Complex Classical Mechanics of a QES Potential

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理论物理通讯 ›› 2015, Vol. 64 ›› Issue (04) : 425-432.

Bhabani Prasad Mandal¹, Sushant S. Mahajan²

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Complex Classical Mechanics of a QES Potential

Bhabani Prasad Mandal¹, Sushant S. Mahajan²

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摘要

We study a combined parity (P) and time reversal (T) invariant non-Hermitian quasi-exactly solvable (QES) potential, which exhibits PT phase transition, in the complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle escapes to an open orbits even with real energy if it is placed beyond a certain distance from the center of the well. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. Our study reveals that spontaneous PT symmetry breaking does not affect the qualitative features of the particle trajectories in the analogous complex classical model.

Abstract

导出引用

Bhabani Prasad Mandal, Sushant S. Mahajan. Complex Classical Mechanics of a QES Potential[J]. 理论物理通讯, 2015, 64(04): 425-432

Bhabani Prasad Mandal, Sushant S. Mahajan. Complex Classical Mechanics of a QES Potential[J]. Communications in Theoretical Physics, 2015, 64(04): 425-432

中图分类号： 03.65.Xp 03.65.Ca 11.30.Er 02.30.Fn 05.45.-a

参考文献

[1] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80(1998) 5243.

[2] C.M. Bender, Rep. Prog. Phys. 70(2007) 947.

[3] A. Mostafazadeh, Int. J. Geom. Math. Mod. Phys. 7(2010) 1191 and references therein.

[4] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 89(1998) 270401-1; C.M. Bender, K. Besseghir, H.F. Jones, and X. Yin, J. Phys. A 42(2009) 355301; C.M. Bender and B. Tan, J. Phys. A 39(2006) 1945; C.M. Bender and H.F. Jones, Phys. Lett. A 328(2004) 102.

[5] A. Khare and B.P. Mandal, Phys. Lett. A 272(2000) 53.

[6] A. Mostafazadeh and F. Zamani, Ann. Phys. 321(2006) 2183; ibid., (2006) 2210; A. Mostafazadeh and H. Mehr-Dehnavi, J. Phys. A 42(2009) 125303; A. Mostafazadeh, Phys. Rev. Lett. 102(2009) 220402; A. Mostafazadeh, J. Phys. A:Math. Theor. 44(2009) 375302.

[7] A. Ghatak, B.K. Mourya, R.D. RayMandal, and B.P. Mandal, Reciprocity in Parity Violating Non-Hermitian systems, IJTP (press) (2014); A. Ghatak, M. Hasan, and B.P. Mandal, Phys. Lett. A 379(2015) 1326; B.P. Mandal, B.K. Mourya, K. Ali, and A. Ghatak, PT Phase Transition in Graphene, (Submitted to a Journal).

[8] A. Khare and B.P. Mandal, Spl. Issue Pramana J. Phys. 73(2009) 387.

[9] B. Basu-Mallick, Int. J. Mod. Phys. B 16(2002) 1875; B. Basu-mallick, T. Bhattacharyya, A. Kundu, and B.P. Mandal, Czech. J. Phys. 54(2004) 5; B. Basu-Mallick and B.P. Mandal, Phys. Lett. A 284(2001) 231; B. Basu-mallick, T. Bhattacharyya, and B.P. Mandal, Mod. Phys. Lett. A 20(2004) 543.

[10] B.P. Mandal and A. Ghatak, J. Phys. A:Math. Gen. 45(2012) 444022.

[11] B.P. Mandal and S. Gupta, Mod. Phys. Lett. A 25(2010) 1723.

[12] A. Ghatak and B.P. Mandal, J. Phys. A:Math. Gen. 45(2012) 355301.

[13] B. Bagchi and C. Quesne, Phys. Lett. A 273(2000) 256.

[14] K. Abhinav and P. K. Panigrahi, Annl. Phys. 326(2011) 538.

[15] B.P. Mandal, Mod. Phys. Lett. A 20(2005) 655.

[16] A. Ghatak, R.D. Ray Mandal, and B.P. Mandal, Ann. Phys. 336(2013) 540.

[17] M. Hasan, A. Ghatak, and B.P. Mandal, Ann. Phys. 344(2014) 17.

[18] B.P. Mandal, B.K. Mourya, and R.K. Yadav, Phys. Lett. A 377(2013) 1043.

[19] A. Ghatak, Z.A. Nathan, B.P. Mandal, and Z. Ahmed, J. Phys. A:Math. Gen. 45(2012) 465305.

[20] C.M. Bender and G.V. Dunne, J. Math. Phys. 40(1999) 4616; C.M. Bender, D.C. Brody, and H.F. Jones, Phys. Rev. Lett. 93(2004) 251601; C.M. Bender and H.F. Jones, Phys. Rev. D 70(2004) 025001; C.M. Bender, S.F. Brandt, J.H. Chen, and Q.Wang, Phys. Rev. D 71(2005) 065010.

[21] J.L. Cardy, Phys. Rev. Lett. 54(1985) 1345; J.L. Cardy and G. Mussardo, Phys. Lett. B 225275(1989); A.B. Zamolodchikov, Nucl. Phys. B 348(1991) 619; R. Brower, M. Furman, and M. Moshe, Phys. Lett. B 76(1978) 213.

[22] C.M. Bender, V. Branchina, and E. Messina, Phys. Rev. D 85(2012) 085001.

[23] Z.H. Musslimani, K.G. Makris, R. El-Ganainy, and D.N. Christodulides, Phys. Rev. Lett. 100(2008) 030402; C.E. Ruter, K.G. Makris, R. El-Ganainy, D.N. Christodulides, M. Segev, and D. Kip, Nature (London) 6(2010) 192; A. Guo and G.J. Salamo, Phys. Rev. Lett. 103(2009) 093902; C.M. Bender, S. Boettcher, and P.N. Meisinger, J. Math. Phys. 40(1999) 2201; C.T. West, T. Kottos, and T. Prosen, Phys. Rev. Lett. 104(2010) 054102; C.M. Bender, B.K. Berntson, D. Parker, and E. Samuel, arXiv:math-ph/1206.4972v1.

[24] J. Schindler, A. Li, M.C. Zheng, F.M. Ellis, and T. Kottos, Phys. Rev. A 84(2011) 040101; H. Ramezani, J. Schindler, F.M. Ellis, and T. Kottos, Phys. Rev. A 85(2012) 062122; Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85(2012) 050101(R).

[25] C.M. Bender, G.V. Dunne, P.N. Meisinger, and M. Simsek, Phys. Lett. A 281(2001) 311; C.M. Bender, M. Berryand, and A. Mandilara, J. Phys. 35(2002) L467; M. Znojil, J. Phys. A 36(2003) 7825; G. Levai, J. Phys. A 41(2008) 244015.

[26] D. Bazeia, A. Das, and L. Losano, Phys. Lett. B 673(2009) 283.

[27] S. Dey, A. Fring, and L. Gouba, J. Phys. A 45(2012) 385302.

[28] B.P. Mandal, B. Mourya, and R.K. Yadav, Phys. Lett. A 377(2013) 1043.

[29] C.M. Bender, D.C. Brody, and D.W. Hook, J. Phys. A:Math. Theor. 41(2008) 352003; C.M. Bender, Peter N. Meisinger, and Qing-Hai Wang, Phys. Rev. Lett. 104(2010) 061601; Ann. Phys. 325(2010) 2332; C.M. Bender and D.W. Hook, J. Phys. A:Math. Theor. 44(2011) 372001; N. Turok, arxiv:1312.1772.

[30] S. Dey and A. Fring, Phys. Rev. A 88(2013) 022116.

[31] A. Nanayakkara, Czech. J. Phys. 54(2004) 101; J. Phys. A:Math. Gen. 37(2004) 4321; C.M. Bender, D.D. Holm, and D.W. Hook, J. Phys. A:Math. Theor. 40(2007) F81; C.M. Bender, D.D. Holm, and D.W. Hook, J. Phys. A:Math. Theor. 40(2007) F793. C.M. Bender, D.C. Brody, and D.W. Hook, J. Phys. A:Math. Theor. 41(2008) 352003. C.M. Bender, D.W. Hook, and K.S. Kooner, J. Phys. A:Math. Theor. 43(2010) 165201.

[32] Anjana Sinha, Eur. Phys. Lett. 98(2012) 60005.

[33] A. Sinha, D. Dutta, and P. Roy, Phys. Lett. A 375(2011) 452.

[34] A.G. Anderson, C.M. Bender, and U.I. Morone, Phys. Lett. A 375(2011) 3399; A. Cavaglia, A. Fring, and B. Bagchi, J. Phys. A:Math. Theor. 44(2011) 325201.

[35] A. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics, Inst. Physics Publishing, Bristol (1994) and references therein.

[36] C.M. Bender and A. Turbiner, Phys. Lett. A 173(1993) 442; P. Assis and A. Fring, J. Phys. A:Math. Theor. 42(2009) 015203.

[37] A. Khare and B.P. Mandal, J. Math. Phys. 39(1998) 3476.

[38] Y. Brihaye, A. Nininahazwe, and B. P. Mandal, J. Phys. A 40(2007) 13063.

[39] C.M. Bender and G.V. Dunne, J. Math. Phys. 37(1996) 6.