Information Entropy of Isospectral Hydrogen Atom

The position and momentum space information entropies of hydrogen atom are exactly evaluated. Using isospectral Hamiltonian approach, a family of isospectral potentials is constructed having same energy eigenvalues as that of the original potential. The information entropy content is obtained in position space as well as in momentum space. It is shown that the information entropy content in each level can be re-arranged as a function of deformation parameter.

Authors:

• Anil Kumar
• C. Nagaraja Kumar

Keywords:

• Information Entropy
• BBM inequality
• Isospectral Potential.

References:

[1] I.I. Hirchman, "A note on entropy," Am. J. Math., vol. 79, 1957, pp. 152-156.
[2] H. Everett, The Many Worlds Interpretation of Quantum Mechanics,
Princeton University Press, Princeton, NJ, 1973.
[3] I. Bialinicki-Birula and J. Mycielski, "Uncertainty relations for information
entropy in wave mechanics," Commun. Math. Phys., vol. 44, 1975, pp. 129-132.
[4] R. Atre, A. Kumar, C.N. Kumar and P.K. Panigrahi, "Quantuminformation
Entropies of the Eigenstates and the Coherent State of the
Poschl-Teller Potential," Phys. Rev. A, vol. 69, 2004, pp. 052107(1-6).
[5] D. Deutsch, "Uncertainty in quantum measurements," Phys. Rev. Lett., vol. 50, 1983, pp. 631-633.
[6] I. Bialinicki-Birula, "Entropic uncertainty relations" Phys. Lett. A, vol.
103, 1984, pp. 253-254.
[7] S. Abe and A.K. Rajagopal, "Information theoretic approach to statistical properties of multivariate Cauchy-Lorentz distributions," J. Phys. A: Math Gen., vol. 34, 2001, pp. 8727-8731.
[8] H. Massen and J.B.M. Uffink, "Generalized entropic uncertainty relations,"
Phys. Rev. Lett., vol. 60, 1990, pp. 1103-1106.
[9] J. Sanchez-Ruiz, "Massen-Uffink entropic uncertainty relation for angular
momentum observables," Phys. Lett. A, vol. 181, 1993, pp. 193-198.
[10] M. Krishna and K.R. Parthasarthy, "An entropic uncertainty principle
for quantum measurement," arXiv:quant-ph/0110025.
[11] S.E. Massen and C.P. Panos, "Universal property of the information
entropy in atoms, nuclei and atomic clusters," Phys Lett. A, vol. 246,
1998, 530-532.
[12] S.R. Gadre and R.D. Bendale, "Regorous relationships among quantum
mechanical kinetic energy and atomic information entropies: Upper and
lower bound," Phys. Rev. A, vol. 36, 1987, pp. 1932-1935.
[13] S.R. Gadre and R.D. Bendale, "Some novel charactristics of atomic
information entropies," Phys. Rev. A, vol. 32, 1995, pp. 2602-2606.
[14] R.J. Yanez, W.V. Assche and J.S. Dehesa, "Position and momentum
information entropies of the D-dimensional harmonic oscillator and
hydrogen atom," Phys. Rev. A, vol. 50, 1994, pp. 3065-3079.
[15] W.V. Assche, R.J. Yang and J.S. Dehesa, "Entropy of orthogonal
polynomials with freud weights and information entropies of harmonic
oscillator potentials," J. Math. Phys. 36, 1995, pp. 4106-4118.
[16] C.P. Panos and S.E. Massen, "Quantum entropy for nuclei," J. Mod.
Phys. E, vol. 6, 1997, pp. 497-505.
[17] V. Majernik and T. Opatrny, "Entropic uncertainty relations for a
quantum oscillator," J. Phys. A: Math. Gen., vol. 29, 1996, pp. 2187-
2197.
[18] V. Majernik and L. Richterek, "Entropic uncertainty relations for the
infinite well," J. Phys A: Math. Gen., vol. 30, 1997, pp. L49-L54.
[19] E. Aydiner, C. Orta and R. Sever, "Quantum information entropies for
the Morse potential," Turk. J. Phys., vol. 30, 2006, pp. 407-410.
[20] K.D. Sen and J. Katriel, "Information entropies for eigendensities of
homogeneous potentials," J. Chem. Phys., vol. 125, 2006, pp. 074117(1-
4).
[21] G.C. Ghirardi, L. Marinatto and R. Romano, "An optimal entropic
uncertainty relation in a two dimensional Hilbert space," Phys. Lett. A,
vol. 317, 2003, pp. 32-36.
[22] V.S. Buyarov, P. Lopez-Artes, A. Martinez-Finkelshtein and W.V. Assche,
"Information entropy of Gegenbauer polynomials," J. Phys. A: Math.
Gen., vol. 33, 2000, pp. 6549-6560.
[23] G.S. Agarwal and J. Benerji, "Spatial coherence and information entropy
in optical vortex fields," Opt. Lett., vol. 27, 2002, pp. 800-802.
[24] S. Abe, "Information entropic uncertainty in the measurement of photon
number and phase in optical states," Phys. Lett. A, vol. 166, 1992, pp.
163-167.
[25] M.W. Coffey, "Semiclassical position entropy for Hydrogen like atoms,"
J. Phys. A: Math. Gen., vol. 36, 2003, pp. 7441-7448.
[26] D.L. Pursey, "New families of isospectral Hamiltonians" Phys. Rev. D,
vol. 33, 1986, pp. 1048-1055.
[27] P.B. Abraham and H.E. Moses, "Changes in potentials due to change in
the point spectram: Anharmonic oscillators with exact solutionss," Phys.
Rev. A, 22, 1980, pp. 1333-1340.
[28] A. Khare and U. Sukhatme, "Phase equivalent potentials obtained from
supersymmetry," J. Phys. A: Math. Gen., vol. 22, 1989, pp. 2847-2860.
[29] B. Mielnik, "Factorization method and new potentials with the oscillator
spectrum," J. Math. Phys. vol. 25, 1984, pp. 3387-3389.
[30] M.M. Neito, "Relationship between supersymmetry and the inverse
methods in quantum mechanics," Phys. Lett. B, vol. 145, 1984, pp. 208-
210.
[31] F. Cooper, A. Khare and U. Sukhatme, "Supersymmetry and quantum
mechanics," Phys. Rep., vol. 251, 1995, pp. 267-385.
[32] B. Chakrabarti, "Use of supersymmetric isospectral formalism to realistic
quantum many body problems," Pramana: J. Phys., vol. 73, 2009,
pp. 405-416.
[33] C.N. Kumar, "Isospectral Hamiltonians: Generation of the soliton profile,"
J. Phys. A, vol. 20, 1987, pp. 5397-5401.
[34] B. Dey and C.N. Kumar, "New set of kink bearing Hamiltonians," Int.
J. Mod. Phys. A, vol. 9, 1994, pp. 2699-2705.
[35] A. Khare and C.N. Kumar, "Landau level spectrum for charged particle
in a class of non-uniform magnetic fields," Mod. Phys. Lett. A, vol. 8,
1993, pp. 523-530.
[36] R. Loudon, "One-dimensional hydrogen atom," Am. J. Phys., vol. 27,
1959, 649-655.
[37] G. Palma and U Raff, "The one-dimensional hydrogen atom revisited,"
Can. J. Phys., vol. 84, 2006, pp. 787-800.