Solvable problems in classical representation

Abstract

Some solvable problems are considered in a classical representation: namely, the harmonic oscillator, the non-relativistic model of confinement of quarks and the time-dependent Feynman model. In a classical representation, the harmonic oscillator $V(x)=\frac{1}{2}m\omega ^2x^2$ transforms into an equation, which formally coincides with the radial Schrödinger equation for Sturmian states of a hydrogen atom with an energy ${\mathcal {E}} = -2 / \omega ^ 2$, an effective charge ${\mathcal {Z}} = 2E_n / \omega ^ 2$ and an orbital quantum number $l = -1 / 2$. The problem of V-type potential $V(x)=F|x|$ describing the confinement of quarks transforms again into the Airy equation. The classical solution of the time-dependent Feynman model $V(x,t)=\frac{1}{2}m\omega ^2x^2+\alpha (t)x$ after projection on the eigen-states in classical representation gives results for inelastic transitions which coincide with the exact results obtained by Feynman in a purely quantum approach. The analysis of harmonic oscillator, V-type potential and the Feynman model gives a new integral relation between the Laguerre polynomials and the square of the Hermite polynomials, the Airy function and the square of the Airy function, quadratic form of the Laguerre polynomials and quadratic form of the generalized Laguerre polynomials.