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The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point , it is one of the most important model systems in quantum mechanics.
6.1 Example of first-order perturbation theory – ground-state energy of the quartic oscillator 6.2 Example of first- and second-order perturbation theory – quantum pendulum 6.3 Potential energy as a perturbation
A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground swing.
The resulting equation is of fourth order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted ...
Leapfrog integration is a second-order method, in contrast to Euler integration, which is only first-order, yet requires the same number of function evaluations per step. Unlike Euler integration, it is stable for oscillatory motion, as long as the time-step Δ t {\displaystyle \Delta t} is constant, and Δ t < 2 / ω {\displaystyle \Delta t<2 ...
The fourth type of possible quartic potential is that of "asymmetric shape" of one of the first two named above. The double-well and other quartic potentials can be treated by a variety of methods—the main methods being (a) a perturbation method (that of B. Dingle and H.J.W. Müller-Kirsten [ 8 ] ) which requires the imposition of boundary ...
In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.
combinatorics, as an example of an Appell sequence, obeying the umbral calculus; numerical analysis as Gaussian quadrature ; physics , where they give rise to the eigenstates of the quantum harmonic oscillator ; and they also occur in some cases of the heat equation (when the term x u x {\displaystyle {\begin{aligned}xu_{x}\end{aligned}}} is ...