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Statistical and strict momentum conservation

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A Correction to this article was published on 10 January 2020

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Abstract

Arguments about the conservation laws of energy and momentum in the microworld being statistical or strict began in 1924, and conflicting viewpoints remain today. The former is mainly supported theoretically, but the latter has been proved by many experiments. Here we explain that in principle, the strict conservation law of momentum always holds in the entangled state form of the momentum eigenstates of a closed composite system with interactions among subsystems, by expanding the total wave function to the sum of the products of momentum eigenstates of subsystems. Common scenario is that one of the two subsystems of a composite system is large or strong, its state remains approximately unchanged in a short time and the entangled state can be approximately written as a product state, which can be easily deduced from the paper of Haroche’s group in 2001. The considered micro-subsystem can be approximately represented as the superposition of its different momentum eigenstates; therefore, the approximation can be used to explain why the law holds statistically for the subsystem and for any single particle due to neglecting the interactions with the large subsystem or the environment. So the two momentum conservation laws reasonably hold without conflicts.

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Change history

  • 10 January 2020

    The author found a mistake in their published article. The eqs. 9 and 10 are incorrect.

  • 10 January 2020

    The author found a mistake in their published article. The eqs. 9 and 10 are incorrect.

References

  1. Bohr, N., Kramers, H.A., Slater, J.C.: Z. Phys. 24, 69 (1924)

    Article  ADS  Google Scholar 

  2. Bothe, W., Geiger, H.: Z. Phys. 32, 639 (1925)

    Article  ADS  Google Scholar 

  3. Bothe, W., Geiger, H.: Die Naturwissenschaften. 13, 440 (1925)

    Article  ADS  Google Scholar 

  4. Shankland, R.S.: Phys. Rev. 49, 8 (1936)

    Article  ADS  Google Scholar 

  5. Dirac, P.A.M.: Nature. 137, 298 (1936)

    Article  ADS  Google Scholar 

  6. Bohr, N.: Nature. 138, 25 (1936)

    Article  ADS  Google Scholar 

  7. Shankland, R.S.: Phys. Rev. 52, 414 (1937)

    Article  ADS  Google Scholar 

  8. Sidharth, B.G.: Chaos Solitons Fractals. 11, 1037 (2000)

    Article  ADS  Google Scholar 

  9. Compton, A.H., Simon, A.W.: Phys. Rev. 26, 289 (1925)

    Article  ADS  Google Scholar 

  10. Alichanian, A.I., Alichanow, A.I., Arzimovitch, L.A.: Nature. 137, 703 (1936)

    Article  Google Scholar 

  11. Williams, E.J., Pickup, E.: Nature. 138, 461 (1936)

    Article  ADS  Google Scholar 

  12. Bothe, W., Maier-Leibnitz, H.: Phys. Rev. 50, 187 (1936)

    Article  ADS  Google Scholar 

  13. Hofstadter, R., McIntyre, J.A.: Phys. Rev. 78, 24 (1950)

    Article  ADS  Google Scholar 

  14. Cross, W.G., Ramsey, N.F.: Phys. Rev. 80, 929 (1950)

    Article  ADS  Google Scholar 

  15. Mendez, E.E., Nocera, J., Wang, W.I.: Phys. Rev. B. 45, 3910 (1992)

    Article  ADS  Google Scholar 

  16. Williams, E.J.: Nature. 137, 614 (1936)

    Article  ADS  Google Scholar 

  17. Jacobsen, J.C.: Nature. 138, 25 (1936)

    Article  ADS  Google Scholar 

  18. Nunn May, A.: Rep. Prog. Phys. 3, 89 (1936)

    Article  ADS  Google Scholar 

  19. Chadwick, J.: Nature. 129, 312 (1932)

    Article  ADS  Google Scholar 

  20. Christy, R.F., Cohen, E.R., Fowler, W.A., Lauritsen, C.C., Lauritsen, T.: Phys. Rev. 72, 698 (1947)

    Article  ADS  Google Scholar 

  21. Smoliner, J., Demmerle, W., Berthold, G., Gornik, E., Weimann, G., Schlapp, W.: Phys. Rev. Lett. 63, 2116 (1989)

    Article  ADS  Google Scholar 

  22. Cho, K., Joannopoulos, J.D., Kleinman, L.: Phys. Rev. E. 47, 3145 (1993)

    Article  ADS  Google Scholar 

  23. Chudnovsky, E.M.: Phys. Rev. B. 54, 5777 (1996)

    Article  ADS  Google Scholar 

  24. Knoll, J., Ivanov Yu, B., Voskresensky, D.N.: Ann. Phys. 293, 126 (2001)

    Article  ADS  Google Scholar 

  25. Bhattacharjee, P.R.: Optik. 120, 642 (2009)

    Article  ADS  Google Scholar 

  26. Busch, P., Loveridge, L.: Phys. Rev. Lett. 106, 110406 (2011)

    Article  ADS  Google Scholar 

  27. Moore, B.E., Noreña, L., Schober, C.M.: J. Comput. Phys. 232, 214 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  28. Lee, J., Kim, S.M.: Phys. Rev. A. 89, 035802 (2014)

    Article  ADS  Google Scholar 

  29. Amelino-Camelia, G., Bianco, S., Brighenti, F., Buonocore, R.J.: Phys. Rev. D. 91, 084045 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  30. Nienhuis, G.: Phys. Rev. A. 93, 23840 (2016)

    Article  ADS  Google Scholar 

  31. Drummond, P.D., Opanchuk, B.: Phys. Rev. A. 96, 043616 (2017)

    Article  ADS  Google Scholar 

  32. Dirac, P.A.M.: The Principle of Quantum Mechanics, pp. 72–76. Science Press, Beijing (2008)

    Google Scholar 

  33. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics, 3rd edn, pp. 7–11. Elsevier Pte Ltd, Singapore (2007)

    Google Scholar 

  34. Wheeler, J.A., Zurek, W.H.: Quantum Theory and Measurement, p. 50. Princeton University Press (1983)

  35. Cohen-Tannoudji, C., Diu, B., Laloё, F.: Quantum Mechanics,translated by Hemley, S.R., Ostrowsky, N., Ostrowsky, D. (WILEY-VCH) p.225-227 (2005)

  36. Shankar, R.: Principles of Quantum Mechanics, p. 116. Plenum Press, New York (1994)

    Book  MATH  Google Scholar 

  37. Zeng, J.Y.: Quantum Mechanics I, 5rd edn, p. 11. Science Press) (in Chinese), Beijing (2013)

    Google Scholar 

  38. Zeng, J.Y.: Quantum Mechanics I (4th ed.) (Beijing: Science Press) (in Chinese) p. 12–13, 27–41, 305 (2007)

  39. Zhang, Y.D.: Quantum Mechanics (4th ed.) (Beijing: Science Press) (in Chinese) p. 157–159, 171–172 (2017)

  40. Griffiths, D.J.: Introduction to Quantum Mechanics, 2nd edn, p. 203. China Machine Press, Prentice-Hall, Inc. and Beijing (2005)

    Google Scholar 

  41. Franck, L.: Do We Really Understand Quantum Mechanics? Cambridge University Press) 6 chapter (2012)

  42. Auletta, G., Fortunato, M., Parisi, G.: Quantum Mechanics, pp. 181–183, 610–619. Cambridge University Press (2014)

  43. Sakurai, J.J., Napolitano, J.: Modern Quantum Mechanics, pp. 175–181. Pearson Education Asia LTD., and Beijing World Publishing Corporation (2016)

  44. Basdevant, J.L.: Lectures on Quantum Mechanics, pp. 434–455. Springer International Publishing, Cham (2016)

    Book  Google Scholar 

  45. Bricmond, J.: Making Sense of Quantum Mechanics, pp. 201–293. Springer International Publishing, Cham (2016)

    Book  Google Scholar 

  46. von Baeyer, H.C.: QBism: The Future of Quantum Physics, pp. 140–169. Harvard University Press, Cambridge (2016)

    Book  Google Scholar 

  47. Norsen, T.: Foundations of Quantum Mechanics, pp. 96–173. Springer International Publishing, Cham (2017)

    Book  MATH  Google Scholar 

  48. Hayshi, M.: Quantum Information Theory, pp. 364–404. Springer-Verlag, Berlin (2017)

    Book  Google Scholar 

  49. Berman, P.R.: Introductory Quantum Mechanics, p. 107. Springer International Publishing, Cham (2018)

    Book  Google Scholar 

  50. Giustina, M., Versteegh, M.A.M., Wengerowsky, S., et al.: Phys. Rev. Lett. 115, 250,401 (2015)

    Article  Google Scholar 

  51. Shalm, L.K., Meyer-Scott, E., Bradley, G., Christensen, B.G., et al.: Phys. Rev. Lett. 115, 250402 (2015)

    Article  ADS  Google Scholar 

  52. Hensen, B., Bernien, H., et al.: Nature. 526, 682 (2015)

    Article  ADS  Google Scholar 

  53. Rosenfeld, W., Burchardt, D., Garthoff, R., Redeker, K., Ortegel, N., Rau, M., Weinfurter, H.: Phys. Rev. Lett. 119, 010402 (2017)

    Article  ADS  Google Scholar 

  54. Li, M.H., Wu, C., Zhang, Y., et al.: Phys. Rev. Lett. 121, 080404 (2018)

    Article  ADS  Google Scholar 

  55. Fayngold, M., Fayngold, V.: Quantum Mechanics and Quantum Information, p. 603. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2013)

    MATH  Google Scholar 

  56. Bertet, P., Osnaghi, S., Rauschenbeutel, A., Nogues, G., Auffeves, A., Brune, M., Raimond, J.M., Haroche, S.: Nature. 411, 166 (2001)

    Article  ADS  Google Scholar 

  57. Zeng, T.H.: arXiv 1307.1851v1 (2013)

  58. Monroe, C., Meekhof, D.M., King, B.E., Wineland, D.J.: Science. 272, 1131 (1996)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgment

Tianhai Zeng thanks Choo Hiap Oh, Berthold-Georg Englert, Gerard’t Hooft, Christopher Monroe, Chinwen Chou, Tongcang Li, Chengzu Li, Linmei Liang, Guojun Jin, Shouyong Pei, Supen Kou, Xiaoming Liu, Haibo Wang, Shidong Liang, Zhibin Li, Keqiu Chen, Xianting Liang, Dianmin Tong, Sixia Yu, Guoyong Xiang, Yinghua Ji, Guiqin Li, Xuexi Yi, Yu Shi, Chun Liu, Fengli Yan, Hui Yan, Haosheng Zeng, Xinqi Li, Chengjie Zhang, Lifan Ying, Peizhu Ding, Xianguo Jiang, Liwen Hu, Chunyu Chang, Liyu Tian and my colleagues: Jian Zou, Xiusan Xing, Feng Wang, Xiangdong Zhang, Yugui Yao, Jungang Li, Hao Wei, Bingcong Gou, Yanxia Xing, Fan Yang, Zhaotan Jiang, Wenyong Su, Changhong Lu, Xinbing Song, Jianfeng He, Rui Wan, Haiyun Hu, Jinfang Cai, Yanquan Feng, Liang Wang, Rongyao Wang, Lin Li, Shaobo Zheng, Chengcheng Liu, Hongkang Zhao, Yongyou Zhang, Lijie Shi, Yulong Liu, Dazhi Xu, Lida Zhang, Shengli Zhang, Ye Cao, Jiangwei Shang, Anning Zhang, Fei Wang, and Hanchun Wu for their discussions and comments; Jinsong Miao for translations of early German literature; and Hao Wei, Guiqin Li, Lifan Ying, Bozhi Sun, Yongjun Lv, and Zhaobin Liu for their help. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11674024 and 11875086.

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Authors and Affiliations

  1. School of Physics, Beijing Institute of Technology, Beijing, 100081, China

    Tianhai Zeng, Zhengzhi Sun & Bin Shao

  2. School of Physical Sciences, University of Chinese Academy of Sciences, P. O. Box 4588, Beijing, 100049, China

    Zhengzhi Sun

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  1. Tianhai Zeng
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Correspondence to Tianhai Zeng.

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Zeng, T., Sun, Z. & Shao, B. Statistical and strict momentum conservation. Int J Theor Phys 59, 229–235 (2020). https://doi.org/10.1007/s10773-019-04315-0

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