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harmonic oscillators

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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 Physical Applications
For a harmonic oscillator, the potential energy is given by …
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 D
  • P. Dean (1966) The constrained quantum mechanical harmonic oscillator. Proc. Cambridge Philos. Soc. 62, pp. 277–286.
  • 6: 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. …