• Previous Article
    Hypocoercive relaxation to equilibrium for some kinetic models
  • KRM Home
  • This Issue
  • Next Article
    Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation
2014, 7(2): 361-379. doi: 10.3934/krm.2014.7.361

A random cloud model for the Schrödinger equation

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39 - 10117 Berlin

Received  October 2013 Revised  January 2014 Published  March 2014

The paper is concerned with the construction of a stochastic model for the spatially discretized time-dependent Schrödinger equation. The model is based on a particle system with a Markov jump evolution. The particles are characterized by a sign (plus or minus), a position (discrete grid) and a type (real or imaginary). The jumps are determined by the creation of offspring. The main result is the construction of a family of complex-valued random variables such that their expected values coincide with the solution of the Schrödinger equation.
Citation: Wolfgang Wagner. A random cloud model for the Schrödinger equation. Kinetic & Related Models, 2014, 7 (2) : 361-379. doi: 10.3934/krm.2014.7.361
References:
[1]

D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden'' variables. I,, Physical Rev. (2), 85 (1952), 166. doi: 10.1103/PhysRev.85.166.

[2]

D. Bohm and J. Bub, A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory,, Rev. Mod. Phys., 38 (1966), 453. doi: 10.1103/RevModPhys.38.453.

[3]

L. Breiman, Probability,, Addison-Wesley Publishing Company, (1968). doi: 10.2307/2285875.

[4]

P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues,, Springer-Verlag, (1999).

[5]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik,, Math. Ann., 100 (1928), 32. doi: 10.1007/BF01448839.

[6]

B. S. DeWitt, Quantum mechanics and reality,, Physics Today, 23 (1970), 155. doi: 10.1063/1.3022331.

[7]

A. Eibeck and W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations,, Ann. Appl. Probab., 13 (2003), 845. doi: 10.1214/aoap/1060202829.

[8]

H. Everett, III, "Relative state" formulation of quantum mechanics,, Rev. Mod. Phys., 29 (1957), 454. doi: 10.1103/RevModPhys.29.454.

[9]

R. P. Feynman, Space-time approach to non-relativistic quantum mechanics,, Rev. Mod. Phys., 20 (1948), 367. doi: 10.1103/RevModPhys.20.367.

[10]

R. P. Feynman, The concept of probability in quantum mechanics,, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (1951), 533.

[11]

R. P. Feynman, Simulating physics with computers,, International Journal of Theoretical Physics, 21 (1982), 467. doi: 10.1007/BF02650179.

[12]

R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics. Vol. 3: Quantum Mechanics,, Addison-Wesley Publishing Co., (1965).

[13]

M. Kac, On distributions of certain Wiener functionals,, Trans. Amer. Math. Soc., 65 (1949), 1. doi: 10.1090/S0002-9947-1949-0027960-X.

[14]

M. Kac, Foundations of kinetic theory,, in Third Berkeley Symposium on Mathematical Statistics and Probability Theory, 3 (1956), 171.

[15]

A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,, Math. Ann., 104 (1931), 415. doi: 10.1007/BF01457949.

[16]

M. A. Leontovich, Basic equations of the kinetic theory of gases from the point of view of the theory of random processes,, Zhurnal Ehksper. Teoret. Fiziki, 5 (1935), 211.

[17]

N. D. Mermin, What's wrong with this pillow?,, Physics Today, 42 (1989), 9. doi: 10.1017/CBO9780511608216.017.

[18]

E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics,, Phys. Rev., 150 (1966), 1079. doi: 10.1103/PhysRev.150.1079.

[19]

E. Nelson, Review of stochastic mechanics,, J. Phys.: Conf. Ser., 361 (2012), 1. doi: 10.1088/1742-6596/361/1/012011.

[20]

T. Norsen and S. Nelson, Yet another snapshot of foundational attitudes toward quantum mechanics,, , (2013), 1.

[21]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation,, Springer, (2005).

[22]

E. Schrödinger, Quantisierung als Eigenwertproblem (Erste Mitteilung),, Ann. d. Phys., 79 (1926), 361.

[23]

E. Schrödinger, Quantisierung als Eigenwertproblem (Vierte Mitteilung),, Ann. d. Phys., 81 (1926), 109.

[24]

W. Wagner, Stochastic models in kinetic theory,, Phys. Fluids, 23 (2011), 1. doi: 10.1063/1.3558866.

show all references

References:
[1]

D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden'' variables. I,, Physical Rev. (2), 85 (1952), 166. doi: 10.1103/PhysRev.85.166.

[2]

D. Bohm and J. Bub, A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory,, Rev. Mod. Phys., 38 (1966), 453. doi: 10.1103/RevModPhys.38.453.

[3]

L. Breiman, Probability,, Addison-Wesley Publishing Company, (1968). doi: 10.2307/2285875.

[4]

P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues,, Springer-Verlag, (1999).

[5]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik,, Math. Ann., 100 (1928), 32. doi: 10.1007/BF01448839.

[6]

B. S. DeWitt, Quantum mechanics and reality,, Physics Today, 23 (1970), 155. doi: 10.1063/1.3022331.

[7]

A. Eibeck and W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations,, Ann. Appl. Probab., 13 (2003), 845. doi: 10.1214/aoap/1060202829.

[8]

H. Everett, III, "Relative state" formulation of quantum mechanics,, Rev. Mod. Phys., 29 (1957), 454. doi: 10.1103/RevModPhys.29.454.

[9]

R. P. Feynman, Space-time approach to non-relativistic quantum mechanics,, Rev. Mod. Phys., 20 (1948), 367. doi: 10.1103/RevModPhys.20.367.

[10]

R. P. Feynman, The concept of probability in quantum mechanics,, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (1951), 533.

[11]

R. P. Feynman, Simulating physics with computers,, International Journal of Theoretical Physics, 21 (1982), 467. doi: 10.1007/BF02650179.

[12]

R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics. Vol. 3: Quantum Mechanics,, Addison-Wesley Publishing Co., (1965).

[13]

M. Kac, On distributions of certain Wiener functionals,, Trans. Amer. Math. Soc., 65 (1949), 1. doi: 10.1090/S0002-9947-1949-0027960-X.

[14]

M. Kac, Foundations of kinetic theory,, in Third Berkeley Symposium on Mathematical Statistics and Probability Theory, 3 (1956), 171.

[15]

A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,, Math. Ann., 104 (1931), 415. doi: 10.1007/BF01457949.

[16]

M. A. Leontovich, Basic equations of the kinetic theory of gases from the point of view of the theory of random processes,, Zhurnal Ehksper. Teoret. Fiziki, 5 (1935), 211.

[17]

N. D. Mermin, What's wrong with this pillow?,, Physics Today, 42 (1989), 9. doi: 10.1017/CBO9780511608216.017.

[18]

E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics,, Phys. Rev., 150 (1966), 1079. doi: 10.1103/PhysRev.150.1079.

[19]

E. Nelson, Review of stochastic mechanics,, J. Phys.: Conf. Ser., 361 (2012), 1. doi: 10.1088/1742-6596/361/1/012011.

[20]

T. Norsen and S. Nelson, Yet another snapshot of foundational attitudes toward quantum mechanics,, , (2013), 1.

[21]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation,, Springer, (2005).

[22]

E. Schrödinger, Quantisierung als Eigenwertproblem (Erste Mitteilung),, Ann. d. Phys., 79 (1926), 361.

[23]

E. Schrödinger, Quantisierung als Eigenwertproblem (Vierte Mitteilung),, Ann. d. Phys., 81 (1926), 109.

[24]

W. Wagner, Stochastic models in kinetic theory,, Phys. Fluids, 23 (2011), 1. doi: 10.1063/1.3558866.

[1]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[2]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878

[3]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[4]

Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235

[5]

Pierre Degond, Simone Goettlich, Axel Klar, Mohammed Seaid, Andreas Unterreiter. Derivation of a kinetic model from a stochastic particle system. Kinetic & Related Models, 2008, 1 (4) : 557-572. doi: 10.3934/krm.2008.1.557

[6]

Wei Wang, Linyi Qian, Xiaonan Su. Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model. Journal of Industrial & Management Optimization, 2015, 11 (2) : 493-514. doi: 10.3934/jimo.2015.11.493

[7]

Brenton LeMesurier. Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 317-327. doi: 10.3934/dcdss.2008.1.317

[8]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689

[9]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161

[10]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

[11]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791

[12]

Michele Gianfelice, Marco Isopi. On the location of the 1-particle branch of the spectrum of the disordered stochastic Ising model. Networks & Heterogeneous Media, 2011, 6 (1) : 127-144. doi: 10.3934/nhm.2011.6.127

[13]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027

[14]

Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061

[15]

Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381

[16]

Veronica Felli, Alberto Ferrero, Susanna Terracini. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3895-3956. doi: 10.3934/dcds.2012.32.3895

[17]

Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2795-2818. doi: 10.3934/dcds.2014.34.2795

[18]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

[19]

Alexander Arbieto, Carlos Matheus. On the periodic Schrödinger-Debye equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 699-713. doi: 10.3934/cpaa.2008.7.699

[20]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094

2016 Impact Factor: 1.261

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]