Direct Fisher Inference of the Quartic Oscillator’s Eigenvalues

ABSTRACT

It is well known that a suggestive connection links Schrödinger’s equation (SE) and the information-optimizing principle based on Fisher’s information measure (FIM). It has been shown that this entails the existence of a Legendre transform structure underlying the SE. Such a structure leads to a first order partial differential equation (PDE) for the SE’s eigenvalues from which a complete solution for them can be obtained. We test this theory with regards to anharmonic oscillators (AHO). AHO pose a long-standing problem and received intense attention motivated by problems in quantum field theory and molecular physics. By appeal to the Cramer Rao bound we are able to Fisher-infer the energy eigenvalues without explicitly solving Schrödinger’s equation. Remarkably enough, and in contrast with standard variational approaches, our present procedure does not involve free fitting parameters.

It is well known that a suggestive connection links Schrödinger’s equation (SE) and the information-optimizing principle based on Fisher’s information measure (FIM). It has been shown that this entails the existence of a Legendre transform structure underlying the SE. Such a structure leads to a first order partial differential equation (PDE) for the SE’s eigenvalues from which a complete solution for them can be obtained. We test this theory with regards to anharmonic oscillators (AHO). AHO pose a long-standing problem and received intense attention motivated by problems in quantum field theory and molecular physics. By appeal to the Cramer Rao bound we are able to Fisher-infer the energy eigenvalues without explicitly solving Schrödinger’s equation. Remarkably enough, and in contrast with standard variational approaches, our present procedure does not involve free fitting parameters.

KEYWORDS

Information Theory, Fisher’s Information Measure, Legendre Transform, Quartic Anharmonic Oscillator

Information Theory, Fisher’s Information Measure, Legendre Transform, Quartic Anharmonic Oscillator

Cite this paper

nullS. Flego, A. Plastino and A. Plastino, "Direct Fisher Inference of the Quartic Oscillator’s Eigenvalues,"*Journal of Modern Physics*, Vol. 2 No. 11, 2011, pp. 1390-1396. doi: 10.4236/jmp.2011.211171.

nullS. Flego, A. Plastino and A. Plastino, "Direct Fisher Inference of the Quartic Oscillator’s Eigenvalues,"

References

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[1] B. R. Frieden, “Science from Fisher Information: A Unification,” Cambridge, University Press, Cambridge, 2004. doi:10.1017/CBO9780511616907

[2] B. R. Frieden, A. Plastino, A. R. Plastino and B. H. Soffer, “Fisher-Based Thermodynamics: Its Legendre Transform and Concavity Properties,” Physical Review E, Vol. 60, 1999, pp. 48-55. doi:10.1103/PhysRevE.60.48

[3] M. Reginatto, “Derivation of the Equations of Nonrelativistic Quantum Mechanics Using the Principle of Minimum Fisher Information,” Physical Review E, Vol. 58, 1998, pp. 1775-1778.

[4] S. P. Flego, B. R. Frieden, A. Plastino, A. R. Plastino and B. H. Soffer, “Non-Equilibrium Thermodynamics and Fisher Information: Sound Waves Propagation in a Dilute Gas,” Physical Review E, Vol. 68, No. 16, 2003, pp. 105- 115.

[5] S. P. Flego, A. Plastino and A. R. Plastino, “Legendre- transform Structure Derived from Quantum Theorems,” Physica A, Vol. 390, 2011, pp. 2276-2281. doi:10.1016/j.physa.2011.02.019

[6] S. P. Flego, A. Plastino and A. R. Plastino, “Special Features of the Relation between Fisher Information and Schroedinger Eigenvalue Equation,” Journal of Mathematical Physics, Vol. 52, No. 8, 2011, pp. 2103-2116. doi:10.1063/1.3625265

[7] S. P. Flego, A. Plastino and A. R. Plastino, “Inferring an Optimal Fisher Measure,” Physica A, Vol. 390, 2011, pp. 4702-4712. doi:10.1016/j.physa.2011.06.050

[8] F. T. Hioe and E. W. Montroll, “Quantum Theory of Anharmonic Oscillators I. Energy Levels of Oscillators with Positive Quartic Anharmonicity,” Journal of Mathematical Physics, Vol. 16, 1975, pp. 1945-1950. doi:10.1063/1.522747

[9] C. M. Bender and T. T. Wu, “Anharmonic Oscillator,” Physical Review, Vol. 184, 1969, pp. 1231-1260. doi:10.1103/PhysRev.184.1231

[10] K. Banerjee. “W. K. B. Approximation and Scaling,” Proceedings of the Royal Society A, Vol. 363, 1978, pp. 147-151.

[11] R. P. Feynman, “Forces in Molecules,” Physical Review, Vol. 56, 1939, pp. 340-343.

[12] A. Desloge, “Thermal Physics,” Holt, Rinehart and Winston, New York, 1968.

[13] W. Greiner and B. Müller, “Quantum Mechanics. An Introduction,” Springer, Berlin, 1988.

[14] P. M. Mathews and K. Venkatesan, “A Textbook of Quantum Mechanics,” Tata McGraw-Hill Publishing Company Limited, New Delhi, 1986.