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June  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), 166-179. doi: 10.1103/PhysRev.85.166.  Google Scholar

[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), 453-469. doi: 10.1103/RevModPhys.38.453.  Google Scholar

[3]

L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass., 1968. doi: 10.2307/2285875.  Google Scholar

[4]

P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer-Verlag, New York, 1999.  Google Scholar

[5]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74. doi: 10.1007/BF01448839.  Google Scholar

[6]

B. S. DeWitt, Quantum mechanics and reality, Physics Today, 23 (1970), 155-165. doi: 10.1063/1.3022331.  Google Scholar

[7]

A. Eibeck and W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab., 13 (2003), 845-889. doi: 10.1214/aoap/1060202829.  Google Scholar

[8]

H. Everett, III, "Relative state" formulation of quantum mechanics, Rev. Mod. Phys., 29 (1957), 454-462. doi: 10.1103/RevModPhys.29.454.  Google Scholar

[9]

R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys., 20 (1948), 367-387. doi: 10.1103/RevModPhys.20.367.  Google Scholar

[10]

R. P. Feynman, The concept of probability in quantum mechanics, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, (1951), 533-541.  Google Scholar

[11]

R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics, 21 (1982), 467-488. doi: 10.1007/BF02650179.  Google Scholar

[12]

R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics. Vol. 3: Quantum Mechanics, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965.  Google Scholar

[13]

M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc., 65 (1949), 1-13. doi: 10.1090/S0002-9947-1949-0027960-X.  Google Scholar

[14]

M. Kac, Foundations of kinetic theory, in Third Berkeley Symposium on Mathematical Statistics and Probability Theory, 3, University of California Press, Berkeley and Los Angeles, (1956), 171-197.  Google Scholar

[15]

A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104 (1931), 415-458. doi: 10.1007/BF01457949.  Google Scholar

[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), 211-231, In Russian. Google Scholar

[17]

N. D. Mermin, What's wrong with this pillow?, Physics Today, 42 (1989), 9-11. doi: 10.1017/CBO9780511608216.017.  Google Scholar

[18]

E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150 (1966), 1079-1085. doi: 10.1103/PhysRev.150.1079.  Google Scholar

[19]

E. Nelson, Review of stochastic mechanics, J. Phys.: Conf. Ser., 361 (2012), 012011, 1-4. doi: 10.1088/1742-6596/361/1/012011.  Google Scholar

[20]

T. Norsen and S. Nelson, Yet another snapshot of foundational attitudes toward quantum mechanics, arXiv:1306.4646, (2013), 1-11. Google Scholar

[21]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation, Springer, Berlin, 2005.  Google Scholar

[22]

E. Schrödinger, Quantisierung als Eigenwertproblem (Erste Mitteilung), Ann. d. Phys., 79 (1926), 361-376. Google Scholar

[23]

E. Schrödinger, Quantisierung als Eigenwertproblem (Vierte Mitteilung), Ann. d. Phys., 81 (1926), 109-139. Google Scholar

[24]

W. Wagner, Stochastic models in kinetic theory, Phys. Fluids, 23 (2011), 030602, 1-14. doi: 10.1063/1.3558866.  Google Scholar

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), 166-179. doi: 10.1103/PhysRev.85.166.  Google Scholar

[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), 453-469. doi: 10.1103/RevModPhys.38.453.  Google Scholar

[3]

L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass., 1968. doi: 10.2307/2285875.  Google Scholar

[4]

P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer-Verlag, New York, 1999.  Google Scholar

[5]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74. doi: 10.1007/BF01448839.  Google Scholar

[6]

B. S. DeWitt, Quantum mechanics and reality, Physics Today, 23 (1970), 155-165. doi: 10.1063/1.3022331.  Google Scholar

[7]

A. Eibeck and W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab., 13 (2003), 845-889. doi: 10.1214/aoap/1060202829.  Google Scholar

[8]

H. Everett, III, "Relative state" formulation of quantum mechanics, Rev. Mod. Phys., 29 (1957), 454-462. doi: 10.1103/RevModPhys.29.454.  Google Scholar

[9]

R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys., 20 (1948), 367-387. doi: 10.1103/RevModPhys.20.367.  Google Scholar

[10]

R. P. Feynman, The concept of probability in quantum mechanics, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, (1951), 533-541.  Google Scholar

[11]

R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics, 21 (1982), 467-488. doi: 10.1007/BF02650179.  Google Scholar

[12]

R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics. Vol. 3: Quantum Mechanics, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965.  Google Scholar

[13]

M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc., 65 (1949), 1-13. doi: 10.1090/S0002-9947-1949-0027960-X.  Google Scholar

[14]

M. Kac, Foundations of kinetic theory, in Third Berkeley Symposium on Mathematical Statistics and Probability Theory, 3, University of California Press, Berkeley and Los Angeles, (1956), 171-197.  Google Scholar

[15]

A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104 (1931), 415-458. doi: 10.1007/BF01457949.  Google Scholar

[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), 211-231, In Russian. Google Scholar

[17]

N. D. Mermin, What's wrong with this pillow?, Physics Today, 42 (1989), 9-11. doi: 10.1017/CBO9780511608216.017.  Google Scholar

[18]

E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150 (1966), 1079-1085. doi: 10.1103/PhysRev.150.1079.  Google Scholar

[19]

E. Nelson, Review of stochastic mechanics, J. Phys.: Conf. Ser., 361 (2012), 012011, 1-4. doi: 10.1088/1742-6596/361/1/012011.  Google Scholar

[20]

T. Norsen and S. Nelson, Yet another snapshot of foundational attitudes toward quantum mechanics, arXiv:1306.4646, (2013), 1-11. Google Scholar

[21]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation, Springer, Berlin, 2005.  Google Scholar

[22]

E. Schrödinger, Quantisierung als Eigenwertproblem (Erste Mitteilung), Ann. d. Phys., 79 (1926), 361-376. Google Scholar

[23]

E. Schrödinger, Quantisierung als Eigenwertproblem (Vierte Mitteilung), Ann. d. Phys., 81 (1926), 109-139. Google Scholar

[24]

W. Wagner, Stochastic models in kinetic theory, Phys. Fluids, 23 (2011), 030602, 1-14. doi: 10.1063/1.3558866.  Google Scholar

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