Abstract
This is an analysis of the statistical nature of the time-independent Schrödinger equation through the use of the information entropy concept. We first study the Schrödinger equation in a general way and then by actually computing entropies of various states of the hydrogen atom for a re-examination of the problem. It is found that there exists a variational procedure involving maximizing entropy for obtaining all solutions once one solution is known. Based on certain observations of the particular single system, some general conclusions can be deduced. First of all, we can safely say that the Schrödinger equation, among many other interpretations, is but the consequence of a principle of minimum potential energy expectation with certain proper constraints imposed. In addition, the ensemble concept in statistical thermodynamics is also useful in understanding microscopic quantum systems and many quantum mechanical concepts such as energy quantization and wave nodal properties can be discussed in the light of information theory and statistics in general.
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References
C.E. Shannon, “A mathematical theory of communication”, Bell Syst. Tech. J. 27 (1948) 379–423,623–656.
N. Wiener,Cybernetics (Wiley, New York, 1948).
N. Wiener, “What is information theory?” IRE Trans. Inform. Theory IT-2 (1956) 48.
R. Ash,Information Theory (Interscience, New York, 1967).
E.T. Jaynes, Phys. Rev. 106 (1957) 620.
S. Kullback,Statistics and Information Theory (Wiley, New York, 1959).
I. Bialynicki-Birula and J. Mycielski, Commum. Math. Phys. 44 (1975) 129.
W. Beckner, Ann. Math. 102 (1975) 159.
S.R. Gadre, Phys. Rev. A30 (1984) 620.
S.R. Gadre, S.J. Chakravorty and R.D. Bendale, Phys. Rev. A32 (1985) 2602.
S.R. Gadre, R.D. Bendale and S.P. Gejji, Chem. Phys. Lett. 117 (1985) 138;
S.R. Gadre and R.D. Bendale, Curr. Sci. 54 (1985) 9970.
G. Maroulis, M. Sana and G. Leroy, Int. J. Quant. Chem. 19 (1981) 43.
S.B. Sears R. Parr and U. Dinur, Israel J. Chem. 19 (1980) 165–173.
B.R. Frieden, J. Mod. Opt. 35 (1988) 1297–1316.
B.R. Frieden, Am. J. Phys. 57 (1989) 1004.
S.R. Gadre and S.B. Sears, J. Chem. Phys. (a) 71 (1979) 4321; 75 (1981) 4626.
T. Koga and M. Morita, J. Chem. Phys. 79 (1983) 1933.
L. Arrachea, N. Canosa, A. Plastino, M. Portesi and R. Rossignoli,Condensed Matter Theories, Vol. 7, eds. A.N. Proto and J.L. Aliaga, (Plenum Press, New York, 1992) p. 63.
N. Canosa, R. Rossignoli and A. Plastino,Condensed Matter Theories, Vol. 7, eds. A.N. Proto and J. L. Aliaga (Plenum Press, New York, 1992) p. 69.
E. Schrödinger,Collected Papers on Wave Mechanics (Blackie, London, 1928; Chelsea, New York, 1982).
V.A. Johnson, Phys. Rev. 60 (1941) 373.
P.R. Auvil and L.M. Brown, Am. J. Phys. 46 (1978) 679.
S.R. Gadre and R.D. Bendale, Phys. Rev. A36 (1987) 1932.
I.N. Levine,Quantum Chemistry, 3rd Ed. (Allyn and Bacon, Boston, 1983) pp. 326–329, 354–365.
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Chen, JC., Yang, YC., Jia, HW. et al. Information theoretical analysis of the hydrogen atom. J Math Chem 16, 125–136 (1994). https://doi.org/10.1007/BF01169201
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DOI: https://doi.org/10.1007/BF01169201