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1: 17.17 Physical Applications
See Kassel (1995). … It involves q -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …
2: 18.39 Applications in the Physical Sciences
argument a) The Harmonic Oscillator
18.39.15 ϵ n = k / m ( n + 1 2 ) .
This is illustrated in Figure 18.39.1 where the first and fourth excited state eigenfunctions of the Schrödinger operator with the rationally extended harmonic potential, of (18.39.19), are shown, and compared with the first and fourth excited states of the harmonic oscillator eigenfunctions of (18.39.14) of paragraph a), above. …
See accompanying text
Figure 18.39.1: Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for = k = m = 1 , of (18.39.13), in ψ 1 ( x ) , ψ 4 ( x ) and those of the rational potential of (18.39.19), in ψ ^ 3 ( x ) , ψ ^ 6 ( x ) . … Magnify
3: 12.17 Physical Applications
Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. …
4: 32.2 Differential Equations
When β = 0 this is a nonlinear harmonic oscillator. …
5: Bibliography M
  • I. Marquette and C. Quesne (2013) New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..
  • I. Marquette and C. Quesne (2016) Connection between quantum systems involving the fourth Painlevé transcendent and k -step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..
  • 6: Bibliography D
  • P. Dean (1966) The constrained quantum mechanical harmonic oscillator. Proc. Cambridge Philos. Soc. 62, pp. 277–286.
  • 7: Bibliography G
  • D. Gómez-Ullate and R. Milson (2014) Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47 (1), pp. 015203, 26 pp..
  • 8: Errata
  • Chapter 18 Additions

    The following additions were made in Chapter 18:

    • §18.2

      In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)18.2(xii), with Equations (18.2.21)–(18.2.46),

    • §18.3

      A new introduction, minor changes in the presentation, and three new paragraphs.

    • §18.5

      Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).

    • §18.8

      Line numbers and two extra rows were added to Table 18.8.1.

    • §18.9

      Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.

    • §18.12

      Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).

    • §18.14

      Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).

    • §18.15

      Equation (18.15.4_5).

    • §18.16

      The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).

    • §18.17

      Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).

    • §18.18

      Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.

    • §18.19

      A new introduction.

    • §18.21

      Equation (18.21.13).

    • §18.25

      Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.

    • §18.26

      In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).

    • §18.27

      Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).

    • §18.28

      A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)18.28(xi), with Equations (18.28.23)–(18.28.34).

    • §18.30

      Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)18.30(viii), with Equations (18.30.8)–(18.30.31).

    • §18.32

      This short section has been expanded, with Equation (18.32.2).

    • §18.33

      Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).

    • §18.34

      This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).

    • §18.35

      This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).

    • §18.36

      This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.

    • §18.38

      The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).

    • §18.39

      This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

      The original text of 18.39(i) Quantum Mechanics was:

      “Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass m with potential energy V ( x ) :

      errata.1 ( 2 2 m 2 x 2 + V ( x ) ) ψ ( x , t ) = i t ψ ( x , t ) ,

      where is the reduced Planck’s constant. On substituting ψ ( x , t ) = η ( x ) ζ ( t ) , we obtain two ordinary differential equations, each of which involve the same constant E . The equation for η ( x ) is

      errata.2 d 2 η d x 2 + 2 m 2 ( E V ( x ) ) η = 0 .

      For a harmonic oscillator, the potential energy is given by

      errata.3 V ( x ) = 1 2 m ω 2 x 2 ,

      where ω is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval < x < , the constant E (the total energy of the particle) must satisfy

      errata.4 E = E n = ( n + 1 2 ) ω , n = 0 , 1 , 2 , .

      The corresponding eigenfunctions are

      errata.5 η n ( x ) = π 1 4 2 1 2 n ( n ! b ) 1 2 H n ( x / b ) e x 2 / 2 b 2 ,

      where b = ( / m ω ) 1 / 2 , and H n is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

      A second example is provided by the three-dimensional time-independent Schrödinger equation

      errata.6 2 ψ + 2 m 2 ( E V ( 𝐱 ) ) ψ = 0 ,

      when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

      For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”

    • Section 18.40

      The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

  • 9: 10.73 Physical Applications
    §10.73(i) Bessel and Modified Bessel Functions
    Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … In the theory of plates and shells, the oscillations of a circular plate are determined by the differential equation
    10.73.3 4 W + λ 2 2 W t 2 = 0 .
    With the spherical harmonic Y , m ( θ , ϕ ) defined as in §14.30(i), the solutions are of the form f = g ( k ρ ) Y , m ( θ , ϕ ) with g = 𝗃 , 𝗒 , 𝗁 ( 1 ) , or 𝗁 ( 2 ) , depending on the boundary conditions. …