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On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.
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Keywords:
- Berry phase /
- Hannay angle /
- generalized gauge transformation /
- quantum-classical correspondence
[1] Makowski A J 2006 Eur. J. Phys. 27 1133
[2] Nielsen J R 1976 Collected Works (Vol. 3): The Correspondence Principle (1918-1923) (Amsterdam: North-Holland)
[3] Bohr N 1922 The Structure of the Atom Nobel Lecture, December 11, 1922
[4] Bohr N 1923 Nature 112 29
[5] Heisenberg W (translated by Eckart C, Hoyt F C) 1949 The Physical Principles of the Quantum Theory (New York: Dover Publications) p116
[6] Schrödinger E 1926 Naturwissenschaften 14 664
[7] Chen Y F 2011 Phys. Rev. A 83 032124
[8] Chen Y F, Lan Y P, Huang K F 2003 Phys. Rev. A 68 043803
[9] Chen Y F, Lu T H, Su K W, Huang K F 2006 Phys. Rev. Lett. 96 213902
[10] Lu T H, Lin Y C, Chen Y F, Huang K F 2008 Phys. Rev. Lett. 101 233901
[11] Xin J L, Liang J Q 2012 Chin. Phys. B 21 040303
[12] Xin J L, Liang J Q 2014 Sci. China: Phys. Mech. Astron. 57 1504
[13] Brack M 1993 Rev. Mod. Phys. 65 677
[14] Heer W A De 1993 Rev. Mod. Phys. 65 611
[15] Berry M V 1984 Proc. R. Soc. A 392 45
[16] Hannay J H 1985 J. Phys. A 18 221
[17] Liu H D 2011 Ph. D. Dissertation (Dalian: Dalian University of technology) (in Chinese) [刘昊迪 2011 博士学位论文(大连: 大连理工大学)]
[18] Xin J L, Liang J Q 2015 Phys. Scr. 90 065207
[19] Wang M H, Wei L F, Liang J Q 2015 Phys. Lett. A 379 1087
[20] Liang J Q, Wei L F 2011 New Advances in Quantum Physics (Beijing: Science Press) (in Chinese) [梁九卿, 韦联福 2011 量子物理新进展(北京: 科学出版社)]
[21] Lai Y Z, Liang J Q, Mller K, Zhou J G 1996 J. Phys. A 29 1773
[22] Chen Y F, Lu T H, Su K W, Huang K F 2005 Phys. Rev. E 72 056210
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[1] Makowski A J 2006 Eur. J. Phys. 27 1133
[2] Nielsen J R 1976 Collected Works (Vol. 3): The Correspondence Principle (1918-1923) (Amsterdam: North-Holland)
[3] Bohr N 1922 The Structure of the Atom Nobel Lecture, December 11, 1922
[4] Bohr N 1923 Nature 112 29
[5] Heisenberg W (translated by Eckart C, Hoyt F C) 1949 The Physical Principles of the Quantum Theory (New York: Dover Publications) p116
[6] Schrödinger E 1926 Naturwissenschaften 14 664
[7] Chen Y F 2011 Phys. Rev. A 83 032124
[8] Chen Y F, Lan Y P, Huang K F 2003 Phys. Rev. A 68 043803
[9] Chen Y F, Lu T H, Su K W, Huang K F 2006 Phys. Rev. Lett. 96 213902
[10] Lu T H, Lin Y C, Chen Y F, Huang K F 2008 Phys. Rev. Lett. 101 233901
[11] Xin J L, Liang J Q 2012 Chin. Phys. B 21 040303
[12] Xin J L, Liang J Q 2014 Sci. China: Phys. Mech. Astron. 57 1504
[13] Brack M 1993 Rev. Mod. Phys. 65 677
[14] Heer W A De 1993 Rev. Mod. Phys. 65 611
[15] Berry M V 1984 Proc. R. Soc. A 392 45
[16] Hannay J H 1985 J. Phys. A 18 221
[17] Liu H D 2011 Ph. D. Dissertation (Dalian: Dalian University of technology) (in Chinese) [刘昊迪 2011 博士学位论文(大连: 大连理工大学)]
[18] Xin J L, Liang J Q 2015 Phys. Scr. 90 065207
[19] Wang M H, Wei L F, Liang J Q 2015 Phys. Lett. A 379 1087
[20] Liang J Q, Wei L F 2011 New Advances in Quantum Physics (Beijing: Science Press) (in Chinese) [梁九卿, 韦联福 2011 量子物理新进展(北京: 科学出版社)]
[21] Lai Y Z, Liang J Q, Mller K, Zhou J G 1996 J. Phys. A 29 1773
[22] Chen Y F, Lu T H, Su K W, Huang K F 2005 Phys. Rev. E 72 056210
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