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 $\beta =0$ this is a nonlinear harmonic oscillator. …

5: Bibliography D

… ►

The constrained quantum mechanical harmonic oscillator. Proc. Cambridge Philos. Soc. 62, pp. 277–286.

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External Links:

Document, MathReview (T. Erber)

Referenced by:

§12.11(i), §12.11(i), §12.17.

Encodings:

BibTeX

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6: 10.73 Physical Applications

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§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 $${\nabla }^{4}W+{\lambda }^2\frac{{\partial }^{2}W}{{\partial t}^2}=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/10.73.E3

Encodings:

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See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …