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Dynamics of entangled trajectories in quantum phase space

Xu Feng Zheng Yu-Jun

Dynamics of entangled trajectories in quantum phase space

Xu Feng, Zheng Yu-Jun
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  • Quantum phase space theory is widely used to investigate physical, and chemical questions. It gives us a powerful tool to study the relations between the classical and quantum world. In quantum phase space theory, entangled molecular dynamics method is developed based on quantum Liouville equation. It is different from classical Hamilton dynamics, the trajectories are entangled i.e. not independent with each other. The quantum effect in the system can be described using the semiclassical theory, and give a vivid physical picture. The quantum tunneling, photodissociation, and self-correlation are investigated using the entangled trajectory. In this paper, we review the recent development in this field.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos.21073110).
    [1]

    Wigner E P 1932 Phys. Rev. 40 749

    [2]

    Glauber R J 1963 Phys. Rev. 130 2593

    [3]

    Glauber R J 1963 Phys. Rev. 131 2766

    [4]

    Moyal J E 1949 Proc. Cambrridge. Philos. Soc. 45 99

    [5]

    Lee H W, Scully M O 1983 Found. Phys. 13 61

    [6]

    Carruthers P, Zachariasen F 1983 Rev. Mod. Phys. 55 245

    [7]

    Berry M V 1977 Phil. Trans. R. Soc. A 287 237

    [8]

    Takahashi K 1989 Prog. Theor. Phys. Suppl. 98 109

    [9]

    Mahmud K W, Perry H, Reinhardt W P 2005 Phys. Rev. A 71 023615

    [10]

    Shuai Z G, Shao J S 2008 Theoretical chemistry and application (Beijing: Science Press) (in Chinese) [帅志刚, 邵久书 2008 理论化学原理和运用 (北京: 科学出版社)]

    [11]

    Allen M P 1987 Computer Simulation of Liquids (Oxford: Clarendon Press)

    [12]

    Heller E J 1991 J. Chem. Phys. 94 2723

    [13]

    Li Z, Zadoyan R, Apkarian V A, Martens C C 1995 J. Chem. Phys. 99 7453

    [14]

    Lee H W, Scully M O 1980 J. Chem. Phys. 73 2238

    [15]

    Geyer T, Rost J M 2002 J. Phys. B 35 1479

    [16]

    Sheppard M G, Walker R B 1983 J. Chem. Phys. 78 7191

    [17]

    Henriksen N E 1985 Chem. Phys. Lett. 121 139

    [18]

    Li Q, Wei Q, Lu L 2004 Phys. Rev. A 70 022105

    [19]

    Heller E J 1976 J. Chem. Phys. 65 1289

    [20]

    Heller E J 1981 J. Chem. Phys. 67 3339

    [21]

    Brown R C, Heller E J 1981 J. Chem. Phys. 75 186

    [22]

    Zhang D, Chu T S, Hao C 2013 Chin. Phys. B 22 063401

    [23]

    Bohm D 1952 Phys. Rev. 85 166

    [24]

    Bohm D 1952 Phys. Rev. 85 180

    [25]

    Miller W H 2001 J. Phys. Chem. A 105 2942

    [26]

    Pollak E, Shao J 2003 J. Phys. Chem. A 107 7112

    [27]

    Donoso A, Martens C C 2001 Phys. Rev. Lett. 87 223202

    [28]

    Li Q S, Hu X G 2000 Reaction scattering theory in quantum phase space (Beijing: Science Press) (in Chinese) [李前树, 胡旭光 2003 量子相空间中的反应散射理论 (北京: 科学出版社)]

    [29]

    Shewell J R 1959 Amer. J. Phys. 27 17

    [30]

    Mehta C L 1964 J. Math. Phys. 5 677

    [31]

    Mehta C L, Sudarshan E C G 1965 Phys. Rev. B 138 274

    [32]

    Cohen L 1966 J. Math. Phys. 7 781

    [33]

    Agarwal G S, Wolf E 1968 Phys. Rev. Lett. 21 180

    [34]

    Cahill K E, Glauber R J 1969 Phys. Rev. 177 1857

    [35]

    Cahill K E, Glauber R J 1969 Phys. Rev. 177 1882

    [36]

    Heller E J 1991 J. Chem. Phys. 94 2723

    [37]

    Broglie L De 1926 C. R. Acad. Sci. Pairs. 183 447

    [38]

    Broglie L De 1927 C. R. Acad. Sci. Pairs. 184 273

    [39]

    Madelung E 1926 Z. Phys. 40 332

    [40]

    Zurek W H, Wheeler J A 1983 Quantum Theory and Measurement (Princeton NJ: Princeton University Press)

    [41]

    Sanz A S, Borondo F, Miret-Artes S 2000 Phys. Rev. B 61 7743

    [42]

    Sanz A S, Borondo F, Miret-Artes S 2004 Phys. Rev. B 69 115413

    [43]

    Guantes R, Sanz A S, Margalef J R, Miret-Artes S 2004 Surf. Sci. Rep. 53 199

    [44]

    Sanz A S, Miret-Artes S 2007 J. Chem. Phys. 126 234106

    [45]

    Sanz A S, Miret-Artes S 2007 Chem. Phys. Lett. 445 350

    [46]

    Wyatt R E 2005 Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics (New York: Springer Press)

    [47]

    Chou C C, Wyatt R E 2006 J. Chem. Phys. 125 174103

    [48]

    Chou C C, Wyatt R E 2008 J. Chem. Phys. 128 154106

    [49]

    Chou C C, Wyatt R E 2008 J. Chem. Phys. 129 124113

    [50]

    Chou C C, Sanz A S, Miret-Artes S, Wyatt R E 2009 Phys. Rev. Lett. 102 250401

    [51]

    Chou C C, Wyatt R E 2012 Ann. Phys. 327 1355

    [52]

    Lai X Y, Cai Q Y, Zhan M S 2010 Chin. Phys. B 19 020302

    [53]

    Donoso A, Zheng Y, Martens C C 2003 J. Chem. Phys. 119 5010

    [54]

    Donoso A, Martens C C 2010 Chem. Phys. 370 20

    [55]

    Hogan P, Wart A A, Donoso A, Martens C C 2002 J. Chem. Phys. 116 10598

    [56]

    Lopez H, Martens C C, Donoso A 2006 J. Chem. Phys. 125 154111

    [57]

    Wang A S, Zheng Y, Martens C C, Ren W Y 2009 Phys. Chem. Chem. Phys. 11 1588

    [58]

    Zhang X F, Zheng Y 2009 Chin. Phys. Lett 26 023404

    [59]

    Wang L, Marten C C, Zheng Y 2012 J. Chem. Phys. 137 034113

    [60]

    Xu F, Wang L, Marten C C, Zheng Y 2013 J. Chem. Phys. 138 024113

    [61]

    Wang L, Xu F, Zheng Y 2013 J. At. & Mol. Sci. (in press)

    [62]

    Sala R, Brouard S, Muga J G 1993 J. Chem. Phys. 99 2708

    [63]

    Zhang S, Pollak E 2004 J. Chem. Phys. 121 3384

    [64]

    Silverman B W 1986 Density Estimation for Statistics and Data Analysis (London: Chapman and Hall)

    [65]

    Fukunaga K 1986 Introduction to Statistical Pattern Recognition (San Diego: Academic, 2nd)

    [66]

    Martens C C, Zheng Y J 2013 personal communication

    [67]

    Nauenberg M 1989 Phys. Rev. A 40 1133

    [68]

    Barnes I M, Nauenberg M, Nockleby M, Tomsovic S 1993 Phys. Rev. Lett. 71 1961

    [69]

    Tomsovic S, Lefebvre J H 1997 Phys. Rev. Lett. 79 3629

    [70]

    Yeazell J A, Stroud C R 1991 Phys. Rev. A 43 5153

    [71]

    Wals J, Fielding H H, Christian J F, Snoek L C, Zande W J, Linden H B 1994 Phys. Rev. Lett. 73 3783

    [72]

    Zdanska P, Moiseyev N 2001 J. Phys. Chem. 115 10608

    [73]

    Wang L, Wang Y, Ran S, Yang G 2009 J. Electron Spectrosc. Relat. Phenom. 173 40

    [74]

    Tannor D J 2006 Introduction to Quantum Mechanics A Time-Dependent Perspective (USA: University Science Books)

    [75]

    Goldfarb Y, Tannor D J 1996 Chem. Phys. Lett. 263 324

    [76]

    Wang L F, Zheng Y J 2013 Chem. Phys. Lett 563 112

    [77]

    Engel V, Schinke R, Staemmler V 1988 J. Chem. Phys. 88 129

    [78]

    Kuhl K, Schinke R 1989 Chem. Phys. Lett. 158 81

    [79]

    Zhang J Z, Imre D G 1989 J. Chem. Phys. 90 1666

    [80]

    Brouard M, Langford S R 1994 J. Chem. Phys. 101 7458

    [81]

    Schinke R, Staemmler V, Wal R L V, Crion F F, Sension R J, Hudson B, Andere-sen P, Silverman R 1992 J. Phys. Chem. 96 3201

    [82]

    Henriken N E, Engel V, Schinke R 1986 J. Phys. Chem. 86 6862

    [83]

    Hupper B E B, Engel V 1997 J. Phys. B 30 3191

    [84]

    Schinke R 1993 Photodissociation Dynamics (England: Cambridge University Press)

  • [1]

    Wigner E P 1932 Phys. Rev. 40 749

    [2]

    Glauber R J 1963 Phys. Rev. 130 2593

    [3]

    Glauber R J 1963 Phys. Rev. 131 2766

    [4]

    Moyal J E 1949 Proc. Cambrridge. Philos. Soc. 45 99

    [5]

    Lee H W, Scully M O 1983 Found. Phys. 13 61

    [6]

    Carruthers P, Zachariasen F 1983 Rev. Mod. Phys. 55 245

    [7]

    Berry M V 1977 Phil. Trans. R. Soc. A 287 237

    [8]

    Takahashi K 1989 Prog. Theor. Phys. Suppl. 98 109

    [9]

    Mahmud K W, Perry H, Reinhardt W P 2005 Phys. Rev. A 71 023615

    [10]

    Shuai Z G, Shao J S 2008 Theoretical chemistry and application (Beijing: Science Press) (in Chinese) [帅志刚, 邵久书 2008 理论化学原理和运用 (北京: 科学出版社)]

    [11]

    Allen M P 1987 Computer Simulation of Liquids (Oxford: Clarendon Press)

    [12]

    Heller E J 1991 J. Chem. Phys. 94 2723

    [13]

    Li Z, Zadoyan R, Apkarian V A, Martens C C 1995 J. Chem. Phys. 99 7453

    [14]

    Lee H W, Scully M O 1980 J. Chem. Phys. 73 2238

    [15]

    Geyer T, Rost J M 2002 J. Phys. B 35 1479

    [16]

    Sheppard M G, Walker R B 1983 J. Chem. Phys. 78 7191

    [17]

    Henriksen N E 1985 Chem. Phys. Lett. 121 139

    [18]

    Li Q, Wei Q, Lu L 2004 Phys. Rev. A 70 022105

    [19]

    Heller E J 1976 J. Chem. Phys. 65 1289

    [20]

    Heller E J 1981 J. Chem. Phys. 67 3339

    [21]

    Brown R C, Heller E J 1981 J. Chem. Phys. 75 186

    [22]

    Zhang D, Chu T S, Hao C 2013 Chin. Phys. B 22 063401

    [23]

    Bohm D 1952 Phys. Rev. 85 166

    [24]

    Bohm D 1952 Phys. Rev. 85 180

    [25]

    Miller W H 2001 J. Phys. Chem. A 105 2942

    [26]

    Pollak E, Shao J 2003 J. Phys. Chem. A 107 7112

    [27]

    Donoso A, Martens C C 2001 Phys. Rev. Lett. 87 223202

    [28]

    Li Q S, Hu X G 2000 Reaction scattering theory in quantum phase space (Beijing: Science Press) (in Chinese) [李前树, 胡旭光 2003 量子相空间中的反应散射理论 (北京: 科学出版社)]

    [29]

    Shewell J R 1959 Amer. J. Phys. 27 17

    [30]

    Mehta C L 1964 J. Math. Phys. 5 677

    [31]

    Mehta C L, Sudarshan E C G 1965 Phys. Rev. B 138 274

    [32]

    Cohen L 1966 J. Math. Phys. 7 781

    [33]

    Agarwal G S, Wolf E 1968 Phys. Rev. Lett. 21 180

    [34]

    Cahill K E, Glauber R J 1969 Phys. Rev. 177 1857

    [35]

    Cahill K E, Glauber R J 1969 Phys. Rev. 177 1882

    [36]

    Heller E J 1991 J. Chem. Phys. 94 2723

    [37]

    Broglie L De 1926 C. R. Acad. Sci. Pairs. 183 447

    [38]

    Broglie L De 1927 C. R. Acad. Sci. Pairs. 184 273

    [39]

    Madelung E 1926 Z. Phys. 40 332

    [40]

    Zurek W H, Wheeler J A 1983 Quantum Theory and Measurement (Princeton NJ: Princeton University Press)

    [41]

    Sanz A S, Borondo F, Miret-Artes S 2000 Phys. Rev. B 61 7743

    [42]

    Sanz A S, Borondo F, Miret-Artes S 2004 Phys. Rev. B 69 115413

    [43]

    Guantes R, Sanz A S, Margalef J R, Miret-Artes S 2004 Surf. Sci. Rep. 53 199

    [44]

    Sanz A S, Miret-Artes S 2007 J. Chem. Phys. 126 234106

    [45]

    Sanz A S, Miret-Artes S 2007 Chem. Phys. Lett. 445 350

    [46]

    Wyatt R E 2005 Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics (New York: Springer Press)

    [47]

    Chou C C, Wyatt R E 2006 J. Chem. Phys. 125 174103

    [48]

    Chou C C, Wyatt R E 2008 J. Chem. Phys. 128 154106

    [49]

    Chou C C, Wyatt R E 2008 J. Chem. Phys. 129 124113

    [50]

    Chou C C, Sanz A S, Miret-Artes S, Wyatt R E 2009 Phys. Rev. Lett. 102 250401

    [51]

    Chou C C, Wyatt R E 2012 Ann. Phys. 327 1355

    [52]

    Lai X Y, Cai Q Y, Zhan M S 2010 Chin. Phys. B 19 020302

    [53]

    Donoso A, Zheng Y, Martens C C 2003 J. Chem. Phys. 119 5010

    [54]

    Donoso A, Martens C C 2010 Chem. Phys. 370 20

    [55]

    Hogan P, Wart A A, Donoso A, Martens C C 2002 J. Chem. Phys. 116 10598

    [56]

    Lopez H, Martens C C, Donoso A 2006 J. Chem. Phys. 125 154111

    [57]

    Wang A S, Zheng Y, Martens C C, Ren W Y 2009 Phys. Chem. Chem. Phys. 11 1588

    [58]

    Zhang X F, Zheng Y 2009 Chin. Phys. Lett 26 023404

    [59]

    Wang L, Marten C C, Zheng Y 2012 J. Chem. Phys. 137 034113

    [60]

    Xu F, Wang L, Marten C C, Zheng Y 2013 J. Chem. Phys. 138 024113

    [61]

    Wang L, Xu F, Zheng Y 2013 J. At. & Mol. Sci. (in press)

    [62]

    Sala R, Brouard S, Muga J G 1993 J. Chem. Phys. 99 2708

    [63]

    Zhang S, Pollak E 2004 J. Chem. Phys. 121 3384

    [64]

    Silverman B W 1986 Density Estimation for Statistics and Data Analysis (London: Chapman and Hall)

    [65]

    Fukunaga K 1986 Introduction to Statistical Pattern Recognition (San Diego: Academic, 2nd)

    [66]

    Martens C C, Zheng Y J 2013 personal communication

    [67]

    Nauenberg M 1989 Phys. Rev. A 40 1133

    [68]

    Barnes I M, Nauenberg M, Nockleby M, Tomsovic S 1993 Phys. Rev. Lett. 71 1961

    [69]

    Tomsovic S, Lefebvre J H 1997 Phys. Rev. Lett. 79 3629

    [70]

    Yeazell J A, Stroud C R 1991 Phys. Rev. A 43 5153

    [71]

    Wals J, Fielding H H, Christian J F, Snoek L C, Zande W J, Linden H B 1994 Phys. Rev. Lett. 73 3783

    [72]

    Zdanska P, Moiseyev N 2001 J. Phys. Chem. 115 10608

    [73]

    Wang L, Wang Y, Ran S, Yang G 2009 J. Electron Spectrosc. Relat. Phenom. 173 40

    [74]

    Tannor D J 2006 Introduction to Quantum Mechanics A Time-Dependent Perspective (USA: University Science Books)

    [75]

    Goldfarb Y, Tannor D J 1996 Chem. Phys. Lett. 263 324

    [76]

    Wang L F, Zheng Y J 2013 Chem. Phys. Lett 563 112

    [77]

    Engel V, Schinke R, Staemmler V 1988 J. Chem. Phys. 88 129

    [78]

    Kuhl K, Schinke R 1989 Chem. Phys. Lett. 158 81

    [79]

    Zhang J Z, Imre D G 1989 J. Chem. Phys. 90 1666

    [80]

    Brouard M, Langford S R 1994 J. Chem. Phys. 101 7458

    [81]

    Schinke R, Staemmler V, Wal R L V, Crion F F, Sension R J, Hudson B, Andere-sen P, Silverman R 1992 J. Phys. Chem. 96 3201

    [82]

    Henriken N E, Engel V, Schinke R 1986 J. Phys. Chem. 86 6862

    [83]

    Hupper B E B, Engel V 1997 J. Phys. B 30 3191

    [84]

    Schinke R 1993 Photodissociation Dynamics (England: Cambridge University Press)

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  • Received Date:  18 June 2013
  • Accepted Date:  27 July 2013
  • Published Online:  05 November 2013

Dynamics of entangled trajectories in quantum phase space

  • 1. School of Physics, Shandong University, Jinan 250100, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos.21073110).

Abstract: Quantum phase space theory is widely used to investigate physical, and chemical questions. It gives us a powerful tool to study the relations between the classical and quantum world. In quantum phase space theory, entangled molecular dynamics method is developed based on quantum Liouville equation. It is different from classical Hamilton dynamics, the trajectories are entangled i.e. not independent with each other. The quantum effect in the system can be described using the semiclassical theory, and give a vivid physical picture. The quantum tunneling, photodissociation, and self-correlation are investigated using the entangled trajectory. In this paper, we review the recent development in this field.

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