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Mathieu function. In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation. where a, q are real -valued parameters. Since we may add π/2 to x to change the sign of q, it is a usual convention to set q ≥ 0.
β = 0 , {\displaystyle \beta =0,} the Duffing equation describes a damped and driven simple harmonic oscillator, γ {\displaystyle \gamma } is the amplitude of the periodic driving force; if. γ = 0 {\displaystyle \gamma =0} the system is without a driving force, and. ω {\displaystyle \omega } is the angular frequency of the periodic driving ...
Poincaré–Lindstedt method. In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms —terms growing without bound—arising in the ...
Quantum pendulum. The quantum pendulum is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the quantized system can be ...
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice. It is a generalization of the free ...
The Mathieu equation is a linear second-order differential equation with periodic coefficients. For q = 0, it reduces to the well-known harmonic oscillator, a being the square of the frequency. [4] The solution of the Mathieu equation is the elliptic-cylinder harmonic, known as Mathieu functions.
Digital ion trap. A digital ion trap mass spectrometer. The digital ion trap (DIT) is an quadrupole ion trap driven by digital signals, typically in a rectangular waveform, generated by switching rapidly between discrete DC voltage levels. The digital ion trap has been mainly developed as a mass analyzer.
Wannier function. Wannier functions of triple- and single-bonded nitrogen dimers in palladium nitride. The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. [ 1 ][ 2 ] Wannier functions are the localized molecular orbitals of crystalline systems.