Fisher information and kinetic energy functionals: a dequantization approach. (English) Zbl 1177.94108
Summary: We strengthen the connection between information theory and quantum-mechanical systems using a recently developed dequantization procedure whereby quantum fluctuations latent in the quantum momentum are suppressed. The dequantization procedure results in a decomposition of the quantum kinetic energy as the sum of a classical term and a purely quantum term. The purely quantum term, which results from the quantum fluctuations, is essentially identical to the Fisher information. The classical term is complementary to the Fisher information and, in this sense, it plays a role analogous to that of the Shannon entropy. We demonstrate the kinetic energy decomposition for both stationary and nonstationary states and employ it to shed light on the nature of kinetic energy functionals.
MSC:
| 94A15 | Information theory (general) |
| 81Q99 | General mathematical topics and methods in quantum theory |
References:
| [1] | Dehesa, J. S.; López-Rosa, S.; Olmos, B.; Yáñez, R. J., Information measures of hydrogenic systems, Laguerre polynomials and spherical harmonics, J. Comput. Appl. Math., 179, 185-194 (2005) · Zbl 1076.33012 |
| [2] | Dehesa, J. S.; Martínez-Finkelshtein, A.; Sorokin, V. N., Information-theoretic measures for Morse and Pöschl-Teller potentials, Mol. Phys., 104, 613-622 (2006) |
| [3] | Dehesa, J. S.; González-Férez, R.; Sánchez-Moreno, P., The Fisher-information-based uncertainty relation, Cramer-Rao inequality and kinetic energy for the D-dimensional central problem, J. Phys. A, 40, 1845-1856 (2007) · Zbl 1114.81015 |
| [4] | Romera, E.; Sánchez-Moreno, P.; Dehesa, J. S., The Fisher information of single-particle systems with a central potential, Chem. Phys. Lett., 414, 468-472 (2005) |
| [5] | Romera, E.; Sánchez-Moreno, P.; Dehesa, J. S., Uncertainty relation for Fisher information of D-dimensional single-particle systems with central potentials, J. Math. Phys., 47, 103504 (2006) · Zbl 1112.81014 |
| [6] | Sánchez-Moreno, P.; González-Férez, R.; Dehesa, J. S., Improvement of the Heisenberg and Fisher-information-based uncertainty relations for D-dimensional central potentials, New J. Phys., 8, 330 (2006) |
| [7] | Mosna, R. A.; Hamilton, I. P.; Delle Site, L., Quantum-classical correspondence via a deformed kinetic operator, J. Phys. A, 38, 3869-3878 (2005) · Zbl 1065.82004 |
| [8] | Mosna, R. A.; Hamilton, I. P.; Delle Site, L., Variational approach to dequantization, J. Phys. A, 39, L229-L235 (2006) · Zbl 1089.81031 |
| [9] | Hamilton, I. P.; Mosna, R. A.; Delle Site, L., Classical kinetic energy quantum fluctuation terms and kinetic-energy functionals, Theor. Chem. Acct., 118, 407-415 (2007) |
| [10] | Fisher, R. A., Theory of statistical estimation, Proc. Cambridge Philos. Soc., 22, 700-725 (1925) · JFM 51.0385.01 |
| [11] | Nagy, A., Fisher information in density functional theory, J. Chem. Phys., 119, 9401-9405 (2003) |
| [12] | Shannon, C. E., A mathematical theory of communication, Bell Syst. Tech. J., 27, 379-423 (1948), 623-656 · Zbl 1154.94303 |
| [13] | Romera, E.; Dehesa, J. S., The Fisher-Shannon information plane an electron correlation tool, J. Chem. Phys., 120, 8906-8912 (2004) |
| [14] | Sen, K. D.; Antolín, J.; Angulo, J. C., Fisher-Shannon analysis of ionization processes and isoelectronic series, Phys. Rev. A, 76, 032502 (2007) |
| [15] | Parr, R. G.; Yang, W., Density Functional Theory of Atoms and Molecules (1989), Oxford University Press: Oxford University Press New York |
| [16] | Dreizler, R. M.; Gross, E. K.U., Density Functional Theory: An Approach to the Quantum Many Body Problem (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0723.70002 |
| [17] | Koch, W.; Holthausen, M. C., A Chemist’s Guide to Density Functional Theory (2000), Wiley-VCH: Wiley-VCH Weinheim |
| [18] | Weizsäcker, C. F.v., Zur Theorie der Kernmassen, Z. Phys., 96, 431-458 (1935) · Zbl 0012.23501 |
| [19] | Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys., 43, 172-198 (1927) · JFM 53.0853.05 |
| [20] | Kennard, E. H., Zur Quantenmechanik einfacher Bewegungstypen, Z. Phys., 44, 326-352 (1927) · JFM 53.0853.02 |
| [21] | Nelson, E., Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150, 1079-1085 (1966) |
| [22] | Nelson, E., Dynamical Theories of Brownian Motion (1967), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0165.58502 |
| [23] | Fényes, I., Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik, Z. Phys., 132, 81-106 (1952) · Zbl 0048.44201 |
| [24] | Weizel, W., Ableitung der Quantentheorie aus einem klassischen kausal determinierten Modell, Z. Phys., 134, 264-285 (1953), Ableitung der Quantentheorie aus einem klassischen Modell. II, Z. Phys. 135 (1953) 270-273; Ableitung der quantenmechanischen Wellengleichung des Mehrteilchensystems aus einem klassischen Modell, Z. Phys. 136 (1954) 582-604 · Zbl 0051.20408 |
| [25] | Goldstein, H., Classical Mechanics (1980), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0491.70001 |
| [26] | Holland, P. R., The Quantum Theory of Motion (1993), Cambridge University Press: Cambridge University Press Cambridge |
| [27] | Thomas, L. H., The calculation of atomic fields, Proc. Cambridge Philos. Soc., 23, 542-548 (1927) · JFM 53.0868.04 |
| [28] | Fermi, E., Un metodo statistico per la determinazione di alcune proprietà dell’atomo, Rend. Accad. Lincei, 6, 602-607 (1927) |
| [29] | Yang, W., Gradient correction in Thomas-Fermi theory, Phys. Rev. A, 34, 4575-4585 (1986) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
