Geodesic
Maximum Entropy Wave functions
Lobachevskii journal of mathematics, Tome 23 (2006), pp. 29-56.

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In this paper we use the classical Maximum Entropy principle to define maximum entropy wave functions. These are wave functions that maximize the entropy among all wave functions satisfying a finite set of constraints in the form of expectation values. This lead to a nonlinear equation for the wave function that reduce to the usual stationary Schrödinger equation if the energy is the only constraint and the value of the constraint is an eigenvalue. We discuss the extension of the thermodynamical formalism to this case and apply our general formalism to several simple quantum systems, the two-level atom, the particle in a box, the free particle and the Harmonic Oscillator and compare with the results obtained by applying the usual von Neumann quantum statistical method to the same systems.
Keywords: Maximum entropy principle, quantum mechanics, wavefunctions, probability theory, density matrix. 
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P. K. Jakobsen; V. V. Lychagin. Maximum Entropy Wave functions. Lobachevskii journal of mathematics, Tome 23 (2006), pp. 29-56. http://geodesic.mathdoc.fr/item/LJM_2006_23_a1/

[1] “Probability, Frequency and Reasonable Expectation”, American Journal of Physics, 14 (1946), 1–13 | DOI | MR

[2] R. T. Cox, The algebra of probable inference, The Johns Hopkins Press, Baltimore, Md, 1961 | MR | Zbl

[3] H. Jeffreys, Theory of Probability, Clarendon Press, Oxford, 1961. | MR | Zbl

[4] E. T. Jaynes, Probability Theory: The Logic of Science, Cambridge University Press, 2003 | MR | Zbl

[5] E. T. Jaynes, “Information Theory and Statistical Mechanics”, The Physical Review (2), 106:4 (1957), 620–630 | DOI | MR | Zbl

[6] I. Białynicki-Birula, J. Mycielski, “Nonlinear wave mechanics”, Ann. Physics, 100:1–2 (1976), 62–93 | MR

[7] Joseph Polchinski, “Weinberg's nonlinear quantum mechanics and the Einstein–Poldolsky–Rosen Paradox”, Phys. Rev. Lett., 66:4 (1991) | MR

[8] S. Weinberg, “Precision tests of Quantum Mechanics”, Phys. Rev. Lett., 62:5 (1989) | DOI

[9] J. von Neuman, Mathematical Foundation of Quantum Mechanics, Princeton University Press, 1955 | MR | Zbl

[10] Per Jakobsen and Valentin Lychagin, “Operator valued probability theory”, Lobachevskii Journal of Mathematics, 16 (2004), 17–56 | MR | Zbl