量子与经典对应
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摘要
1926年，量子力学建立。迄今为止，量子力学取得了巨大的成功，还没有发现有与其相违背的实验，但不是量子力学所有的基础性问题都已经很好的解决。
本文研究了量子力学的基本性问题中的两个具体问题。一个是海森堡对应原理在半空间谐振子中的应用的问题。另一个问题是约束体系的量子哈密顿中涉及到的算符次序问题。
在本文中，我们首先对于半空间谐振子给出了包括位置、动量算符及其平方的矩阵元，定态上的不确定关系等等量子力学的基本结果。由于位置矩阵元的结果较复杂，我们借助海森堡对应原理对半空间谐振子的位置矩阵元及其平方的矩阵元给出了很好的近似表达式。
然后，我们讨论了约束体系中的算符次序问题。对于非约束体系，量子力学动能表达式为，其中p_i为笛卡尔动量算符，这一结论与坐标的选取无关。但是，对于约束体系这一结论不再成立。当我们将二维椭球面嵌入三维平直空间后，就可以在三维直角坐标系中描述在这个二维椭球面上的运动。动能的正确形式为，其中p_i是厄密的动量算符，f_i(x，y，z)为坐标x，y，z的非平凡函数。于是，我们在二维椭球约束体系中扁椭球和长椭球情况下得到了函数f_i(x，y，z)，给出了动能算符的明确形式，并讨论了相关问题。
The quantum mechanics was founded in 1926. Since then, it has achieved huge success and no experiment we have found is disobedient with it. However, not every fundamental question is studied enough and adequately.
In this paper two concrete problems concerned fundamental questions in quantum mechanics are discussed. One is the use of the Heisenberg correspondence for the harmonic oscillator in half space. The other is the operator ordering problem in quantum harniton of constrained systems.
Firstly, the elementary quantum-mechanical results for the harmonic oscillator in half space are carried out. These results include expectation values for position, momentum and their square, the uncertainty relation in the eigenstates, etc. Since the result of expectation value for position of the harmonic oscillator in half space is quite complicated, the Heisenberg correspondence principle is used to give the approximate expressions for position and its square of the harmonic oscillator in half space, and the expressions prove to be very accurate by numerical calculations.
Secondly, the operator ordering problem in quantum hamiton of constrained systems is discussed. For an unconstrained system, the quantum kinetic energy operator can be written in terms of where pi are Cartesian momentum, that is
irrespective of the choosing in coordinate. But, the same result cannot be applied to the constrained system. Since the motion on an ellipsoid surface is representable in 3-dimensional Cartesian coordinate, the quantum kinetic operator turns to be
where Cartesian momentum Pi are hermitian operators and functions fi(x,y,z) are now nontrivial in quantum mechanics. So we have the fractions fi(x,y,z) and the specific form of the quantum kinetic operator on
the oblate ellipsoid surface and the prolate ellipsoid surface, and we also discuss the interrelated problem.
本文研究了量子力学的基本性问题中的两个具体问题。一个是海森堡对应原理在半空间谐振子中的应用的问题。另一个问题是约束体系的量子哈密顿中涉及到的算符次序问题。
在本文中，我们首先对于半空间谐振子给出了包括位置、动量算符及其平方的矩阵元，定态上的不确定关系等等量子力学的基本结果。由于位置矩阵元的结果较复杂，我们借助海森堡对应原理对半空间谐振子的位置矩阵元及其平方的矩阵元给出了很好的近似表达式。
然后，我们讨论了约束体系中的算符次序问题。对于非约束体系，量子力学动能表达式为，其中p_i为笛卡尔动量算符，这一结论与坐标的选取无关。但是，对于约束体系这一结论不再成立。当我们将二维椭球面嵌入三维平直空间后，就可以在三维直角坐标系中描述在这个二维椭球面上的运动。动能的正确形式为，其中p_i是厄密的动量算符，f_i(x，y，z)为坐标x，y，z的非平凡函数。于是，我们在二维椭球约束体系中扁椭球和长椭球情况下得到了函数f_i(x，y，z)，给出了动能算符的明确形式，并讨论了相关问题。
The quantum mechanics was founded in 1926. Since then, it has achieved huge success and no experiment we have found is disobedient with it. However, not every fundamental question is studied enough and adequately.
In this paper two concrete problems concerned fundamental questions in quantum mechanics are discussed. One is the use of the Heisenberg correspondence for the harmonic oscillator in half space. The other is the operator ordering problem in quantum harniton of constrained systems.
Firstly, the elementary quantum-mechanical results for the harmonic oscillator in half space are carried out. These results include expectation values for position, momentum and their square, the uncertainty relation in the eigenstates, etc. Since the result of expectation value for position of the harmonic oscillator in half space is quite complicated, the Heisenberg correspondence principle is used to give the approximate expressions for position and its square of the harmonic oscillator in half space, and the expressions prove to be very accurate by numerical calculations.
Secondly, the operator ordering problem in quantum hamiton of constrained systems is discussed. For an unconstrained system, the quantum kinetic energy operator can be written in terms of where pi are Cartesian momentum, that is
irrespective of the choosing in coordinate. But, the same result cannot be applied to the constrained system. Since the motion on an ellipsoid surface is representable in 3-dimensional Cartesian coordinate, the quantum kinetic operator turns to be
where Cartesian momentum Pi are hermitian operators and functions fi(x,y,z) are now nontrivial in quantum mechanics. So we have the fractions fi(x,y,z) and the specific form of the quantum kinetic operator on
the oblate ellipsoid surface and the prolate ellipsoid surface, and we also discuss the interrelated problem.
引文
[1] 王竹溪．统计物理学导论第2版，北京：高等教育出版社，1965：1-89
[2] L. E. Ballentine, The Statistical Interpretation of Quantum Mechanics. Rev. Mod. Phys., 1970,41 (4):358-381
[3] J. Bell. On the Problem of Hidden Variable in Quantum Mechanics, Rev. Mod. Phys., 1966,38(3):447-452
[4] A. Aspect, J. Dalibard, and G.Roger.. "Experimental Test of Bell's Inequalities Using Time-varying Analyzers" , Phys. Rev. Lett., 1982,49(25): 1804-1807
[5] Heisenberg W.The Physical Principles of the Quantum Theory(Translated by Eckart C, and Hoyt F C). New York:Dover. 1949
[6] Naccache P F.Matrix Elements and Correspondence Principles. J.Phys.B, 1972,5(7): 1308-1319
[7] Percival I C, Richards D. A Correspondence Principle for Strongly Coupled States. J.Phys.B, 1970,3(8): 1035-1046
[8] Percival I C, Kichards D.Classical Collisions and Correspondence Principle. J.Phys.B, 1970,3(3):315-327
[9] Percival I C,Richards D. The Theory of Collisions between Charged Particles and Highly Excited Atoms. Advances in Atomic and Molecular Physics, 1975, (11):1-82
[10] Landau L. D., Lifshitz E. M..Quantum Mechanics Oxford:Pergamon, 1965
[11] Liu Q. H., Wang X., Qi W. H.,et al. The Fejér Average and the Mean Value of A Quantity in a Quasiclassical Wave Packet. J. Math. Phys.,2002,43(6):3411-3418
[12] Morehead J. J.. Semiclassical Integrable Matrix Elements. Phys.Rev.A., 1996,53 (3): 1285-1294
[13] Kasperkovitz P., Peev M.. Semiclassical Mechanics of Periodic Motion: Ⅰ. General scheme. J.Phys.A, 1998, 31 (9):2197-2227
[14] Greenberg W. R., A. Klein, C.-T. Li .Invariant Tori and Heisenberg Matrix Mechanics: A New Window on the Quantum- Classical Correspondence. Phys. Rev. Lett., 1995,75(7): 1244-1247
[15] Greenberg W. R., A. Klein, C.-T. Li .From Heisenberg Matrix Mechanics to Semiclassical Quantization: Theory and First Applications. Phys. Rev.A., 1996, 54(3): 1820-1837
[16] Liu Q. H., B. Hu.The Hydrogen Atom's Quantum-to-Classical Correspondence in Heisenberg's Correspondence Principle. J.Phys. A,2001,34(28):5713-5719
[17] Born M..The Physical Principles of the Quantum Theory.New York:Dove, 1927
[18] Child M.S..Semiclassical Mechanics with Molecular Application. Oxford: Clarendon,1991
[19] Kasperkovitz P. ,Peev M.. Semiclassical Mechanics of Periodic Motion: I. General scheme. J. Phys. A, 1998,31 (9):2196-2225
[20] Nieto M. M..Displaced and Squeezed Number States. Phys. Lett. A,1997, (229): 135-143
[21] Lin Q. G..Wave Packets of Harmonic Oscillator with Various Degrees of Rigidity. Chin. J. Phys.,2002,40:387-394
[22] Schiff L. I.. Quantum Mechanics.New York:McGraw-Hill, 1968
[23] 林琼桂．一般波包在均匀场中的运动．大学物理，2003，22(2)：3-6
[24] Green L.C., Rush P. P., Chandler C. D..Oscillator Strengths and Matrix Elements for the Electric Dipole Moment for Hydrogen. Astrophysical Journal Supplement, 1957, (3):37-45
[25] P.A.M.Dirac,The Principles of Quantum Mechanics, :Oxford Univ. Press, 1958
[26] P.A.M.Dirac,Lectures on Quantum Mechanics,New York:Yeshiva university, 1964
[27] P.A.M.Dirac.Generalized Hamiltonian Dynamics. Can.J.Math., 1950,(2): 129-143
[28] 李政道．场论与粒子物理学(上册)．第一版，北京：科学出版社，1980，7
[29] 周世勋．量子力学教程．第一版，北京：高等教育出版社，1997：54-57
[30] Boris Podolsky.Quantum-Mechanically Correct Form of Hamiltonian Function For Conservative Systems.Physics Review, 1928,32(5): 812-816
[31] Liu Q. H.,Liu T. G..Quantum Hamiltonian for the Rigid Rotator.International Journal of Theoretical Physics,2003,42(12):2877-2880
[32] M. Moshinsky. The Harmonic Oscillator in Modern Physics:from Atoms to Quarks, New York:Gordon and Breach, 1969
[33] 文根旺．谐振子任意次幂坐标算符矩阵元的计算．大学物理，1991，10(3)：20-21
[34] 卢书城．线性谐振子力学量算符矩阵元通式的研究．上海师范大学学报，1994，23(4)：39—46
[35] 贾祥富．谐振子任意次幂动量算符矩阵元的计算．大学物理，1994，13(6)：10-11
[36]张文英．三维各向同性谐振子的新径向阶梯算符．广西大学学报，1994，19(3)： 197-201
[37]王敬山，钱尚武．含时谐振子在磁偶极近似下的规范不变跃迁几率幅．北京大学学报，1994，30(3)：347-350
[38]程衍富，陈定君，韦联福．能量算符与广义含时谐振子巡回演化的位相．四川师范大学学报，1994，17(4)：50-52
[39]刘绍谅，曹阳素．密度矩阵与线性谐振子．湘潭大学自然科学学报，1994，16(2)： 23-28
[40]龙姝明．同调谐振子参数与基态能量的关系．物理学报，2002，51(10)： 2256-2260
[41]Dai Z. Stroud Jr C. R. Classical and Quatunm-Mechanical Dynamics of A Quasiclassical State of the Hydrogen Atom. Phys. Rev. A, 1990,42(11):6308-6313
[42]Ensslin K. Heitmann D. Ploog K.. Determination of Subband Structure, Depolarization Shift, and Depletion Charge in An AlxGal-xAs-GaAs HerteroStructure. Phys. R.ev. B, 1989,39(15): 10879-10886
[43]曾谨言．量子力学．第二版，北京：科学出版社，1997，101-107
[44]王竹溪，郭敦仁．特殊函数概论．第一版，北京：北京大学出版社，2000， 657-660
[45]Cohen-Tannoudji C., Diu. B., Lao F..Quantum Mechanics, 1977, 1:71
[46]刘全慧，刘天贵．在三维直角坐标中研究量子转子体系的动能表达式．大学物理，2003，23(3)：32-33
[47]贾艳伟，刘全慧，彭解华．氢原子的1/r矩阵元的量子-经典对应．物理学报，2002，51(2)：0365-0368
[48]钱伯初，曾谨言．量子力学习题精选与剖析(上册)．第二版，北京：科学出版社，2001，62-63
[2] L. E. Ballentine, The Statistical Interpretation of Quantum Mechanics. Rev. Mod. Phys., 1970,41 (4):358-381
[3] J. Bell. On the Problem of Hidden Variable in Quantum Mechanics, Rev. Mod. Phys., 1966,38(3):447-452
[4] A. Aspect, J. Dalibard, and G.Roger.. "Experimental Test of Bell's Inequalities Using Time-varying Analyzers" , Phys. Rev. Lett., 1982,49(25): 1804-1807
[5] Heisenberg W.The Physical Principles of the Quantum Theory(Translated by Eckart C, and Hoyt F C). New York:Dover. 1949
[6] Naccache P F.Matrix Elements and Correspondence Principles. J.Phys.B, 1972,5(7): 1308-1319
[7] Percival I C, Richards D. A Correspondence Principle for Strongly Coupled States. J.Phys.B, 1970,3(8): 1035-1046
[8] Percival I C, Kichards D.Classical Collisions and Correspondence Principle. J.Phys.B, 1970,3(3):315-327
[9] Percival I C,Richards D. The Theory of Collisions between Charged Particles and Highly Excited Atoms. Advances in Atomic and Molecular Physics, 1975, (11):1-82
[10] Landau L. D., Lifshitz E. M..Quantum Mechanics Oxford:Pergamon, 1965
[11] Liu Q. H., Wang X., Qi W. H.,et al. The Fejér Average and the Mean Value of A Quantity in a Quasiclassical Wave Packet. J. Math. Phys.,2002,43(6):3411-3418
[12] Morehead J. J.. Semiclassical Integrable Matrix Elements. Phys.Rev.A., 1996,53 (3): 1285-1294
[13] Kasperkovitz P., Peev M.. Semiclassical Mechanics of Periodic Motion: Ⅰ. General scheme. J.Phys.A, 1998, 31 (9):2197-2227
[14] Greenberg W. R., A. Klein, C.-T. Li .Invariant Tori and Heisenberg Matrix Mechanics: A New Window on the Quantum- Classical Correspondence. Phys. Rev. Lett., 1995,75(7): 1244-1247
[15] Greenberg W. R., A. Klein, C.-T. Li .From Heisenberg Matrix Mechanics to Semiclassical Quantization: Theory and First Applications. Phys. Rev.A., 1996, 54(3): 1820-1837
[16] Liu Q. H., B. Hu.The Hydrogen Atom's Quantum-to-Classical Correspondence in Heisenberg's Correspondence Principle. J.Phys. A,2001,34(28):5713-5719
[17] Born M..The Physical Principles of the Quantum Theory.New York:Dove, 1927
[18] Child M.S..Semiclassical Mechanics with Molecular Application. Oxford: Clarendon,1991
[19] Kasperkovitz P. ,Peev M.. Semiclassical Mechanics of Periodic Motion: I. General scheme. J. Phys. A, 1998,31 (9):2196-2225
[20] Nieto M. M..Displaced and Squeezed Number States. Phys. Lett. A,1997, (229): 135-143
[21] Lin Q. G..Wave Packets of Harmonic Oscillator with Various Degrees of Rigidity. Chin. J. Phys.,2002,40:387-394
[22] Schiff L. I.. Quantum Mechanics.New York:McGraw-Hill, 1968
[23] 林琼桂．一般波包在均匀场中的运动．大学物理，2003，22(2)：3-6
[24] Green L.C., Rush P. P., Chandler C. D..Oscillator Strengths and Matrix Elements for the Electric Dipole Moment for Hydrogen. Astrophysical Journal Supplement, 1957, (3):37-45
[25] P.A.M.Dirac,The Principles of Quantum Mechanics, :Oxford Univ. Press, 1958
[26] P.A.M.Dirac,Lectures on Quantum Mechanics,New York:Yeshiva university, 1964
[27] P.A.M.Dirac.Generalized Hamiltonian Dynamics. Can.J.Math., 1950,(2): 129-143
[28] 李政道．场论与粒子物理学(上册)．第一版，北京：科学出版社，1980，7
[29] 周世勋．量子力学教程．第一版，北京：高等教育出版社，1997：54-57
[30] Boris Podolsky.Quantum-Mechanically Correct Form of Hamiltonian Function For Conservative Systems.Physics Review, 1928,32(5): 812-816
[31] Liu Q. H.,Liu T. G..Quantum Hamiltonian for the Rigid Rotator.International Journal of Theoretical Physics,2003,42(12):2877-2880
[32] M. Moshinsky. The Harmonic Oscillator in Modern Physics:from Atoms to Quarks, New York:Gordon and Breach, 1969
[33] 文根旺．谐振子任意次幂坐标算符矩阵元的计算．大学物理，1991，10(3)：20-21
[34] 卢书城．线性谐振子力学量算符矩阵元通式的研究．上海师范大学学报，1994，23(4)：39—46
[35] 贾祥富．谐振子任意次幂动量算符矩阵元的计算．大学物理，1994，13(6)：10-11
[36]张文英．三维各向同性谐振子的新径向阶梯算符．广西大学学报，1994，19(3)： 197-201
[37]王敬山，钱尚武．含时谐振子在磁偶极近似下的规范不变跃迁几率幅．北京大学学报，1994，30(3)：347-350
[38]程衍富，陈定君，韦联福．能量算符与广义含时谐振子巡回演化的位相．四川师范大学学报，1994，17(4)：50-52
[39]刘绍谅，曹阳素．密度矩阵与线性谐振子．湘潭大学自然科学学报，1994，16(2)： 23-28
[40]龙姝明．同调谐振子参数与基态能量的关系．物理学报，2002，51(10)： 2256-2260
[41]Dai Z. Stroud Jr C. R. Classical and Quatunm-Mechanical Dynamics of A Quasiclassical State of the Hydrogen Atom. Phys. Rev. A, 1990,42(11):6308-6313
[42]Ensslin K. Heitmann D. Ploog K.. Determination of Subband Structure, Depolarization Shift, and Depletion Charge in An AlxGal-xAs-GaAs HerteroStructure. Phys. R.ev. B, 1989,39(15): 10879-10886
[43]曾谨言．量子力学．第二版，北京：科学出版社，1997，101-107
[44]王竹溪，郭敦仁．特殊函数概论．第一版，北京：北京大学出版社，2000， 657-660
[45]Cohen-Tannoudji C., Diu. B., Lao F..Quantum Mechanics, 1977, 1:71
[46]刘全慧，刘天贵．在三维直角坐标中研究量子转子体系的动能表达式．大学物理，2003，23(3)：32-33
[47]贾艳伟，刘全慧，彭解华．氢原子的1/r矩阵元的量子-经典对应．物理学报，2002，51(2)：0365-0368
[48]钱伯初，曾谨言．量子力学习题精选与剖析(上册)．第二版，北京：科学出版社，2001，62-63