quantum stationary states

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1: T. Mark Dunster

… … ►is a Professor in the Department of Mathematics and Statistics, San Diego State University, California. …He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. …

2: 18.39 Applications in the Physical Sciences

… ►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form … ►

1D Quantum Systems with Analytically Known Stationary States

… ►By Table 18.3.1#12 the normalized stationary states and corresponding eigenvalues are … ►The orthonormal stationary states and corresponding eigenvalues are then of the form … ►

The Quantum Coulomb Problem: Scattering States

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3: 13.28 Physical Applications

… ►For potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000). … ►

§13.28(iii) Other Applications

►For dynamics of many-body systems see Meden and Schönhammer (1992); for tomography see D’Ariano et al. (1994); for generalized coherent states see Barut and Girardello (1971); for relativistic cosmology see Crisóstomo et al. (2004).

4: 17.17 Physical Applications

… ►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys. ►Quantum groups also apply $q$-series extensively. Quantum groups are really not groups at all but certain Hopf algebras. They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. … ►A substantial literature on $q$-deformed quantum-mechanical Schrödinger equations has developed recently. …

5: 25.17 Physical Applications

§25.17 Physical Applications

►Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. …See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). …

6: 32.16 Physical Applications

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Statistical Physics

►Statistical physics, especially classical and quantum spin models, has proved to be a major area for research problems in the modern theory of Painlevé transcendents. … ►For the Ising model see Barouch et al. (1973), Wu et al. (1976), and McCoy et al. (1977). ►For applications in 2D quantum gravity and related aspects of the enumerative topology see Di Francesco et al. (1995). …

7: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). ►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).

8: Bibliography B

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Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.

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External Links:

Document

Referenced by:

§24.18.

Encodings:

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Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin.

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External Links:

MathReview (L. Van Hove), ZentralBlatt

Referenced by:

§18.39(ii), §18.39(ii).

Encodings:

BibTeX

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Quantum Mechanics of One- and Two-electron Atoms. Rosetta edition, Plenum Publishing Corp., New York.

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Notes:

Reprint (with corrections) of the original edition published in 1957 by Springer and Academic Press

External Links:

MathReview, ZentralBlatt (D. Geissler)

Referenced by:

§1.18(viii), §33.22(ii), §33.22(ii), §33.22(v).

Encodings:

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Angular Momentum in Quantum Physics: Theory and Application. Encyclopedia of Mathematics and its Applications, Vol. 8, Addison-Wesley Publishing Co., Reading, M.A..

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External Links:

ISBN 0-201-13507-8, MathReview (Kurt Bernardo Wolf), ZentralBlatt

Referenced by:

§34.14.

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Quantum Theory of Angular Momentum. A Collection of Reprints and Original Papers. Academic Press, New York.

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External Links:

ISBN 0-12-096056-7, MathReview (R. Mirman)

Referenced by:

§34.12, §34.3(v), §34.5(v), §34.7(v).

Encodings:

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9: 24.18 Physical Applications

… ►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

10: 36.14 Other Physical Applications

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§36.14(iii) Quantum Mechanics

►Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems. …