Arquímedes, Huygens, Schrödinger

P. Enders

Huygens' Principle (HP) contains both the principle of action-at-proximity and the superposition principle. Although\nthe propagation of sharp, non-spreading wave fronts is included in Huygens' original formulation, it can be left\nout without touching those principles. The formulation of HP by means of the Chapman-Kolmogorov equation\n(following Feynman ) comprises both versions and overcomes misunderstandings like "Huygens" principle is not\nexactly obeyed in Optics' and "HP is incompatible with Green's functions'. This way, HP applies not only to\nthe propagation of light, but also to heat and matter diffusion, Schrödinger matter waves, i. e., to virtually all\npropagation phenomena, which can be described through explicit linear differential and difference equations,\nrespectively. HP for Maxwell's equations is discussed in terms of the Helmholtz-decomposed fields and currents. The\nappearances of HP in mechanics and in fractional Fourier transformation being exploited in optics are also mentioned.

Henk W. de Regt

Hichem Eleuch, Yuri V. Rostovtsev

We have found an analytical solution of the 3D stationary Schrödinger equation for any arbitrary potential. The solution has been obtained by transforming the Schrödinger equation to the generalized Ricatti equation, and it has the form of Huygens’ principle for matter waves. We have shown that the solution is more accurate than the JWKB approximation. We discuss possible applications of the solution to molecular dynamics and dynamics of charges in nanostructures.

Dieter Hoffmann

M A Cirone, J P Dahl, M Fedorov +2 more

Huygens' principle following from the d'Alembert wave equation is not valid in two-dimensional space. A Schrödinger particle of vanishing angular momentum moving freely in two dimensions experiences an attractive force - the quantum anti-centrifugal force - towards its centre. We connect these two phenomena by comparing and contrasting the radial propagators of the d'Alembert wave equation and of a free non-relativistic quantum mechanical particle in two and three dimensions.