. C. Adorno, M. C. Baldiotti, M. Chaichian, D. M. Gitman and A. Tureanu, Phys. Lett. B 682, 235 (2009);
S. Zaim, L. Khodja and Y. Delenda, Int. J. Mod. Phys. A 26, 4133 (2011);
L. Khodja and S. Zaim, Int. J. Mod. Phys. A 27, 1250100 (2012);
M. Moumni and A. BenSlama, Int. J. Mod. Phys. A 28, 1350139 (2013);
W. O. Santos and A. M. C. Souza, Int. J. Mod. Phys. A 29, 1450177 (2014); Int. J. Theor. Phys. 51, 3882 (2012).
[22] K. Li and S. Dulat, Eur. Phys. J. C 46, 825 (2006).