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A path integral approach to quantum fluid dynamics: application to double well potential

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Abstract

In this work we develop an alternative approach for solution of Quantum Trajectories using the Path Integral method. The state-of-the-art technique in the field is to solve a set of nonlinear, coupled partial differential equations simultaneously. We opt for a fundamentally different route. We first derive a general closed form expression for the Path Integral propagator valid for any general potential as a functional of the corresponding classical path. The method is exact and is applicable in many dimensions as well as multi-particle cases. This, then, is used to compute the Quantum Potential, which, in turn, can generate the Quantum Trajectories. As a model application to illustrate the method, we solve for the double-well potential, both analytically (using a perturbative approach) and numerically (exact solution). Using this we delve into seeking insight into Quantum Tunnelling. The work formally bridges the Path Integral approach with Quantum Fluid Dynamics, an issue of fundamental importance.

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Notes

  1. The radius of convergence depends on the V(x), its higher derivatives and the convergence is not guaranteed in general.

  2. Assuming convergence of the individual integrals.

  3. Although the initial value problem does, [36, 37]. The BVP solution is not even unique.

References

  1. Madelung E (1927) Quantentheorie in hydrodynamischer form. Zeitschrift für Physik 40:322–326

    Article  Google Scholar 

  2. De Broglie L (1926) CR Acad Sci Paris 183:447

    Google Scholar 

  3. Bohm D (1952) A suggested interpretation of the quantum theory in terms of “hidden’’ variables, i. Phys Rev 85:166–179

    Article  CAS  Google Scholar 

  4. Bohm D (1952) A suggested interpretation of the quantum theory in terms of “hidden’’ variables. ii. Phys Rev 85:180–193

    Article  CAS  Google Scholar 

  5. Takabayasi T (1952) On the formulation of quantum mechanics associated with classical pictures*. Prog Theor Phys 8(2):143–182

    Article  Google Scholar 

  6. Ghosh SK, Deb BM (1982) Densities, density functionals and electron fluids. Phys Rep 92:1–44

    Article  CAS  Google Scholar 

  7. Mayor FS, Askar A, Rabitz H (1999) Quantum fluid dynamics in the Lagrangian representation and applications to photodissociation problems. J Chem Phys 111:2423

    Article  CAS  Google Scholar 

  8. Feynman RP (1948) Space-time approach to non-relativistic quantum mechanics. Rev Mod Phys 20:367–387

    Article  Google Scholar 

  9. Feynman RP, Hibbs AR, Styer DF (2010) Quantum mechanics and path integrals. Courier Corporation

  10. Nelson E (1966) Derivation of the Schrödinger equation from Newtonian mechanics. Phys Rev 150:1079

    Article  CAS  Google Scholar 

  11. Nelson E (1985) Quantum fluctuations. Princeton University Press, Princeton

    Book  Google Scholar 

  12. Parr R, Yang W (1989) Density-functional theory of atoms and molecules. Oxford Univ Press, Oxford

    Google Scholar 

  13. Ghosh SK, Dhara AK (1988) Density-functional theory of many-electron systems subjected to time-dependent electric and magnetic fields. Phys Rev A 38:1149–1158

    Article  CAS  Google Scholar 

  14. Chattaraj PK, Maity B (2001) Reactivity dynamics in atom field interactions: a quantum fluid density functional study. J Phys Chem A 105(1):169–183

    Article  CAS  Google Scholar 

  15. Ghosh SK, Berkowitz M, Parr RG (1984) Transcription of ground state density functional theory into a local thermodynamics. Proc Natl Acad Sci USA 81:8028

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Ghosh SK, Deb BM (1987). In: March NH, Deb DM (eds) The single-particle density in physics and chemistry, Chemistry. Academic Press, New York

    Google Scholar 

  17. Wyatt RE (2006) Quantum dynamics with trajectories: introduction to quantum hydrodynamics, vol 28. Springer, New York

    Google Scholar 

  18. Lopreore CL, Wyatt RE (1999) Quantum wave packet dynamics with trajectories. Phys Rev Lett 82:5190–5193

    Article  CAS  Google Scholar 

  19. Dirac PAM (1945) On the analogy between classical and quantum mechanics. Rev Mod Phys 17:195–199

    Article  Google Scholar 

  20. Covers the detail nitty-gritties of the mathematical foundations of Feynman path integrals [37]. For the physical theory [9] is the standard introduction

  21. Ursell HD (1927) The evaluation of Gibbs’ phase-integral for imperfect gases. Math Proc Camb Philos Soc 23(6):685–697

    Article  CAS  Google Scholar 

  22. Kulkarni D, Schmidt D, Tsui S (1999) Eigenvalues of tridiagonal pseudo Toeplitz matrices. Linear Algebra Appl 297(1):63–80

    Article  Google Scholar 

  23. Landau L, Lifshitz E (1965) Quantum mechanics; non-relativistic theory. Pergamon Press

    Google Scholar 

  24. Schulman LS (2012) Techniques and applications of path integration. Courier Corporation

  25. Carusotto S (1988) Theory of a quantum anharmonic oscillator. Phys Rev A 38:3249–3257

    Article  CAS  Google Scholar 

  26. Hazra A, Soudackov AV, Hammes-Schiffer S (2010) Role of solvent dynamics in ultrafast photoinduced proton-coupled electron transfer reactions in solution. J Phys Chem B 114(38):12319–12332

    Article  CAS  PubMed  Google Scholar 

  27. Lin C-K, Chang H-C, Lin SH (2007) Symmetric double-well potential model and its application to vibronic spectra: Studies of inversion modes of ammonia and nitrogen-vacancy defect centers in diamond. J Phys Chem A 111(38):9347–9354

    Article  CAS  PubMed  Google Scholar 

  28. Quhe R, Nava M, Tiwary P, Parrinello M (2015) Path integral metadynamics. J Chem Theory Comput 11(4):1383–1388

    Article  CAS  PubMed  Google Scholar 

  29. Although the initial value problem does [38,39]. The BVP solution is not even unique

  30. Landau L, Lifshitz E (1960) “28, course of theoretical physics. vol 1: Mechanics”. Pergamon Press, Oxford

  31. Gradshteyn I, Ryzik I (1980) Tables of integrals, series and products. Academic, New York

    Google Scholar 

  32. Byrd PF, Friedman MD (2013) Handbook of elliptic integrals for engineers and physicists, vol 67. Springer, New York

    Google Scholar 

  33. Sornborger AT (2012) Quantum simulation of tunneling in small systems. Sci Rep 8

  34. Karmakar S, Datta A (2014) Tunneling assists the 1,2-hydrogen shift in n-heterocyclic carbenes. Angewandte Chemie Int Edn 53(36):9587–9591

    Article  CAS  Google Scholar 

  35. Albeverio S, Høegh-Krohn R, Mazzucchi S (2008) Mathematical theory of Feynman path integrals: an introduction, vol 523. Springer, New York

    Google Scholar 

  36. Onodera Y (1970) Dynamic susceptibility of classical Anharmonic oscillator: a unified oscillator model for order disorder and displacive ferroelectrics. Prog Theor Phys 44(6):1477–1499

    Article  Google Scholar 

  37. Codaccioni JP, Caboz R (1984) Anharmonic oscillators and generalized hypergeometric functions. J Math Phys 25(8):2436–2438

    Article  Google Scholar 

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Acknowledgements

The present paper reports a part of the work done at the Homi Bhabha Centre for Science Education (Tata Institute of Fundamental Research), Mumbai. We are indebted to the National Initiative on Undergraduate Science (NIUS), Chemistry fellowship of HBCSE (Batch XIII, 2016–2018) for their tremendous and inspiring support. We acknowledge the support of the Government of India, Department of Atomic Energy, under Project No. 12-R &D-TFR\(-\)6.04-0600. Many of the colleagues at the programme and the institute have enriched us. SG wishes to thank Dr. Anirban Hazra for several stimulating discussions at the initial parts of the project. No language of acknowledgement is enough for the utmost care that Dr. Amrita B. Hazra has provided to SG. Without the thorough support she has offered throughout, it would have not been possible to conceive the project to its present extent.

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Authors and Affiliations

  1. Indian Institute of Science Education and Research, Pune, 411008, India

    Sagnik Ghosh

  2. UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Kalina, Santacruz, Mumbai, 400098, India

    Swapan K. Ghosh

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  1. Sagnik Ghosh
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  2. Swapan K. Ghosh
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Sagnik Ghosh and Swapan K. Ghosh wrote the main manuscript text and Sagnik Ghosh prepared all the Figures. All authors reviewed the manuscript.

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Correspondence to Swapan K. Ghosh.

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Ghosh, S., Ghosh, S.K. A path integral approach to quantum fluid dynamics: application to double well potential. Theor Chem Acc 142, 57 (2023). https://doi.org/10.1007/s00214-023-02995-w

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