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21: 31.17 Physical Applications
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§31.17(i) Addition of Three Quantum Spins
โบThe problem of adding three quantum spins , , and can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. … โบ§31.17(ii) Other Applications
โบHeun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)). … โบMore applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). …22: Bibliography G
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials.
J. Phys. A 47 (1), pp. 015203, 26 pp..
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Quantum mechanics: fundamentals.
Second edition, Springer-Verlag, New York.
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General form of the quantum-defect theory.
Phys. Rev. A 19 (4), pp. 1485–1509.
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Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics.
Texts and Monographs in Physics, Springer.
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23: 9.16 Physical Applications
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โบAiry functions are applied in many branches of both classical and quantum physics.
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โบThe frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other.
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โบThe KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechanics.
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โบThis reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point.
A study of the semiclassical description of quantum-mechanical scattering is given in Ford and Wheeler (1959a, b).
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24: Bibliography V
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Quantum Theory of Angular Momentum.
World Scientific Publishing Co. Inc., Singapore.
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Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions.
Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.
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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
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25: 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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Quantum mechanics of one- and two-electron atoms.
Springer-Verlag, Berlin.
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Quantum Mechanics of One- and Two-electron Atoms.
Rosetta edition, Plenum Publishing Corp., New York.
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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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Quantum Theory of Angular Momentum. A Collection of Reprints and Original Papers.
Academic Press, New York.
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26: 20 Theta Functions
Chapter 20 Theta Functions
…27: 13.28 Physical Applications
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โบFor potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000).
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28: 14.31 Other Applications
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§14.31(iii) Miscellaneous
โบMany additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …29: Bibliography W
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Lectures on Quantum Mechanics.
Cambridge University Press, Cambridge, UK.
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Group Theory and its Application to the Quantum Mechanics of Atomic Spectra.
Pure and Applied Physics. Vol. 5, Academic Press, New York.
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Elliptic genera and quantum field theory.
Comm. Math. Phys. 109 (4), pp. 525–536.
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30: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Confluent hypergeometric equations and related solvable potentials in quantum mechanics.
J. Math. Phys. 41 (12), pp. 7964–7996.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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