广义测不准关系及其应用
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摘要
首先介绍了位置与动量的广义测不准关系，由此得到了位置的最小不确定度，与此相应的微观状态的态密度必须作出修正，该修正对我们当前前沿科学所遇到的许多问题将产生广泛的影响。例如对黑体辐射问题，我们的讨论显示，与Planck温度比较在温度比较高的情况下斯特藩一玻耳兹曼定律并不成立，辐射场的总能量应该与温度成正比。
其次，将广义测不准关系的应用拓展到黑洞熵的计算，利用经广义测不准关系改进的薄层brick-wall方法计算了黑洞熵。结果表明，由这种方法得到的黑洞熵上限与它的外视界和宇宙视界面积之和成正比，和人们预期的结果相符，从中揭示了黑洞熵与视界面积之间的内在联系，也进一步表明了黑洞熵是视界面上量子态的熵，是一种量子效应。广义测不准关系的引入使我们看到，brick-wall方法与引力场量子化可能存在着一些内在的联系。
最后，利用量子力学的能量与时间的测不准关系，计算了光频的引力红移，结果表明，由这种方法得到的引力红移量与人们预期的结果相符。
First of all, the generalized position-momentum uncertainty relation,which leads to a finite minimum uncertainty of position, is introduced.Accordingly, the density of quantum states needs to be modified. This isbelieved to have a very broad effect on our current knowledge of variousproblems. In the specific example of the blackbody radiation, we havefound that the Stephan-Boltzmann law needs to be modified at very hightemperatures compared with the Planck temperature, i.e., the energy of theradiation field as a whole will be proportional to the temperature.
Second, the generalized uncertainty relation is applied to thecalculation of the entropy as well. The entropy is calculated by theimproved brick-wall method due to the generalized uncertainty relation.The entropy bound of this system not only includes the contribution of theblack hole horizon, but also includes the contribution of the cosmologicalhorizon, which is consistent with what people anticipate. It is found thatthere is an internal relation between the event horizon and the entropy; itis further revealed that the black hole entropy is an entropy which belongsto event horizon and is a kind of quantum effect. It is also apparent thatthe cut-off in brick-wall model is something related to the quantum theoryof gravity.
Last but not least, we also calculate the red-shift of light frequency for gravitation by using the energy-time of the uncertainty relation ofquantum mechanical, It is shown that the result of the red-shift of lightfrequency for gravitation obtained by this method accords with the oneexpected by some people.
其次，将广义测不准关系的应用拓展到黑洞熵的计算，利用经广义测不准关系改进的薄层brick-wall方法计算了黑洞熵。结果表明，由这种方法得到的黑洞熵上限与它的外视界和宇宙视界面积之和成正比，和人们预期的结果相符，从中揭示了黑洞熵与视界面积之间的内在联系，也进一步表明了黑洞熵是视界面上量子态的熵，是一种量子效应。广义测不准关系的引入使我们看到，brick-wall方法与引力场量子化可能存在着一些内在的联系。
最后，利用量子力学的能量与时间的测不准关系，计算了光频的引力红移，结果表明，由这种方法得到的引力红移量与人们预期的结果相符。
First of all, the generalized position-momentum uncertainty relation,which leads to a finite minimum uncertainty of position, is introduced.Accordingly, the density of quantum states needs to be modified. This isbelieved to have a very broad effect on our current knowledge of variousproblems. In the specific example of the blackbody radiation, we havefound that the Stephan-Boltzmann law needs to be modified at very hightemperatures compared with the Planck temperature, i.e., the energy of theradiation field as a whole will be proportional to the temperature.
Second, the generalized uncertainty relation is applied to thecalculation of the entropy as well. The entropy is calculated by theimproved brick-wall method due to the generalized uncertainty relation.The entropy bound of this system not only includes the contribution of theblack hole horizon, but also includes the contribution of the cosmologicalhorizon, which is consistent with what people anticipate. It is found thatthere is an internal relation between the event horizon and the entropy; itis further revealed that the black hole entropy is an entropy which belongsto event horizon and is a kind of quantum effect. It is also apparent thatthe cut-off in brick-wall model is something related to the quantum theoryof gravity.
Last but not least, we also calculate the red-shift of light frequency for gravitation by using the energy-time of the uncertainty relation ofquantum mechanical, It is shown that the result of the red-shift of lightfrequency for gravitation obtained by this method accords with the oneexpected by some people.
引文
[1] 周世勋．量子力学教程．北京．高等教育出版社，1979．
[2] 曾谨言．量子力学导论．第二版．北京．北京大学出版社，1998．
[3] 喀兴林．高等量子力学．北京．高等教育出版社，1999．
[4] L. J. Garay, Quantum gravity and minimum length, Int. J. Mod. Phys.1995, A 10:145.
[5] L. N. Chang et al., The Effect of the Minimal Length Uncertainty Relation on the Density of States and the Cosmological Constant Problem, Phys. Rev.2002, D 65:125 028-125 032.
[6] N．Bohr．原子论和自然的描述，商务印书馆，1964，43-46．
[7] David Bohm.Quantum theory. 1961.
[8] W.Heiserberg.Zeit.physik. 1927,43:172.
[9] W.Heiserberg. The physical of Quantum Theory. 1930.
[10] Kennard E H.Zeits of physic. 1927,44:326.
[11] Robertson H P.Phys Rev. 1929.34:163.
[12] Larsen U.Superspace geometry.the exact uncertainty relationship between complementary aspects. J Phys. 1990.A23:1041.
[13] 绰英超．广义Heiserberg不确定关系．大学物理，1997，16(2)：31-33．
[14] 李家宝、邝安祥．广义坐标与广义动量的研究．湖北大学学报(自然科学版)，1988，1：63-67．
[15] 高守恩，关于时间-能量的测不准关系，杭州师范学院学报。1990，6：32-36．
[16] Li Xiang. Black hole entropy without brick walls. Physics Letter.2002,540B(1-2):9-13.
[17] 张子珍等，柱黑洞的熵，数学物理学报，2005，25(21)：1061-1066．
[18] 刘成周、周宙安、李翔、赵峥．广义不确定原理对一般静态黑洞熵的影响．北京师范大学学报(自然科学版)，2004，40(4)：487-492．
[19] 叶鑫、桂元星等．一维Rinder谐振子广义测不准关系．大连理工大学学报，2001，41(5)：523-526．
[20] 刘全慧等．Wey1本征微分和连续定态体系的精确不确定关系．黑龙江大学自然科学版，1993，10(2)：69-74．
[21] 林景、涂宜炎．微观物理量不确定关系的信息熵表示．福州大学学报(自然科学版)，1992，20(4)：36-39．
[22] 周宙安、索标、刘文彪．广义测不准关系与Reissner-Nordstrom de Sitter黑洞熵．数学物理学报，2005，25A(3)：381-385．
[23] 牛振风、李翔、赵峥．利用广义不确定关系计算Vaidya-de Sitter黑洞的熵．北京师范大学学报(自然科学版)，2003，39(6)：751-755．
[24] 景玲、周宙安、刘文彪．广义测不准关系与Schwarzschild-de Sitter黑洞熵．北京师范大学学报(自然科学版)，2003，39(5)：623-627．
[25] 周宙安、唐卫华、周石安．广义测不准关系与de Sitter时空背景下的熵．湘潭师院学报(自然科学版)，2003，25(3)：16-19．
[26] Bekenstein J D. Black hole and entropy. Phys Rev, 1973, 7D(8): 2333-2346.
[27] Hawking S W. Particle creation by black holes. Commun Math Phys, 1975, 43(3):199-220.
[28] G 't Hooft. On the quantum structure of a black hole. Nuclear Physics, 1985, 256B:727-745.
[29] Ghosh A, Mitra P. Entropy in Dilatonic black hole background. Phys Rev Lett, 1994, 73(19):2521-2523.
[30] Lee M H, Kim J K. Entropy of a quantum field in rotating black hole. Phys Rev, 1996, 54D(6):3904-3914.
[31] Cai R G, Zhu L B. Entropy of scalar fields and its duality invariance in three-dimensional spacetimes. Phys Lett, 1996, 219A(3-4): 191-198.
[32] Lee H, Kim S W, Kim W T. Nonvanishing entropy of extremal charged black holes.Phys Rev, 1996,54D(10):6559-6562.
[33] Ho J, Kim W T, Park Y J, et al. Entropy in the Kerr-Newman black hole.Class Quantum Grav, 1997,14(9):2617-2626.
[34] 罗智坚，朱建阳．Schwarzschild黑洞背景下Dirac场的熵．物理学报，1999，48(3)：395-401．
[35] 刘文彪，朱建阳，赵峥．Nernst定理与R-N黑洞Dirac场的熵．物理学报，2000，49(3)：581-585．
[36] Liu Wenbiao, Zhao Zheng. Entropy of the Dirac field in a Kerr-Newman black hole.Phys Rev,2000, 61D(6): 63-69.
[37] Li X,Zhao Z.Entropy of Vaidya-de Sitter Spacetime. Chin Phys Lett,2001,18(3): 463-465.
[38] Ronald J.Adler et al.The Generalized Uncertainty Principle and Black Hole Remnants. Gen.Rel. Grav,2001,33(6): 2101-2106.
[39] Bombelli L, Koul R K, Lee J, et al.Quantum source of entropy for black holes.Phys Rev,1986,34D: 373-378.
[40] Srednicki M. Entropy and area. Phys Rev Lett, 1993, 71(5):666-669.
[41] 曾谨言，裴寿镛．量子力学新进展(第一辑)．北京：北京大学出版社，2000．
[42] 赵峥，刘辽．一般稳态时空视界的确定．物理学报，1991，40(10)：1564-1567．
[43] 赵峥．黑洞温度、熵变化率和时间尺度的压缩．北京师范大学学报(自然科学版)，1995，31(4)：476-480．
[44] Einstein A.Annalen der physic, 1911,35(4):35,898-908.
[45] S．温伯格．引力论和宇宙论[M]．北京：科学出版社，1980．
[46] 刘辽．广义相对论[M]．北京：高等教育出版社，1987．
[47] 俞允强．广义相对论引论(第二版)[M]．北京：北京大学出版社，1997．
[48] 赵峥．黑洞的热性质与时空奇异性[M]．北京：北京师范大学出版社，1999．
[2] 曾谨言．量子力学导论．第二版．北京．北京大学出版社，1998．
[3] 喀兴林．高等量子力学．北京．高等教育出版社，1999．
[4] L. J. Garay, Quantum gravity and minimum length, Int. J. Mod. Phys.1995, A 10:145.
[5] L. N. Chang et al., The Effect of the Minimal Length Uncertainty Relation on the Density of States and the Cosmological Constant Problem, Phys. Rev.2002, D 65:125 028-125 032.
[6] N．Bohr．原子论和自然的描述，商务印书馆，1964，43-46．
[7] David Bohm.Quantum theory. 1961.
[8] W.Heiserberg.Zeit.physik. 1927,43:172.
[9] W.Heiserberg. The physical of Quantum Theory. 1930.
[10] Kennard E H.Zeits of physic. 1927,44:326.
[11] Robertson H P.Phys Rev. 1929.34:163.
[12] Larsen U.Superspace geometry.the exact uncertainty relationship between complementary aspects. J Phys. 1990.A23:1041.
[13] 绰英超．广义Heiserberg不确定关系．大学物理，1997，16(2)：31-33．
[14] 李家宝、邝安祥．广义坐标与广义动量的研究．湖北大学学报(自然科学版)，1988，1：63-67．
[15] 高守恩，关于时间-能量的测不准关系，杭州师范学院学报。1990，6：32-36．
[16] Li Xiang. Black hole entropy without brick walls. Physics Letter.2002,540B(1-2):9-13.
[17] 张子珍等，柱黑洞的熵，数学物理学报，2005，25(21)：1061-1066．
[18] 刘成周、周宙安、李翔、赵峥．广义不确定原理对一般静态黑洞熵的影响．北京师范大学学报(自然科学版)，2004，40(4)：487-492．
[19] 叶鑫、桂元星等．一维Rinder谐振子广义测不准关系．大连理工大学学报，2001，41(5)：523-526．
[20] 刘全慧等．Wey1本征微分和连续定态体系的精确不确定关系．黑龙江大学自然科学版，1993，10(2)：69-74．
[21] 林景、涂宜炎．微观物理量不确定关系的信息熵表示．福州大学学报(自然科学版)，1992，20(4)：36-39．
[22] 周宙安、索标、刘文彪．广义测不准关系与Reissner-Nordstrom de Sitter黑洞熵．数学物理学报，2005，25A(3)：381-385．
[23] 牛振风、李翔、赵峥．利用广义不确定关系计算Vaidya-de Sitter黑洞的熵．北京师范大学学报(自然科学版)，2003，39(6)：751-755．
[24] 景玲、周宙安、刘文彪．广义测不准关系与Schwarzschild-de Sitter黑洞熵．北京师范大学学报(自然科学版)，2003，39(5)：623-627．
[25] 周宙安、唐卫华、周石安．广义测不准关系与de Sitter时空背景下的熵．湘潭师院学报(自然科学版)，2003，25(3)：16-19．
[26] Bekenstein J D. Black hole and entropy. Phys Rev, 1973, 7D(8): 2333-2346.
[27] Hawking S W. Particle creation by black holes. Commun Math Phys, 1975, 43(3):199-220.
[28] G 't Hooft. On the quantum structure of a black hole. Nuclear Physics, 1985, 256B:727-745.
[29] Ghosh A, Mitra P. Entropy in Dilatonic black hole background. Phys Rev Lett, 1994, 73(19):2521-2523.
[30] Lee M H, Kim J K. Entropy of a quantum field in rotating black hole. Phys Rev, 1996, 54D(6):3904-3914.
[31] Cai R G, Zhu L B. Entropy of scalar fields and its duality invariance in three-dimensional spacetimes. Phys Lett, 1996, 219A(3-4): 191-198.
[32] Lee H, Kim S W, Kim W T. Nonvanishing entropy of extremal charged black holes.Phys Rev, 1996,54D(10):6559-6562.
[33] Ho J, Kim W T, Park Y J, et al. Entropy in the Kerr-Newman black hole.Class Quantum Grav, 1997,14(9):2617-2626.
[34] 罗智坚，朱建阳．Schwarzschild黑洞背景下Dirac场的熵．物理学报，1999，48(3)：395-401．
[35] 刘文彪，朱建阳，赵峥．Nernst定理与R-N黑洞Dirac场的熵．物理学报，2000，49(3)：581-585．
[36] Liu Wenbiao, Zhao Zheng. Entropy of the Dirac field in a Kerr-Newman black hole.Phys Rev,2000, 61D(6): 63-69.
[37] Li X,Zhao Z.Entropy of Vaidya-de Sitter Spacetime. Chin Phys Lett,2001,18(3): 463-465.
[38] Ronald J.Adler et al.The Generalized Uncertainty Principle and Black Hole Remnants. Gen.Rel. Grav,2001,33(6): 2101-2106.
[39] Bombelli L, Koul R K, Lee J, et al.Quantum source of entropy for black holes.Phys Rev,1986,34D: 373-378.
[40] Srednicki M. Entropy and area. Phys Rev Lett, 1993, 71(5):666-669.
[41] 曾谨言，裴寿镛．量子力学新进展(第一辑)．北京：北京大学出版社，2000．
[42] 赵峥，刘辽．一般稳态时空视界的确定．物理学报，1991，40(10)：1564-1567．
[43] 赵峥．黑洞温度、熵变化率和时间尺度的压缩．北京师范大学学报(自然科学版)，1995，31(4)：476-480．
[44] Einstein A.Annalen der physic, 1911,35(4):35,898-908.
[45] S．温伯格．引力论和宇宙论[M]．北京：科学出版社，1980．
[46] 刘辽．广义相对论[M]．北京：高等教育出版社，1987．
[47] 俞允强．广义相对论引论(第二版)[M]．北京：北京大学出版社，1997．
[48] 赵峥．黑洞的热性质与时空奇异性[M]．北京：北京师范大学出版社，1999．
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