Hamiltonian structure

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1: 32.6 Hamiltonian Structure

§32.6 Hamiltonian Structure

… ►For Hamiltonian structure for ${\text{P}}_{\text{IV}}$ see Jimbo and Miwa (1981), Okamoto (1986); also Forrester and Witte (2001). ►For Hamiltonian structure for ${\text{P}}_{\text{V}}$ see Jimbo and Miwa (1981), Okamoto (1987b); also Forrester and Witte (2002). ►For Hamiltonian structure for ${\text{P}}_{\text{VI}}$ see Jimbo and Miwa (1981) and Okamoto (1987a); also Forrester and Witte (2004).

2: 29.19 Physical Applications

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§29.19(i) Lamé Functions

… ►Brack et al. (2001) shows that Lamé functions occur at bifurcations in chaotic Hamiltonian systems. …

3: 34.12 Physical Applications

§34.12 Physical Applications

… ►For applications in nuclear structure, see de-Shalit and Talmi (1963); in atomic spectroscopy, see Biedenharn and van Dam (1965, pp. 134–200), Judd (1998), Sobelman (1992, Chapter 4), Shore and Menzel (1968, pp. 268–303), and Wigner (1959); in molecular spectroscopy and chemical reactions, see Burshtein and Temkin (1994, Chapter 5), and Judd (1975). …

4: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(i) Schrödinger Equation

►With $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. … ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. … ►

§33.22(iv) Klein–Gordon and Dirac Equations

… ►The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic structure of heavy elements (Johnson (2007)), is described by a Dirac equation. …

5: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►Physical scientists use the $n$ of Bohr as, to $0$th and $1$st order, it describes the structure and organization of the Periodic Table of the Chemical Elements of which the Hydrogen atom is only the first. … ►A relativistic treatment becoming necessary as $Z$ becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order ${\left(\alpha Z\right)}^2\sim {\left(Z/137\right)}^2$, $\alpha$ being the dimensionless fine structure constant $e^2/(4\pi {\epsilon }_0\mathrm{\hslash }c)$, where $c$ is the speed of light. …

6: Bibliography

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Study of nuclear structure by electromagnetic excitation with accelerated ions. Rev. Mod. Phys. 28, pp. 432–542.

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External Links:

ISSN 0034-6861, Document, MathReview, ZentralBlatt

Referenced by:

§33.22(ii).

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BibTeX

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The Riemann Hypothesis and the Hamiltonian of a Quantum Mechanical System. In Number Theory and Dynamical Systems (York, 1987), M. M. Dodson and J. A. G. Vickers (Eds.), London Math. Soc. Lecture Note Ser., Vol. 134, pp. 153–172.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.17.

Encodings:

BibTeX

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7: 26.20 Physical Applications

… ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). …

8: 26.22 Software

… ►For algorithms for counting and analyzing combinatorial structures see Knuth (1993), Nijenhuis and Wilf (1975), and Stanton and White (1986).

9: Alexander I. Bobenko

… ► Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y. …

10: Bibliography Y

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$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

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Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

Encodings:

BibTeX

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