Demos.GeneralPendulum.Main

WavePacket demo example: Generalized pendulum

Here we treat the case of a generalized pendulum where the potential energy function is represented by a sum of a linear (∝ η) and a quadratic (∝ ζ) term in a trigonometric (cosine) function. For each of the two terms alone, the corresponding Schrödinger equations can be mapped onto a Mathieu equation, see the work by E. U. Condon in 1928 who recognized that the eigenfunctions of the time-independent Schrödinger equation with a simple cosine potential can be expressed in terms of Mathieu' (co-)sine elliptic functions, see also our Wiki page on simple pendula. In contrast, the time-independent Schrödinger equation for the combined potential

V(θ) = − η cos θ − ζ cos² θ

can be mapped onto a Whittaker-Hill equation which belongs to the class of conditionally quasi-exactly solvable problems. A finite number (quasi-exactly solvable) of analytic solutions can be obtained for certain relations between the two parameters η and ζ (conditionally exactly solvable). Note that the generalized pendulum is anti-isospectral to the hyperbolic Razavy single/double well problem which is also conditionally quasi-exactly solvable.

Planar case

For the planar case the wavefunctions live on a unit circle, i.e. the Schrödinger equation of the generalized planar pendulum is solved for a single periodic coordinate 0 ≤ θ ≤ 2π. Note that also solutions anti-periodic in 2π (with 0 ≤ θ ≤ 4π) exist; they may of interest, e.g., in connection with Berry's geometric phase. Exact solutions are obtained for κ ≡ η / √ ζ being integer. Learn more ...

Spherical case

For the spherical case the wavefunctions live on the surface of a unit sphere. Because the interaction is not depending on the azimuthal angle φ, the (m-dependent!) effective Schrödinger equation of the generalized planar pendulum is solved for the polar angle 0 ≤ θ ≤ π, and the solutions are then multiplied with a phase factor exp(imφ). Exact solutions are obtained for k ≡ η / ( 2 √ ζ ) being integer. Learn more ...