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One can easily understand the transition from special relativity to Newton mechanics under the condition of v/c 1. But it is not so easy to understand the transition from quantum representation to classical representation from the point of view of wave mechanics. We define such a quantum state as near classical state (NCS), in which the mean value of coordinates equals the classical solution on a macroscopic scale. We take the NCS for three-dimensional isotropic harmonic oscillator in a spherical coordinate system for example. We take and choose cnl =(1/(2N+1))(1/(2lM+1)). The mean values of coordinates are r2 =(Ecl)/(2)(1+1-((2Lcl2)/(Ecl2)cos(2t)) and tg = (Ecl/lcl)[1-1-((Lcl)/(Ecl)2]tg(t)) in this NCS, which are in agreement with the classical solution on a macroscopic scale, where N/N1, lM/lM1. N and lM are determined by the macroscopic state. N =[(Ecl)/(ħ)], Ecl = 1/22(a2+ b2) , lM= [Lcl}/ħ], and Lcl = ab. Here , Ecl and Lcl respectively denote the mass, the energy and the angular momentum of harmonic oscillator. And the bracket [c] means taking the integer part of the number c, for example [2.78]=2. It is also emphasized that for a definite macro state, there are many NCS corresponding to a macro state; just like the case in statistical physics, many micro dynamical states correspond to a macro thermodynamic state. Thus the transition from quantum representation to classical representation is a coarse-graining process and also an information losing process.
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Keywords:
- quantum-classical correspondence /
- near classical states /
- three dimensional isotropic harmonic oscillator
[1] Feynman R P, Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York: McGraw-Hill)
[2] Zeng J Y 2002 Quantum Mechanics Vol I (3nd Ed.) (Beijing: Science Perss) p14 (in Chinese) [曾谨言 2002 量子力学卷I (第3版) (北京: 科学出版社) 第14页]
[3] Bohr N 1920 Z. Phys. 2 423
[4] Bohr N 1992 The Theory of Spectra and Atmic Constitution (Cambridge: Cambridge University Press)
[5] Zeng J Y 2000 Quantum Mechanics Vol II (3nd Ed.) (Beijing: Science Perss) pp73, 74 (in Chinese ) [曾谨言 2000 量子力学 卷II(第3版) (北京: 科学出版社) 第73, 74页]
[6] Schrodinger E 1926 Naturwissenchaften 14 664
[7] Howard S, Ray S K 1987 Am. J. Phys. 55 1109
[8] Li X H, Yang Y T, Xu G O 2009 Acta Phys. Sin. 58 7466 (in Chinese) [李兴华, 杨亚天, 徐躬耦 2009 58 7466]
[9] Xu B W, Ding G H, Kong F M 2000 Phys. Rev. A 62 022106
[10] Lu J, Du M L 2004 Acta Phys. Sin. 53 2450 (in Chinese) [陆军, 杜孟利 2004 53 2450]
[11] Du M L, Delos J B 1987 Phys. Rev. Lett. 58 1731
[12] Du M L, Delos J B 1988 Phys. Rev. A 38 1896
[13] Ma Z Q, Xu B W 2006 Acta Phys. Sin. 55 1571 (in Chinese) [马中骐, 许伯威 2006 55 1571]
[14] Berry M V, Balazs N L 1979 J. Phys. A 12 625
[15] Berry M V 1977 J. Phys. A 10 2083
[16] Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer-Verlag)
[17] Berry M V 1986 Quantum Chaos and Statistical Nuclear Physics (Berlin: Springer-Verlag) pp1-17
[18] Zeng J Y 2002 Quantum Mechanics Vol I (3nd Ed.) (Beijing: Science Perss) pp318, 319 (in Chinese) [曾谨言 2002 量子力学 卷I (第3版) (北京: 科学出版社) 第318, 319页]
[19] Li X H, Yang Y T 2003 China J. College Phys. 22 10 (in Chinese) [李兴华, 杨亚天 2003 大学物理 22 10]
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[1] Feynman R P, Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York: McGraw-Hill)
[2] Zeng J Y 2002 Quantum Mechanics Vol I (3nd Ed.) (Beijing: Science Perss) p14 (in Chinese) [曾谨言 2002 量子力学卷I (第3版) (北京: 科学出版社) 第14页]
[3] Bohr N 1920 Z. Phys. 2 423
[4] Bohr N 1992 The Theory of Spectra and Atmic Constitution (Cambridge: Cambridge University Press)
[5] Zeng J Y 2000 Quantum Mechanics Vol II (3nd Ed.) (Beijing: Science Perss) pp73, 74 (in Chinese ) [曾谨言 2000 量子力学 卷II(第3版) (北京: 科学出版社) 第73, 74页]
[6] Schrodinger E 1926 Naturwissenchaften 14 664
[7] Howard S, Ray S K 1987 Am. J. Phys. 55 1109
[8] Li X H, Yang Y T, Xu G O 2009 Acta Phys. Sin. 58 7466 (in Chinese) [李兴华, 杨亚天, 徐躬耦 2009 58 7466]
[9] Xu B W, Ding G H, Kong F M 2000 Phys. Rev. A 62 022106
[10] Lu J, Du M L 2004 Acta Phys. Sin. 53 2450 (in Chinese) [陆军, 杜孟利 2004 53 2450]
[11] Du M L, Delos J B 1987 Phys. Rev. Lett. 58 1731
[12] Du M L, Delos J B 1988 Phys. Rev. A 38 1896
[13] Ma Z Q, Xu B W 2006 Acta Phys. Sin. 55 1571 (in Chinese) [马中骐, 许伯威 2006 55 1571]
[14] Berry M V, Balazs N L 1979 J. Phys. A 12 625
[15] Berry M V 1977 J. Phys. A 10 2083
[16] Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer-Verlag)
[17] Berry M V 1986 Quantum Chaos and Statistical Nuclear Physics (Berlin: Springer-Verlag) pp1-17
[18] Zeng J Y 2002 Quantum Mechanics Vol I (3nd Ed.) (Beijing: Science Perss) pp318, 319 (in Chinese) [曾谨言 2002 量子力学 卷I (第3版) (北京: 科学出版社) 第318, 319页]
[19] Li X H, Yang Y T 2003 China J. College Phys. 22 10 (in Chinese) [李兴华, 杨亚天 2003 大学物理 22 10]
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