Abstract
A viable methodology for the exact analytical solution of the multiparticle Schrodinger and Dirac equations has long been considered a holy grail of theoretical chemistry. Since a benchmark work by Torres-Vega and Frederick in the 1990s, the QPSR (Quantum Phase Space Representation) has been explored as an alternate method for solving various physical systems. Recently, the present author has developed an exact analytical symbolic solution scheme for broad classes of differential equations utilizing the HOA (Heaviside Operational Ansatz). An application of the scheme to chemical systems was initially presented in Journal of Mathematical Chemistry (Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1/r(I) radial potential in quantum phase space. Journal of Mathematical Chemistry, 2009; 45(1):129–140). It is believed that the coupling of HOA with QPSR represents not only a fundamental breakthrough in theoretical physical chemistry, but it is promising as a basis for exact solution algorithms that would have tremendous impact on the capabilities of computational chemistry/physics. The novel methods allow the exact determination of the momentum [and configuration] space wavefunction from the QPSR wavefunction by way of a Fourier transform. In this note some remarks, examples and further directions, concerning HOA as a tool to solve and provide analytical insight into solutions of dynamical systems occurring in, but not limited to Mathematical Chemistry, are also posited.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Torres-Vega G., Frederick J.H.: A quantum-mechanical representation in phase space. J. Chem. Phys. 98(4), 3103–3120 (1993)
Li Q.S., Lu J.: Rigorous solutions of diatomic molecule oscillator with empirical potential function in phase space. J. Chem. Phys. 113(11), 4565–4571 (2000)
Hu X.G., Li Q.S.: Morse oscillator in a quantum phase-space representation: rigorous solutions. J. Phys. A Math. Gen. 32(1), 139–146 (1999)
Li Q.S., Lu J.: One-dimensional hydrogen atom in quantum phase-space representation: rigorous solutions. Chem. Phys. Lett. 336(1,2), 118–122 (2001)
Li Q.S., Wei G.M., Lu L.Q.: Relationship between the Wigner function and the probability density function in quantum phase space representation. Phys. Rev. A At. Mol. Opt. Phys. 70(2), 022105/1–022105/5 (2004)
Simpao V.A.: Electron. J. Theor. Phys. 1, 10–16 (2004)
V.A. Simpao, Toward chemical applications of Heaviside operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1/rij radial potential in quantum phase space. J. Math. Chem. 2009 45(1): 129–140
Simpao V.A.: Electron. J. Theor. Phys. 3(10), 239–247 (2006)
Moss R.E.: Advanced Molecular Quantum Mechanics. Wiley, NY (1973)
I. Lindgren, in Lecture Notes in Chemistry, Many-Body Methods in Quantum Chemistry, vol. 52, ed. by U. Kaldor (Springer, Berlin, 1989), pp. 293–306
Goldstein H., Poole C.P. Jr., Safko J.L.: Classical Mechanics, 3rd edn. Prentice Hall, Englewood Cliffs (2002)
Whittaker E.T.: Analytical Dynamics, 4th edn. Dover, New York (1944)
Weinstock R.: Calculus of Variations. Dover, New York (1974)
de Gossen M., Luef F.: Lett. Math. Phys. 85, 173–183 (2008)
Zachos C., Fairlie D.B., Thomas L.: Curtright, Quantum Mechanics in Phase Space. World Scientific, Singapore (2005)
P.S. Addison, The illustrated wavelet transform handbook: introductory theory and applications in science, engineering, medicine and finance (CRC Institute of Physics Publishing, Bristol). ISBN 0-7503-0692-0 (2002)
B.P. Bogert, M.J.R. Healy, J.W. Tukey, The quefrency alanysis of time series for echoes: Cepstrum, pseudo autocovariance, cross-cepstrum and saphe cracking. in Proceedings of the Symposium on Time Series Analysis, Chap. 1, ed. by M. Rosenblatt (Wiley, New York, 1963), pp. 209–243
V.A. Simpao, Mathematical Chemistry, Chapt. 6. (NovaScience Publishers, Commack) (cross- published in Int. J. Theoretical Phys. Group Theory NonLinear Opt. 14(2), Nova Science (2010))
Shi Q., Kais S.: Finite size scaling for the atomic Shannon-information entropy?. J. Chem. Phys. 121, 5611–5617 (2004)
Nalewajski R.F.: Entropy/information descriptors of the chemical bond revisited. J. Math. Chem. 49(10), 2308–2329 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Simpao, V.A. HOA (Heaviside Operational Ansatz) revisited: recent remarks on novel exact solution methodologies in wavefunction analysis. J Math Chem 50, 1931–1972 (2012). https://doi.org/10.1007/s10910-012-0012-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-012-0012-z
Keywords
- Heaviside operators
- Integral transforms
- Quantum dynamics
- Classical dynamics: quantum phase space
- Differential equations
- Exact analytical solution
- Quantum chemistry
- Molecular Hamiltonian
- Dirac
- Majorana
- Functional differential equations
- Relativistic
- Schrodinger equation
- Generalised Hamiltonian principle
- Inhomogeneous Lagrange equation
- Equations of motion
- Shannon entropy
- Wavelet transform
- Cepstral analysis
- Ehrenfest’s theorem