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Deducing the Schrödinger equation from minimum χ2

Abstract

The most unbiased probabilistic model for the possible values of a characteristic of a quantum system subject to the constraints represented by some known mean values characterizes the system in a steady-state condition. We suppose that random fluctuations alter such a steady-state condition. The probability distribution of the possible deviations from the steady-state condition is estimated by minimizing Pearson's ϰ2 subject to the mean fluctuations available. The optimum Pearson function ϰ* may be interpreted as the wave function of the system and in the case of the harmonic oscillator, the free particle in a box, and the hydrogen atom, the prediction based on it is compatible with that provided by the solution of the corresponding Schrödinger equations.

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References

  1. Abramowitz, M., and Stegun, I. A., eds. (1972).Handbook of Mathematical Functions, Dover, New York, pp. 773–783.

  2. Baublitz, Jr., M. (1988).Progress of Theoretical Physics,80, 232.

  3. Bohm, D. (1984).Causality and Chance in Modern Physics, 3rd ed., University of Pennsylvania Press, Philadelphia, Pennsylvania.

  4. Fisher, R. A. (1947).The Design of Experiments, Oliver and Boyd, Edinburgh.

  5. Frieden, B. R. (1989).American Journal of Physics,57, 1004.

  6. Guiasu, S. (1977).Information Theory with Applications, McGraw-Hill, New York.

  7. Guiasu, S. (1987).Physical Review A,36, 1971.

  8. Guiasu, S., and Shenitzer, A. (1985).Mathematical Intelligencer,7, 42.

  9. Jaynes, E. T. (1957).Physical Review,106, 620.

  10. Justice, J. H., ed. (1986).Maximum Entropy and Bayesian Methods in Applied Statistics, Cambridge University Press, Cambridge.

  11. Kullback, S., and Leibler, R. A. (1951).Annals of Mathematical Statistics,32, 79.

  12. Levine, R. D., and Tribus, M., eds. (1979).The Maximum Entropy Formalism, MIT Press, Cambridge, Massachusetts.

  13. McQuarrie, D. A. (1983).Quantum Chemistry, University Science Books, Mill Valley, California.

  14. Mehra, J., and Rechenberg, H. (1987).The Historical Development of Quantum Theory, Springer-Verlag, New York, Vol. 5, Part 2, pp. 827–828.

  15. Nelson, E. (1985).Quantum Fluctuations, Princeton University Press, Princeton, New Jersey.

  16. Pauling, L., and Wilson, Jr., E. B. (1935).Introduction to Quantum Mechanics, McGraw-Hill, New York.

  17. Pearson, K. (1900).Philosophical Magazine (5th Series),50, 157.

  18. Polkinghorne, J. C. (1986).The Quantum World, Penguin Books, Middlesex.

  19. Skilling, J., ed. (1989).Maximum Entropy and Bayesian Methods, Kluwer, Dordrecht, Holland.

  20. Von Neumann, J. (1932).Mathematische Grundlagen der Quantenmechanik, Springer-Verlag, Berlin.

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Guiasu, S. Deducing the Schrödinger equation from minimum χ2 . Int J Theor Phys 31, 1153–1176 (1992). https://doi.org/10.1007/BF00673918

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Keywords

  • Hydrogen
  • Wave Function
  • Probability Distribution
  • Field Theory
  • Hydrogen Atom