relation to quantum eigenvalues
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1: 25.17 Physical Applications
§25.17 Physical Applications
…2: Bibliography W
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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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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials.
Ph.D. Thesis, University of Wisconsin, Madison, WI.
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Computation with Recurrence Relations.
Pitman, Boston, MA.
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Elliptic genera and quantum field theory.
Comm. Math. Phys. 109 (4), pp. 525–536.
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3: 18.38 Mathematical Applications
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Supersymmetric Quantum Mechanics (SUSY)
… ►EOP’s, Painlevé Transcendents, and Quantum Mechanics
►EOP’s are the subject of recent work on rational solutions to the fourth Painlevé equation, see Clarkson (2003a) and Marquette and Quesne (2016),where use of Hermite EOP’s makes a connection to quantum mechanics. …4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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, corresponding to distinct eigenvalues, are orthogonal: i.
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►The special form of (1.18.28) is especially useful for applications in physics, as the connection to non-relativistic quantum mechanics is immediate: being proportional to the kinetic energy operator for a single particle in one dimension, being proportional to the potential energy, often written as , of that same particle, and which is simply a multiplicative operator.
The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator.
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►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum.
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5: Bibliography R
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Normal limit theorems for symmetric random matrices.
Probab. Theory Related Fields 112 (3), pp. 411–423.
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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators.
Z. Angew. Math. Mech. 78 (1), pp. 66–72.
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Calculation of - symbols by Labarthe’s method.
International Journal of Quantum Chemistry 63 (1), pp. 57–64.
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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent.
Proc. Roy. Soc. London Ser. A 361, pp. 265–275.
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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media.
In Differential Operators and Related Topics, Vol. I (Odessa,
1997),
Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
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6: Bibliography V
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An integral transform involving Heun functions and a related eigenvalue problem.
SIAM J. Math. Anal. 17 (3), pp. 688–703.
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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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Integral relations for Lamé functions.
SIAM J. Math. Anal. 13 (6), pp. 978–987.
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On the growth of convergence radii for the eigenvalues of the Mathieu equation.
Math. Nachr. 192, pp. 239–253.
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7: Bibliography L
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Quantum Mechanics: Non-relativistic Theory.
Pergamon Press Ltd., Oxford.
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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics.
J. Math. Phys. 27 (5), pp. 1238–1265.
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Recurrence relations for hypergeometric functions of unit argument.
Math. Comp. 45 (172), pp. 521–535.
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Kinetic Theory: Classical, Quantum, and Relativistic Descriptions.
third edition, Springer, New York.
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Monotonic sequences related to zeros of Bessel functions.
Numer. Algorithms 49 (1-4), pp. 221–233.
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8: Bibliography D
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The constrained quantum mechanical harmonic oscillator.
Proc. Cambridge Philos. Soc. 62, pp. 277–286.
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Computational properties of three-term recurrence relations for Kummer functions.
J. Comput. Appl. Math. 233 (6), pp. 1505–1510.
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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes.
Phys. Rev. E (3) 57 (1), pp. 252–275.
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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2.
American Mathematical Society, Providence, RI.
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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions.
Methods Appl. Anal. 6 (3), pp. 21–56.
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9: 18.39 Applications in the Physical Sciences
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§18.39(i) Quantum Mechanics
… ► … ►The Quantum Coulomb Problem
… ►with eigenvalues … ►b) The Bohr Quantum Number
…10: 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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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind.
U.S. Government Printing Office, Washington, D.C..
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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind.
U.S. Government Printing Office, Washington, D.C..
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Concerning the zeros of some functions related to Bessel functions.
J. Mathematical Phys. 10 (9), pp. 1729–1744.
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