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There are two approaches to investigating the quantum mechanics for a particle constrained on a curved hypersurface, namely the Schrödinger formalism and the Dirac theory.#br#The Schrödinger formalism utilizes the confining potential technique to lead to a unique form of geometric kinetic energy T that contains the geometric potential VS and the geometric momentum p,#br#T=-ħ2/(2m)▽2+VS=-ħ2/(2m)[▽2+(M2-K)],p=-iħ(▽2+Mn),#br#where ▽2 is the gradient operator on the two-dimensional surface. Both the kinetic energy and momentum are geometric invariants. The geometric potential has been experimentally confirmed in two systems.#br#The Dirac's canonical quantization procedure assumes that the fundamental quantum conditions involve only the canonical position x and momentum p, which are in general given by#br#[xi,xj]=iħÂij,[pi,pj]=iħΩij,[xi,pj]=iħΘij#br#where Âij, Ωij, and Θijare all antisymmetric tensors. It does not always produce a unique form of momentum or Hamiltonian after quantization. An evident step is to further introduce more commutation relations than the fundamental ones, and what we are going to do is to add those between Hamiltonian and positions x, and between Hamiltonian and momenta p, i.e.,#br#[x,Ĥ]=iħÔ({x,HC}c) and [p,Ĥ]=iħÔ({p,HC}c)#br#where {f,g}c denotes the Poisson or Dirac bracket in classical mechanics, and Ô({f,g}c) means a construction of operator based on the resulting {f,g}c, and in general we have [f,ĝ]≠Ô({f,g}c). The association between these two sets of relations means that the operators {x,p,H must be simultaneously quantized. This is the basic framework of the so-called enlarged canonical quantization scheme.#br#For particles constrained on the minimum surface, momentum and kinetic energy are assumed to be dependent on purely intrinsic geometric quantity. Whether the intrinsic geometry offers a proper framework for the canonical quantization scheme is then an interesting issue. In the present paper, we take the catenoid to find whether the quantum theory can be established satisfactorily. Results show that the theory is not self-consistent. In contrast, in the threedimensional Euclidean space, the geometric momentum and geometric potential are then in agreement with those given by the Schrödinger theory.
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Keywords:
- quantization /
- intrinsic geometry /
- geometric potential /
- geometric momentum
[1] Jensen H, Koppe H 1971 Ann. Phys. (N.Y.) 63 586
[2] da Costa R C T 1981 Phys. Rev. A 23 1982
[3] Ferrari G, Cuoghi G 2008 Phys. Rev. Lett. 100 230403
[4] Liu Q H 2013 J. Phys. Soc. Jpn. 82 104002
[5] Liu Q H, Tong C L, Lai M M 2007 J. Phys.: Math. Theor. 40 4161
[6] Liu Q H, Tang L H, Xun D M 2011 Phys. Rev. A 84 042101
[7] Liu Q H 2014 Phys. Lett. A 378 785
[8] Dirac P A M 1950 Can. J. Math. 2 129
[9] Dirac P A M 1964 Lectures on Quantum Mechanics (New York: Yeshiva University) p40
[10] Dirac P A M 1967 The Principles of Quantum Mechanics (Vol. 4) (Oxford: Oxford University Press) p114
[11] Kleinert H, Shabanov S V 1997 Phys. Lett. A 232 327
[12] Liu Q H 2013 J. Math. Phys. 54 122113
[13] Dirac P A M 1925 Proc. R. Soc. London Ser. A 109 642
[14] Szameit A, Dreisow F, Heinrich M, Keil R, Nolte S, Tnnermann A, Longhi S 2010 Phys. Rev. Lett. 104 150403
[15] Onoe J, Ito T, Shima H, Yoshioka H, Kimura S 2012 Epl-Europhys. Lett. 98 27001
[16] Xun D M, Liu Q H 2014 Ann. Phys. (N.Y.) 341 132
[17] Xun D M, Liu Q H 2013 Int. J. Geom. Methods M. 10 1220031
[18] Zhang Z S, Xiao S F, Xun D M, Liu Q H 2015 Commun. Theor. Phys. 63 19
[19] Xun D M, Liu Q H, Zhu X M 2013 Ann. Phys. (N.Y.) 338 123
[20] Chen W H 1999 The Elementary Differential Geometry(Beijing: Beijing University Press) pp121-123 (in Chinese) [陈维桓 1999 微分几何初步 (北京: 北京大学出版社) 第121–123页]
[21] Miao Y G 1993 Acta Phys. Sin. 42 536 (in Chinese) [缪炎刚 1993 42 536]
[22] Miao Y G, Liu Y Y 1993 Chin. Phys. Lett. 10 5
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[1] Jensen H, Koppe H 1971 Ann. Phys. (N.Y.) 63 586
[2] da Costa R C T 1981 Phys. Rev. A 23 1982
[3] Ferrari G, Cuoghi G 2008 Phys. Rev. Lett. 100 230403
[4] Liu Q H 2013 J. Phys. Soc. Jpn. 82 104002
[5] Liu Q H, Tong C L, Lai M M 2007 J. Phys.: Math. Theor. 40 4161
[6] Liu Q H, Tang L H, Xun D M 2011 Phys. Rev. A 84 042101
[7] Liu Q H 2014 Phys. Lett. A 378 785
[8] Dirac P A M 1950 Can. J. Math. 2 129
[9] Dirac P A M 1964 Lectures on Quantum Mechanics (New York: Yeshiva University) p40
[10] Dirac P A M 1967 The Principles of Quantum Mechanics (Vol. 4) (Oxford: Oxford University Press) p114
[11] Kleinert H, Shabanov S V 1997 Phys. Lett. A 232 327
[12] Liu Q H 2013 J. Math. Phys. 54 122113
[13] Dirac P A M 1925 Proc. R. Soc. London Ser. A 109 642
[14] Szameit A, Dreisow F, Heinrich M, Keil R, Nolte S, Tnnermann A, Longhi S 2010 Phys. Rev. Lett. 104 150403
[15] Onoe J, Ito T, Shima H, Yoshioka H, Kimura S 2012 Epl-Europhys. Lett. 98 27001
[16] Xun D M, Liu Q H 2014 Ann. Phys. (N.Y.) 341 132
[17] Xun D M, Liu Q H 2013 Int. J. Geom. Methods M. 10 1220031
[18] Zhang Z S, Xiao S F, Xun D M, Liu Q H 2015 Commun. Theor. Phys. 63 19
[19] Xun D M, Liu Q H, Zhu X M 2013 Ann. Phys. (N.Y.) 338 123
[20] Chen W H 1999 The Elementary Differential Geometry(Beijing: Beijing University Press) pp121-123 (in Chinese) [陈维桓 1999 微分几何初步 (北京: 北京大学出版社) 第121–123页]
[21] Miao Y G 1993 Acta Phys. Sin. 42 536 (in Chinese) [缪炎刚 1993 42 536]
[22] Miao Y G, Liu Y Y 1993 Chin. Phys. Lett. 10 5
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