resonance in forced oscillators motion of ions in a quadrupole ion trap the Stark effect for a rotating electric dipole the Floquet theory of the stability of limit cycles Quantum mechanics Mathieu functions play a role in certain quantum mechanical systems, particularly those with spatially periodic potentials such as the quantum pendulum and crystalline lattices. The modified Mathieu equation also arises when describing the quantum mechanics of singular potentials. For the particular singular potential the radial Schrödinger equation can be converted into the equation The transformation is achieved with the following substitutions By solving the Schrödinger equation (for this particular potential) in terms of solutions of the modified Mathieu equation, scattering properties such as the S-matrix and the absorptivity can be obtained. See also List of mathematical functions Hill differential equation Lamé function Monochromatic electromagnetic plane wave Inverted pendulum Bessel function Notes References Note: Reprinted lithographically in Great Britain at the University Press, Oxford, 1951 from corrected sheets of the (1947) first edition. (free online access to the appendix on Mathieu functions) External links List of equations and identities for Mathieu Functions functions.wolfram.com Timothy Jones, Mathieu's Equations and the Ideal rf-Paul Trap (2006) Mathieu equation, EqWorld NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation Ordinary differential equations Special functions